diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzijeq" "b/data_all_eng_slimpj/shuffled/split2/finalzzijeq" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzijeq" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nConsistency relations between cosmological observables exist for any underlying physical model class \\cite{Mortonson:2008qy,Mortonson:2009hk}. This means that the combination of observables pertaining to the expansion history and those pertaining to structure growth could potentially falsify a given dark energy paradigm, such as the standard $\\Lambda$CDM model of cold dark matter and a cosmological constant with Gaussian initial conditions, or smooth dark energy models with equation of state parameter $-1\\le w(z)\\le 1$, known as quintessence models.\nFor instance, even a single massive high-redshift cluster could falsify all $\\Lambda$CDM\\ and quintessence models, if its mass falls significantly outside of what we predict based on Type Ia supernovae (SNe), the cosmic microwave background (CMB), baryon acoustic oscillations (BAO), and the local measurement of the Hubble constant ($H_0$) \\cite{Mortonson:2010mj}.\n\nWeak gravitational lensing, whereby distant galaxy images are distorted due to the gravitational effects of matter lying along the line of sight, is another key probe of cosmic structure growth, with promising results in recent years, e.g.~\\cite{Fu:2007qq,Kilbinger:2008gk,Massey:2007gh,Massey:2007wb,Schrabback:2009ba}. \nSee Refs.~\\cite{Takada:2003ef,Hoekstra:2008db,Massey:2010hh} for recent reviews. However, current cosmological constraints are very limited; since we do not know the intrinsic shapes of galaxies, weak lensing is by its very nature a statistical measure and thus the power of a survey is directly related to its sky coverage \\cite{Hu:1998az,Amara:2006kp}. Given future large-area surveys on the horizon, for instance from the ground with the Dark Energy Survey (DES) \\cite{des_url} and the Large Synoptic Survey Telescope (LSST) \\cite{lsst_url} or from space with {\\it Euclid} \\cite{euclid_url} and the Wide-Field Infrared Survey Telescope ({\\it WFIRST}) \\cite{wfirst_url}, it is particularly timely to determine our expectations \\cite{Weinberg:2012es}.\n\nUsing the wealth of data in hand from distance measures and the CMB, we expand upon Ref.~\\cite{Mortonson:2010mj} to predict these weak lensing observables within the\n$\\Lambda$CDM and quintessence paradigms. We focus on the shear power spectrum and show how the predictions relax as we generalize the model beyond flat $\\Lambda$CDM and allow for curvature, an arbitrary dark energy equation of state, and early dark energy (EDE). Once the upcoming large-area weak lensing surveys are completed, we can compare their results to these predictions and possibly falsify the $\\Lambda$CDM model or perhaps the entire quintessence paradigm. In this way weak lensing could provide a smoking gun for new physics such as primordial non-Gaussianity, phantom dark energy ($w(z)<-1$), sub-horizon dark energy clustering, an interacting dark sector, or even a modification to general relativity.\n\nThis paper is organized as follows. In \\S \\ref{sec:methadology} we describe our methodology, including the datasets we use, the parameter spaces we explore, our Markov Chain Monte Carlo (MCMC) likelihood analysis, and how we compute the ingredients needed to predict the weak lensing shear power spectrum and two point correlation functions (2PCFs) for source redshift distributions typical of current and future surveys. In \\S \\ref{sec:wl} we present our weak lensing predictions and discuss how they depend on source redshift and dark energy model class. In \\S \\ref{sec:uncertainties} we explore uncertainties related to the matter power spectrum and SN light-curve fitting. We discuss and conclude in \\S \\ref{sec:discussion}.\n\n\\section{Methodology}\n\\label{sec:methadology}\n\nOur methodology is based on that of Refs.~\\cite{Mortonson:2008qy,Mortonson:2009hk,Mortonson:2010mj}. Briefly, we take current CMB constraints on the initial power spectrum plus current data related to the overall geometry and expansion history of the Universe, determine parameter constraints within a class of dark energy models using an MCMC analysis, and then compute the weak lensing predictions which can be used to falsify the class. The observational data we use come from local distance measures of $H_0$, the Type Ia SNe distance-redshift relation, BAO distance measures, and the CMB temperature and polarization power spectra.\n\n\\subsection{Data sets}\n\\label{sec:data}\n\nThe Type Ia SN sample we use is the compilation of 288 SNe from\nRef.~\\cite{SDSS_SN}, consisting of data from the first season of the Sloan\nDigital Sky Survey-II (SDSS-II) Supernova Survey, the ESSENCE survey\n\\cite{WoodVasey_2007}, the Supernova Legacy Survey \\cite{Astier}, Hubble Space\nTelescope SN observations \\cite{Riess_2006}, and a collection of nearby SN\ndata \\cite{Jha:2006fm}. The light curves of these SNe have been uniformly\nanalyzed by \\cite{SDSS_SN} using both the MLCS2k2 \\cite{Jha:2006fm} \nand SALT2 \\cite{Guy:2007dv} methods.\nWe use the SALT2 method for our default analysis but discuss the impact of\nswitching to MLCS2k2 in \\S \\ref{sec:sn}.\n\nFor the CMB, we use the most recent, 7-year release of data from the {\\it WMAP}\nsatellite (WMAP7) \\cite{Larson:2010gs} employing a modified version of the likelihood\ncode available at the LAMBDA web site \\cite{WMAP_like} which is substantially\nfaster than the standard version while remaining sufficiently \naccurate \\cite{Dvorkin:2010dn,wmapfast_url}. We\ncompute the CMB angular power spectra using the code CAMB\n\\cite{Lewis:1999bs,camb_url} modified with the parametrized post-Friedmann\n(PPF) dark energy module \\cite{PPF,ppf_url} to include models with general\ndark energy equation of state evolution where $w(z)$ may cross $w=-1$.\n\nWe use the BAO constraints from Ref.~\\cite{PercivalBAO}, which combines \ndata from SDSS and the 2-degree Field Galaxy Redshift Survey that\ndetermine the ratio of the sound horizon at last scattering to the \nquantity $D_V(z)\\equiv [zD^2(z)\/H(z)]^{1\/3}$ at redshifts $z=0.2$ and $z=0.35$.\nSince these constraints actually come from galaxies spread over a range \nof redshifts, and our most general dark energy model classes allow the \npossibility of high frequency variations in the expansion rate $H(z)$ \nacross this\nrange, we implement the constraints by taking the volume average of $D_V$ \nover $0.1 1.7) = w_{\\infty}$, the $\\alpha_i$ are the PC amplitudes, the $e_i(z)$ are the PC functions, and $N_{\\rm{max}}=10$ has been found to provide a complete basis for our purposes here \\cite{Mortonson:2009hk}. \nThe PCs we use here are the eigenfunctions of the Fisher matrix for the \ncombination of CMB constraints from {\\it Planck} and {\\it WFIRST}-like \nSN data, as described in Ref.~\\cite{Mortonson:2008qy}.\nWe will fix $w_{\\infty}=-1$ for the model classes which exclude EDE, as a constant equation of state dark energy component quickly becomes negligible at early times, and allow it to vary otherwise with a flat prior on $e^{w_{\\infty}}$ between $e^{-1}$ and 1. A detailed discussion of this EDE approach and its relation to CMB\nconstraints can be found in Appendix B of Ref.~\\cite{Mortonson:2010mj}.\n\nQuintessence models describe dark energy as a scalar field with kinetic and potential contributions to energy and pressure. Barring models where large kinetic and (negative)\npotential contributions cancel (e.g.~\\cite{Mortonson:2009qq}), quintessence equations of state are restricted to $-1\\leq w(z)\\leq 1$. Following \nRef.~\\cite{Mortonson:2008qy}, this bound is conservatively implemented\nwith uncorrelated top-hat priors on the PC amplitudes $\\alpha_i$. \nAny combination of PC amplitudes that is\nrejected by these priors must arise from an equation of state $w(z)$ that\nviolates the bound on $w(z)$, but not all models that are allowed by the priors\nstrictly satisfy this bound.\n\nIn summary, the full set of parameters we will consider here are \n\\begin{equation}\n\\bm{\\theta} = \\{\\bm{\\theta}_{0} , \\Omega_{\\rm{K}}, \\alpha_1,\\ldots, \\alpha_{10}, e^{w_{\\infty}}\\}~,\n\\label{eq:parametersfull}\n\\end{equation}\nwhere we will look at models of increasing generality by exploring increasingly larger subsets of this parameter set. \n\n\\subsection{Observables}\n\nOnce we have sampled the joint posterior distribution of the cosmological parameters, we can then compute the posterior probabilities for any derived statistic, in particular cosmic shear observables. \nAs an intermediate step, it is useful to first consider the two basic ingredients that the\ntwo-point shear observables are constructed from: the comoving angular diameter distance\n$D$ and the nonlinear matter power spectrum $\\Delta^2_{\\rm NL}$. \nThe former is related to the cosmological parameters through \n\\begin{equation}\nD(\\chi)=\\frac{1}{\\sqrt{|\\Omega_{\\rm{K}}|H_0^2}}S_{\\rm{K}}\\left[\\chi\\sqrt{|\\Omega_{\\rm{K}}|H_0^2}\\right]~,\n\\end{equation}\nwhere $S_{\\rm{K}}(x)$ equals $x$ in a flat universe ($\\Omega_{\\rm{K}}=0$), $\\sinh x$ in an open universe ($\\Omega_{\\rm{K}}>0$), and $\\sin x$ in a closed universe ($\\Omega_{\\rm{K}}<0$). Here the comoving radial coordinate is\n\\begin{equation}\n\\chi(z) = \\int_0^z\\frac{dz'}{H(z')},\n\\end{equation}\nwhere the Hubble expansion rate is\n\\begin{equation}\nH(z)=H_0\\left[\\Omega_{\\rm{m}}(1+z)^3+\\Omega_{\\rm{DE}}f(z)+\\Omega_{\\rm{K}}(1+z)^2\\right]^{1\/2}\n\\end{equation}\nwith\n\\begin{equation}\nf(z)=\\exp\\left[3\\int^z_0 dz'\\frac{1+w(z')}{1+z'}\\right]\\,,\n\\label{eqn:fz}\n\\end{equation}\nand the contribution from radiation is assumed to be negligible.\n\nThe linear matter power spectrum depends on the initial matter power spectrum \nand the growth function of linear density perturbations\n$\\delta\\propto G a$, where $a$ is the scale factor. Given the smooth dark energy paradigm, the growth function is given by\n\\begin{equation}\nG'' + \\left(4+\\frac{H'}{H}\\right)G'+\\left[3+\\frac{H'}{H}-\\frac{3\\Omega_{\\rm{m}} H_0^2(1+z)^3}{2 H^2(z)}\\right]G = 0~,\n\\end{equation}\nwhere primes denote derivates with respect to $\\ln a$ and the function is normalized so\nthat $G=1$ in the matter dominated limit. \nWe then compute the linear power spectrum at redshift $z$ by rescaling the \n$z=0$ spectrum $\\Delta_{\\rm L}^2(k;0)$ (computed using CAMB with the \nPPF dark energy module) by the growth function:\n\\begin{equation}\n\\Delta_{\\rm L}^2(k; z) = \\left[ {G(z) \\over (1+z)G(0) } \\right]^2 \\Delta_{\\rm L}^2(k; 0) \\,.\n\\end{equation}\n\nWe shall see that current CMB constraints on $\\Omega_{\\rm{m}} h^2$ still allow substantial variation in the matter radiation equality scale and hence the\nshape of the linear power spectrum.\n\nCosmic shear observables however depend on the full nonlinear power spectrum.\nWe compute this quantity using the Halofit fitting function \\cite{Smith:2002dz}, which we will now describe. \nThe halo model decomposes the nonlinear power spectrum into the sum of two contributions,\n\\begin{equation}\n\\Delta_{\\rm{NL}}^2(k) = \\Delta_{\\rm{Q}}^2(k) + \\Delta_{\\rm{H}}^2(k)~,\n\\end{equation}\nwhere $\\Delta_{\\rm{Q}}^2(k)$ is the ``two-halo\" term, which encapsulates quasi-linear power from large-scale halo placement, and $\\Delta_{\\rm{H}}^2(k)$ is the ``one-halo\" term, which arises due to correlations within the haloes themselves. The Halofit fitting functions depend on parameters based on Gaussian filtering of the linear power spectrum with variance\n\\begin{equation}\n\\sigma^2(R)\\equiv \\int \\Delta_{\\rm{L}}^2(k)\\exp\\left(-k^2R^2\\right)d\\ln k~:\n\\end{equation}\nthe nonlinear scale $k_{\\rm{NL}}$ defined such that $\\sigma\\left(k_{\\rm{NL}}^{-1}\\right) \\equiv 1$, the effective spectral index,\n\\begin{equation}\nn \\equiv -3 -\\left.\\frac{d\\ln\\sigma^2(R)}{d\\ln R}\\right|_{\\sigma=1}~,\n\\end{equation}\nand the spectral curvature,\n\\begin{equation}\nC\\equiv -\\left.\\frac{d^2\\ln\\sigma^2(R)}{d\\ln R^2}\\right|_{\\sigma=1}~.\n\\end{equation}\nThen the nonlinear power spectrum is parameterized by a set of coefficients $(a_n,b_n,c_n,\\gamma_n,\\alpha_n,\\beta_n,\\mu_n,\\nu_n)$ which are allowed to vary as a function of the aforementioned spectral index and curvature so as to fit N-body simulation data. In terms of these coefficients,\n\\begin{equation} \n\\Delta^2_{\\rm{Q}}(k)= \\Delta^2_{\\rm{L}}(k) \\left\\{\\frac{\\left[1+\\Delta_{\\rm{L}}^2(k)\\right]^{\\beta_n}}{1+\\alpha_n\\Delta^2_{\\rm{L}}(k)} \\right\\}\\exp{[-f(y)]}~,\n\\end{equation}\nwhere $y\\equiv k\/k_{\\rm{NL}}$ and $f(y)=y\/4+y^2\/8$ is a function introduced to truncate at high $k$, and \n\\begin{equation} \n\\Delta^2_{\\rm{H}}(k) = \\frac{\\Delta^{2\\ \\prime}_{\\rm{H}}(k)}{1+\\mu_ny^{-1}+\\nu_ny^{-2}}~,\n\\end{equation}\nwith\n\\begin{equation} \n\\Delta^{2\\ \\prime}_{\\rm{H}}(k) = \\frac{a_n y^{3f_1(\\Omega_{\\rm{m}})}}{1+b_ny^{f_2(\\Omega_{\\rm{m}})}+\\left[ c_nf_3(\\Omega_{\\rm{m}}) y\\right]^{3-\\gamma_n}}~.\n\\label{deltah}\n\\end{equation}\nThe coefficients are\n\\begin{eqnarray}\n\\log_{10} a_n &=& 1.4861 + 1.8369n + 1.6762n^2 + 0.7940n^3\\nonumber\\\\ \n& &+ 0.1670n^4 -0.6206C~,\\nonumber\\\\\n\\log_{10} b_n &=& 0.9463 + 0.9466n + 0.3084n^2 - 0.9400C~,\\nonumber\\\\\n\\log_{10} c_n &=& -0.2807 + 0.6669n + 0.3214n^2 - 0.0793C~,\\nonumber\\\\\n\\gamma_n &=& 0.8649 + 0.2989n + 0.1631C~,\\nonumber\\\\\n\\alpha_n &=& 1.3884 + 0.3700n - 0.1452n^2~,\\nonumber\\\\\n\\beta_n &=& 0.8291 + 0.9854n + 0.3401n^2~,\\nonumber\\\\\n\\log_{10}\\mu_n &=& -3.5442+0.1908n~,\\nonumber\\\\\n\\log_{10}\\nu_n &=& 0.9589+1.2857n~.\n\\end{eqnarray}\nThe functions $(f_1,f_2,f_3)$ are power laws, with the exponents fit to N-body data for either matter-only open models\n\\begin{equation} \n\\left. \n\\begin{array}{lll}\nf_{1}(\\Omega_{\\rm m}) & = & \\Omega_{\\rm m}^{\\;-0.0732} \\\\\nf_{2}(\\Omega_{\\rm m}) & = & \\Omega_{\\rm m}^{\\;-0.1423}\\\\\nf_{3}(\\Omega_{\\rm m}) & = & \\Omega_{\\rm m}^{\\;0.0725}\\\\\n\\end{array}\n\\right\\} \\ \\ \\Omega_{\\rm m}\\le1\n\\end{equation}\nor flat $\\Lambda$CDM models\n\\begin{equation} \n\\left. \n\\begin{array}{lll}\nf_{1}(\\Omega_{\\rm m}) & = & \\Omega_{\\rm m}^{\\;-0.0307}\\\\\nf_{2}(\\Omega_{\\rm m}) & = & \\Omega_{\\rm m}^{\\;-0.0585}\\\\\nf_{3}(\\Omega_{\\rm m}) & = & \\Omega_{\\rm m}^{\\;0.0743}\\\\\n\\end{array}\n\\right\\} \\ \\ \\Omega_{\\rm m}+\\Omega_{\\rm DE}=1~.\n\\end{equation}\nNote that we will use these fitting functions for all of our model classes, despite that fact that they were calibrated on simulations where the dark energy equation of state never deviated from $w=-1$.\nWe comment on this simplification in \\S \\ref{sec:nonlinearpk}. For our main result that\n$\\Lambda$CDM\\ sets an upper bound on shear statistics, we expect that this approximation suffices.\n\nAs suggested in Appendix C of Ref.~\\cite{Smith:2002dz}, we use linear interpolation for models in which $\\Omega_{\\rm{DE}}$ is neither zero nor $1-\\Omega_{\\rm{m}}$. We further use the high-$k$ correction \\cite{halofit_correction_url} \n\\begin{equation}\n\\Delta^2_{\\rm NL}(k)-\\Delta^2_{\\rm{L}}(k) \\rightarrow \\left[\\Delta^2_{\\rm NL}(k)-\\Delta^2_{\\rm{L}}(k)\\right]\\left(\\frac{1+2x^2}{1+x^2}\\right)~,\n\\label{halofitcorrection}\n\\end{equation}\nwhere $x\\equiv k\/(10\\,h~{\\rm Mpc}^{-1})$.\nThese fitting functions have been found to be inaccurate (even for flat $\\Lambda$CDM) at up to the $5$--$10\\%$ level, for instance with the Coyote universe project \\cite{Heitmann:2008eq,Heitmann:2009cu,Lawrence:2009uk}. In \\S \\ref{sec:uncertainties} we will explore how our predictions depend on the accuracy of the one-halo term amplitude $a_n$. We will also show (and exploit) how one can use $c_n$ to parameterize uncertainties due to warm dark matter and baryonic processes.\n\nFrom the distance-redshift relation and the nonlinear matter power spectrum, we can then compute the shear power spectrum, equal to the convergence power spectrum\n\\begin{equation}\n{l^2 P_{\\kappa} \\over 2\\pi} = {9\\pi \\over 4 c^4 l} {\\Omega_{\\rm{m}}^2H_0^4 } \\int_0^{\\infty}dz {D^3 \\over H} \\frac{g^2(z)}{a^2}\\Delta^2_{\\rm NL}\\left(\\frac{l}{D(\\chi)},z\\right)~,\n\\label{shearpower}\n\\end{equation}\nwhere $k \\approx l\/D$ in units of Mpc$^{-1}$ in the Limber approximation and we have defined the geometric lensing efficiency factor\n\\begin{equation}\ng(z)\\equiv\\int^{\\infty}_{z} dz' n(z')\\frac{D(\\chi'-\\chi)}{D(\\chi')}~.\n\\end{equation}\nThe efficiency factor is weighted according to the source distribution in a given survey, $n(z)$, normalized such that $\\int_0^{\\infty}n(z) dz =1$. For this paper we will use the simple model\n\\begin{equation}\nn(z)\\propto \\left(\\frac{z}{z_0}\\right)^{\\alpha}\\exp\\left[-\\left(\\frac{z}{z_0}\\right)^{\\beta}\\right]~,\n\\end{equation}\nwith parameters $z_0$, $\\alpha$, and $\\beta$. This parameterization is similar to what has been used in both the COSMOS \\cite{Massey:2007gh} and CFHTLS \\cite{Benjamin:2007ys} analyses, with $(z_0,\\alpha,\\beta)=(0.894,2.0,1.5)$ for COSMOS and $(z_0,\\alpha,\\beta)=(0.555,1.197,1.193)$ for CFHTLS. This leads to approximate median redshifts of $1.3$ and $0.8$, respectively, for our simplified COSMOS and CFHTLS surveys.\nNote that these distributions closely approximate the ones\nexpected for {\\it WFIRST} \\cite{Jouvel:2009mh,Green:2011zi} and DES. \nIn the work that follows we will plot results for both of these simple source distributions, but will specialize to CFHTLS (as in \\cite{Eifler:2010kt}) when results for the two are \nsimilar.\n\nFinally, we also compute the 2PCFs \n \\begin{equation}\n\\xi_{+\/-}(\\theta) = \\frac{1}{2\\pi}\\int_0^{\\infty}dl \\, l \\, J_{0\/4}(l\\theta)P_{\\kappa}(l)~\n\\end{equation}\nof the shear components from the power spectrum. Being the real-space correlation\nand the easier quantity to measure on small scales, the 2PCFs are better suited to compare against current measurements than $P_\\kappa$ itself.\n\n\\section{Weak lensing predictions}\n\\label{sec:wl}\n\nCurrent distance constraints are highly predictive for growth of structure statistics such as\nthe weak lensing power spectrum in the flat $\\Lambda$CDM context.\nSince this is the standard model of cosmology and our baseline case, we shall use it to illustrate the steps in our chain of inference in \\S \\ref{sec:lcdm}. We then discuss observations that would falsify \n$\\Lambda$CDM in favor of quintessence and those that would falsify the whole quintessence class\nof $-1 \\le w(z) \\le 1$ smooth dark energy models in \\S \\ref{sec:quintessence}.\n\n\\subsection{$\\Lambda$CDM}\n\\label{sec:lcdm}\n\nIn Fig.~\\ref{fig:growth}, we start with the growth function prediction. \nUnless stated otherwise, henceforth shaded regions will correspond to 68\\% confidence level (CL) regions and curves will bound 95\\% CL regions defined to have equal posterior probabilities in the two tails. \nWe will often plot results as fractional differences from the prediction for the maximum likelihood (ML) model.\nThe growth predictions are most precise at high redshifts, but even at $z=0$ the allowed range in the growth function is only $1$--$2\\%$ \\cite{Mortonson:2009hk}.\n\n\\begin{figure}[t]\n\\includegraphics[width=8cm]{gz_f}\n\\caption{Flat $\\Lambda$CDM prediction for the growth factor (upper panel) and its deviation from the ML flat $\\Lambda$CDM model (lower panel). \nThe shaded regions correspond to 68\\% CL and the curves bound the 95\\% CL regions.}\n\\label{fig:growth}\n\\end{figure}\n\nAs described in the previous section, we can combine the growth function with the initial\npower spectrum and transfer function to predict the linear matter power spectrum \nat $z=0$; see the upper and lower panels of Fig.~\\ref{fig:pklin}. \nFor plotting purposes we follow the usual convention of taking $P(k) = (2\\pi^2\/k^3)\\Delta^2(k)$\nwith $k$ defined in $h$ Mpc$^{-1}$ for both linear and nonlinear power spectra.\nHere the predictions\ncarry $\\sim 10\\%$ errors in spite of the precise growth results. \nThe dominant\nsource of uncertainty is from the measurements of the \nmatter density and shape of the initial power spectrum from WMAP7. In\nparticular, uncertainties in $\\Omega_{\\rm{m}} h^2$ correspond to shifts in\nmatter-radiation equality which cause left-right\nshifts in the power spectrum. There are also contributions from tilt ($n_s$) uncertainties\nthat pivot the spectrum\naround the best constrained {\\it WMAP} scale of $k \\sim 0.02$ Mpc$^{-1} \\approx 0.03 h$ Mpc$^{-1}$. Both of these types of uncertainties should be reduced by a factor of a few with\n{\\it Planck} CMB data and allow the full precision of the growth predictions to be utilized\n(see \\S \\ref{sec:linearpk} and Fig.~\\ref{fig:planck}). \n\nAlso shown in Fig.~\\ref{fig:pklin} is the full nonlinear matter power spectrum as computed with Halofit \n(grey hatched curve in the upper panel), along with its deviation from the ML prediction in the middle panel.\nFor the nonlinear power spectrum, the uncertainties are the same as the linear one for \n$k < k_{\\rm NL} \\sim 0.3~h~\\rm{Mpc}^{-1}$. For smaller scales, the nonlinear power spectrum\nuncertainties no longer reflect the linear uncertainties. Whereas the tilt $n_s$ makes the\nlinear uncertainties continue to grow larger, the nonlinear ones become saturated \nreflecting the fixed nature of the one-halo piece of the Halofit prescription. We shall see that uncertainties here\nare dominated by the accuracy of Halofit and the ability of adjustments in its parameters to model baryonic physics (see \\S \\ref{sec:nonlinearpk}). \n\n\\begin{figure}[t]\n\\includegraphics[width=8cm]{pk_f}\n\\caption{Flat $\\Lambda$CDM prediction for the $z=0$ matter power spectrum, showing the 68\\% and 95\\% CL regions as in Fig.~\\ref{fig:growth}.\nUpper panel: Linear (blue solid) and nonlinear (grey hatched) matter power spectra.\nMiddle panel: Fractional confidence range in the nonlinear spectrum around the ML flat $\\Lambda$CDM model.\nLower panel: The same for the linear power spectrum. Note $k$ is in units of $h$ Mpc$^{-1}$.}\n\\label{fig:pklin}\n\\end{figure}\n\nUncertainties in $P_{\\rm NL}(k,z)$ propagate into the shear statistics.\nWe show the shear power spectrum as computed from Eq.~(\\ref{shearpower})\nin Fig.~\\ref{fig:shear} for both the COSMOS and CFHTLS source distributions.\nThe confidence intervals mainly reflect those of the nonlinear matter power spectrum. The interval is actually slightly narrower at low $l$ than at low $k$ in \nFig.~\\ref{fig:pklin}. \nSignificant contributions to $P_\\kappa$ come from $z>0.5$, where both linear growth function uncertainties diminish and \nthe linear power spectrum of the gravitational potential is better fixed by the CMB (in the relevant units of Mpc$^{-1}$ rather than\nin $h$ Mpc$^{-1}$).\nLikewise, at a fixed angular scale CFHTLS predictions tend to be slightly weaker than those from COSMOS due to the lower source redshifts typical of ground-based surveys.\n\nWe can make this redshift dependence explicit by replacing the realistic redshift distributions with\nidealized single source planes.\nFigure~\\ref{fig:shear_z} shows the 95\\% CL region widths for sources at $z=0.5$, 2, and 3.5.\nNote that the well-constrained pivot in the power spectrum projects to higher multipole\nat higher redshift leading to tighter predictions at a fixed multipole. \nGiven the tighter predictions, high redshift cosmic shear measurements provide an interesting opportunity to falsify the flat $\\Lambda$CDM model.\n\nIn Fig.~\\ref{fig:xip_comp} we show the 2PCF $\\xi_+$ which is more useful for comparison with the relevant observations from COSMOS \\cite{Schrabback:2009ba} and CFHTLS \\cite{Benjamin:2007ys}. The displayed $1\\sigma$ error bars are computed from the full covariance matrices estimated for each survey, as described in Refs.~\\cite{Schrabback:2009ba,Benjamin:2007ys}. The predicted range of flat $\\Lambda$CDM models appears to be consistent with the observations. However, the error bars at different angular scales are heavily correlated, and therefore do not represent the actual uncertainty at any individual scale. Further note that these small-volume surveys are not well-suited for making statements for or against ruling out the $\\Lambda$CDM model; COSMOS results\nuse a 1.64 ${\\rm deg}^2$ field containing 76 galaxies per ${\\rm arcmin}^2$, and CFHTLS results use 22 ${\\rm deg}^2$ containing 12 galaxies per ${\\rm arcmin}^2$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=8cm]{shear_f}\n\\caption{Upper panel: Flat $\\Lambda$CDM predictions for the shear power spectrum, showing the 68\\% and 95\\% CL regions as in Fig.~\\ref{fig:growth} for COSMOS (upper, grey hatched) and CFHTLS (lower, blue solid). Lower panel: CFHTLS shear power spectrum prediction plotted with respect to the ML flat $\\Lambda$CDM model.}\n\\label{fig:shear}\n\\end{figure}\n\n\\begin{figure}[t]\n\\includegraphics[width=8cm]{shear_z}\n\\caption{Single source plane, 95\\% CL full-width extent for the shear power spectrum $\\Delta P_\\kappa\/P_\\kappa$ (as plotted in the lower panel of Fig.~\\ref{fig:shear}) as a function of $l$, for sources at $z=0.5$ (top), 2 (middle), and 3.5 (bottom).}\n\\label{fig:shear_z}\n\\end{figure}\n\n\\begin{figure}[t]\n\\includegraphics[width=8cm]{xip_comp}\n\\caption{Flat $\\Lambda$CDM predictions for $\\xi_{+}$, showing the 68\\% and 95\\% CL regions as in Fig.~\\ref{fig:shear} for both the COSMOS (grey hatched) and CFHTLS (blue solid) source redshift distributions. Also shown are the data points from Refs.~\\cite{Schrabback:2009ba} (black points) and \\cite{Benjamin:2007ys} (magenta points) with $1\\sigma$ error bars. Note that the error bars at different angular separations are correlated.}\n\\label{fig:xip_comp}\n\\end{figure}\n\nIf future observations falsify these predictions, then one would need to generalize the\ncosmological model class.\nThe next simplest class of models retains $\\Lambda$ as the dark energy but allows\nfor non-vanishing spatial curvature $\\Omega_{\\rm{K}}$ in the $\\Lambda$CDM context. We find a minimal error increase in this class, as shown in the upper panel of Fig.~\\ref{fig:shear_gen}. This is because spatial curvature is well constrained in the $\\Lambda$CDM paradigm. Thus a measurement that falsifies the flat $\\Lambda$CDM model would also falsify the $\\Lambda$CDM assumption itself, indicating that the dark sector is more complicated.\n\n\\subsection{Quintessence}\n\\label{sec:quintessence}\n\nMeasurements of shear observables outside the bounds shown in the previous subsection would be in statistical conflict with $\\Lambda$CDM. Barring systematic errors and unknown astrophysical effects, some of which we address in \\S \\ref{sec:uncertainties},\nsuch a measurement would indicate that the true cosmology belongs to a wider class of models. \n\n\\begin{figure}[t]\n\\includegraphics[width=8cm]{shear_gen}\n\\caption{Generalizing the model class for shear power spectrum predictions for the CFHTLS source redshift distribution. In all cases, the range is shown relative to \nthe ML flat $\\Lambda$CDM model with blue regions indicating the wider model class\nand gray hatched regions the narrower: (upper)\n $\\Lambda$CDM flat vs non-flat predictions; (middle) flat $\\Lambda$CDM vs flat quintessence without EDE; (lower) flat quintessence without EDE vs nonflat quintessence with EDE (blue solid).\nRed lines correspond to the maximum likelihood model of the more general of the two\nclasses compared.}\n\\label{fig:shear_gen}\n\\end{figure}\n\n\nIf we relax the equation of state to allow all 10 PC amplitudes to vary but revert to the flatness assumption, we find that the contours shift toward lower power as shown in the middle panel of Fig.~\\ref{fig:shear_gen}. \nNevertheless, the ML quintessence model is \nquite similar to the flat $\\Lambda$CDM\\ ML model and is near the 68\\% CL upper limit \nof the quintessence model class. This is an \nartifact of the parameter priors: given the freedom in $w(z)$ there are many ways to reduce\nthe growth in ways unconstrained by distance measures (see Fig.~3 of \\cite{Mortonson:2009hk}). The converse is not true as there are no models with good likelihood values\nin the quintessence class with more power than already allowed in flat $\\Lambda$CDM, an effect also seen in Ref.~\\cite{Mortonson:2010mj}. The effective data-driven upper bound on shear statistics for this quintessence class remains that of flat $\\Lambda$CDM\\ in spite of the downward shift in posterior contours.\n\nThe lower panel of Fig.~\\ref{fig:shear_gen} overlays the results for flat quintessence with $w_{\\infty}=-1$, and quintessence with both curvature and $w_{\\infty}$ allowed to vary (the latter generalization potentially allowing for EDE). This additional freedom further reduces growth, in accordance with the results of Ref.~\\cite{Mortonson:2010mj}. This is one of our main results, namely that generalizing from $\\Lambda$CDM\\ to quintessence models only serves to reduce cosmic shear. \nWe will show below in \\S~\\ref{sec:nonlinearpk} that warm dark matter could also only explain a power deficit; likewise, massive neutrinos would only decrease power.\\footnote{Massive neutrinos would also shift parameter values due to their effect on the CMB, so accurately quantifying the suppression of the predicted shear power requires including neutrino mass as a parameter in the initial MCMC analysis, which we leave for future work.} It is possible for baryonic effects to increase power in either $\\Lambda$CDM\\ or quintessence, but only at high multipoles.\nIn general, observations which rule out the $\\Lambda$CDM model by finding a shear excess that cannot be explained by astrophysical uncertainties also falsify \nthe entire\nquintessence paradigm. \n\n\\section{Systematic and Other Uncertainties}\n\\label{sec:uncertainties}\n\nWe now turn to exploring various factors that can impact the statistical predictions\nshown in the previous section. These uncertainties largely stem from uncertainties in the matter power spectrum and systematic uncertainties in the SNe data. In this section we will discuss the impact of an improved initial linear matter power spectrum from {\\it Planck} (\\S \\ref{sec:linearpk}), uncertainties in the nonlinear matter power spectrum from small-scale physics and the Halofit fitting function (\\S \\ref{sec:nonlinearpk}), tomography\n(\\S \\ref{sec:tomography}), and supernova light-curve fitting (\\S \\ref{sec:sn}).\n\n\\subsection{Linear Matter Power Spectrum}\n\\label{sec:linearpk}\n\nMuch of the residual statistical uncertainty in the flat $\\Lambda$CDM predictions comes from \nuncertainties in the linear matter power spectrum from WMAP7. The main sources\nare the matter density $\\Omega_{\\rm{m}} h^2$, tilt $n_s$ and normalization $\\ln A_s$. \nThese uncertainties should soon be improved by {\\it Planck}. We show how the predictions change from importance sampling to {\\it Planck}-like priors on the parameters $\\Omega_{\\rm{b}}h^2$, $\\Omega_{\\rm{m}}h^2$, $n_s$, and $\\ln A_s$, using the DETF projected covariance matrix \\cite{DETF}\nand assuming the mean corresponds to the ML flat $\\Lambda$CDM model. \n\nIn Fig.~\\ref{fig:planck} we show the improvement in the fractional constraints (grey hatched curves) on the linear power spectrum (upper panel) and on the shear power spectrum of the CFHTLS source redshift distribution (lower panel). One typically should expect Planck to improve the precision of predictions by more than a factor of 2 to the level of $\\pm 3$--$5\\%$, which will greatly enhance the prospects for \nfalsifying the standard $\\Lambda$CDM paradigm.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=8cm]{planck}\n\\caption{Expected improvements from {\\it Planck} (grey hatched) over WMAP7 (blue solid) \nfor flat $\\Lambda$CDM predictions. Upper panel shows the linear matter power spectrum and the lower panel shows the shear power spectrum for the CFHTLS source redshift distribution, with the 68\\% and 95\\% CL regions relative to the ML flat $\\Lambda$CDM\\ model.}\n\\label{fig:planck}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=8cm]{shear_halofit}\n\\caption{Upper panel: Flat $\\Lambda$CDM prediction for the CFHTLS shear power spectrum in blue, shown with the prediction for when the parameter $c_n$ is amplified by $10\\%$ (green curves). The altered 68\\% CL regions are bounded by solid lines and the corresponding altered 95\\% CL regions are bounded by dashed lines.\nLower panel: Same as above, but now instead the one-halo term amplitude $a_n$ is amplified by $10\\%$ (magenta curves).}\n\\label{fig:shear_halofit}\n\\end{figure}\n\n\\subsection{Nonlinear Matter Power Spectrum}\n\\label{sec:nonlinearpk}\n\nWe now explore the impact of a few sources of uncertainty in the nonlinear matter power spectrum on our predictions for the shear power spectrum ---\nthe Halofit fitting function \\cite{Smith:2002dz},\n warm dark matter \\cite{Viel:2011bk}, and baryonic physics \\cite{Rudd:2007zx,vanDaalen:2011xb}. We will find it most convenient to parameterize the latter two (physics-based) uncertainties in terms of the former. For these studies we employ our CFHTLS source redshift distribution and the flat $\\Lambda$CDM model class.\n\nHalofit itself has been found to be only accurate at up to the $5$--$10\\%$ level in the nonlinear regime for the $\\Lambda$CDM cosmological models for which it was designed \\cite{Heitmann:2008eq,Heitmann:2009cu,Lawrence:2009uk}, even with the correction factor Eq.~(\\ref{halofitcorrection}). While these inaccuracies are smaller than our current statistical errors in the same regime, they are comparable to the expected {\\it Planck} errors.\n\nFurthermore, we have employed the $\\Lambda$CDM-calibrated Halofit parameters in our predictions for the more general quintessence\nclass. While we expect that the increased statistical uncertainties in those classes are again currently larger than Halofit errors, this expectation remains largely untested by simulations.\nTwo of the Halofit parameters in particular describe the characteristic changes in the the nonlinear power spectrum: $a_n$ controls the amplitude of the one-halo term and $c_n$\ndescribes its shape. We explore how variations in these two parameters affect our\npredictions as a means of quantifying how well Halofit parameters must be calibrated in the $\\Lambda$CDM\\ and quintessence $w(z)$ classes.\n\nIn Fig.~\\ref{fig:shear_halofit},\nwe show once again our flat $\\Lambda$CDM predictions for the CFHTLS shear power spectrum where in the upper panel we show what happens when $c_n$ is amplified and in the lower panel we show what happens when $a_n$ is amplified. In each case we are only altering the nonlinear matter power spectrum, and thus only the high multipole regime of the shear power spectrum. In the case of amplifying $a_n$, we are simply enhancing the one-halo term. We conclude that $a_n$ and $c_n$ must be calibrated in $\\Lambda$CDM\\ to the $\\sim 10\\%$ level to make use of current statistical predictions for $\\ell < 1000$ whereas in quintessence models with EDE and curvature creating large power deficits the tolerances can be up to double that.\n\nWhile Halofit was \nconstructed to fit N-body simulations with cold dark matter and no baryonic effects,\nwe find that in particular $c_n$ is a useful parameter\nfor describing typical deviations from these assumptions.\n\nWarm dark matter (WDM) reduces the small-scale matter power spectrum in a way that increases with decreasing WDM particle mass. By using high-resolution N-body\/hydrodynamic simulations, Ref.~\\cite{Viel:2011bk} constructs a fitting function to describe the modified matter power spectrum $P_{\\Lambda\\rm{WDM}}(k)$ in terms of that of the corresponding $\\Lambda$CDM model $P_{\\Lambda\\rm{CDM}}(k)$,\n\\begin{equation}\nP_{\\Lambda\\rm{WDM}}(k)=\\left[1+(\\alpha k)^{1.8}\\right]^{-2\/15} P_{\\Lambda\\rm{CDM}}(k)~,\n\\end{equation}\nwhere\n\\begin{equation}\n\\alpha(m_{\\rm{WDM}},z)=0.0476\\left(\\frac{1~\\rm{keV}}{m_{\\rm{WDM}}}\\right)^{1.85}\\left(\\frac{1+z}{2}\\right)^{1.3}\n\\end{equation}\nin units of $h^{-1}$~Mpc and $m_{\\rm{WDM}}$ is the mass of the warm dark matter particle. This fitting function is accurate at the $\\sim 2\\%$ level below $z=3$ and for masses $m_{\\rm{WDM}} > 0.5$ keV. On the other hand, this form assumes a fixed set of $\\Lambda$CDM\\ parameters associated\nwith the simulations and thus effects should be taken as illustrative of the magnitude of the effect.\nWe find that the WDM effects corresponding to $0.5$ keV dark matter can be mimicked by the Halofit parameter $c_n$\nwith $c_n\\rightarrow 1.01c_n$ (Figure~\\ref{fig:shear_cn}, upper panel). \nWe thus see that, even for this fairly extreme particle mass, warm dark matter contributes negligibly to the error budget for this particular statistic, in agreement with Ref.~\\cite{Viel:2011bk}. \n\nBaryonic effects, such as radiative heating and cooling, star formation, and supernova and AGN feedback, affect our predictions through their impact on the small-scale matter power spectrum. Given the inherent difficulties in adding baryons to large-scale structure simulations, the degree to which these various processes impact structure growth remains highly uncertain. However we can make a conservative assessment by using the most extreme example, namely the results of Ref.~\\cite{vanDaalen:2011xb} which found a significant decrease in power ranging from (at $z=0$) $1\\%$ at $k\\approx 0.3~h~{\\rm Mpc}^{-1}$ to $10\\%$ at $k\\approx 1~h~{\\rm Mpc}^{-1}$ and $30\\%$ at $k\\approx 10~h~{\\rm Mpc}^{-1}$. To find the rough impact on the nonlinear matter power spectrum, we interpolate Fig.~8 of Ref.~\\cite{vanDaalen:2011xb}.\nAgain we find that the Halofit $c_n$ accurately mimics this effect with $c_n\\rightarrow 1.15c_n$ corresponding to the simulations of Ref.~\\cite{vanDaalen:2011xb} which include AGN feedback (Figure~\\ref{fig:shear_cn}, lower panel). A more detailed study of the effect on weak lensing observations is presented in Ref.~\\cite{Semboloni:2011fe}.\n\nThus current uncertainties in baryonic physics at their most extreme are as large as current statistical ranges in the predictions. Hence the modeling of baryonic effects on structure formation must improve if we are to make use of future constraints above $l=10^3$, as found in Refs.~\\cite{Rudd:2007zx,vanDaalen:2011xb}. Note also that, although the particular model we have shown here suppresses scale-scale power, baryons can have the opposite effect of potentially the same magnitude, by increasing small-scale power via cooling \\cite{Rudd:2007zx}. \nIn the future, one approach to accounting for these systematic errors is to marginalize $a_n$ and $c_n$ as model parameters given\na prior range appropriate to the state-of-the-art simulations including baryonic and other effects.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=8cm]{shear_cn}\n\\caption{Demonstration of our parameterization of WDM (upper panel) and baryonic effects (lower panel) with the Halofit parameter $c_n$, using the CFHTLS source distribution.\nUpper panel: Flat $\\Lambda$CDM prediction for the shear power spectrum in blue, shown with predictions for $0.5$~keV WDM as in Ref.~\\cite{Viel:2011bk} with grey dashed lines and predictions for $c_n\\rightarrow 1.01 c_n$ with red solid lines.\nLower panel: Flat $\\Lambda$CDM prediction for the shear power spectrum in blue, shown with predictions for baryons (including AGN feedback) as in Ref.~\\cite{vanDaalen:2011xb} with grey dashed lines and predictions for $c_n\\rightarrow 1.15 c_n$ with red solid lines.}\n\\label{fig:shear_cn}\n\\end{figure}\n\n\\begin{figure}[t]\n\\includegraphics[width=8cm]{shear_ratio}\n\\caption{The ratio $f(l)$ of the COSMOS and CFHTLS cosmic shear power spectrum predictions for the flat $\\Lambda$CDM (upper panel) and nonflat quintessence with EDE (lower panel) model classes, plotted with respect to the ratio of the ML COSMOS and CFHTLS power spectra. Once again shaded regions correspond to 68\\% CL and the curves bound the 95\\% CL regions. \nThe upper panel also shows how the flat $\\Lambda$CDM ratio changes when either $c_n$ (upper green contours) or $a_n$ (lower magenta contours) are amplified by $10\\%$, where the altered 68\\% CL regions are bounded by solid lines and the corresponding altered 95\\% CL regions are bounded by dashed lines.}\n\\label{fig:shear_ratio}\n\\end{figure}\n\n\\subsection{Tomography and Ratio Tests}\n\\label{sec:tomography}\n\nOne means of reducing both the statistical and systematic uncertainty is to employ\ntomographic probes that compare the shear at different source redshifts\n\\cite{Hu:1999ek}. Uncertainties remaining from the primordial power spectrum \ndiscussed in \\S \\ref{sec:linearpk} largely\ndrop out and some of the effects of baryonic physics may be ``self-calibrated\" through \njointly determining the concentration-mass relation \\cite{Rudd:2007zx,Zentner:2007bn}.\n\nAs a simple demonstration and a proxy for tomography, we can define the statistic $f(l)$ (not to be confused with $f(z)$ from Eq.~\\ref{eqn:fz}) for a given model class as the ratio of the shear power spectrum $P_{\\kappa}(l)$ predicted for COSMOS (median redshift of $1.3$) to that predicted for CFHTLS (median redshift of $0.8$). The resulting prediction is shown in Fig.~\\ref{fig:shear_ratio} for our most restrictive (flat $\\Lambda$CDM, upper panel) and most general (nonflat quintessence with EDE, lower panel) model classes, where now the $\\sim 1\\sigma$ uncertainties are reduced to the $1\\%$ level for the flat $\\Lambda$CDM model. This sharp prediction offers another opportunity to falsify flat $\\Lambda$CDM if systematics in the measurement and ratio prediction can be made comparably precise.\nIn the lower panel of Fig.~\\ref{fig:shear_ratio}, the features at small scales are related to the asymmetry of the quintessence growth predictions and the difference in the nonlinear scales for the source distributions of COSMOS ($l\\gtrsim 200$) and CFHTLS ($l\\gtrsim 500$).\n\nHowever, present uncertainties in the nonlinear power spectrum arising from Halofit render predictions in the nonlinear regime unreliable. \nIn the upper panel of Fig.~\\ref{fig:shear_ratio} we show how the uncertainties in $c_n$ (green, upper curves) and $a_n$ (magenta, lower curves), respectively, affect the flat $\\Lambda$CDM\\ predictions for $f(l)$. These uncertainties affect the quintessence predictions similarly. We see that $\\sim 10\\%$ level calibration of these Halofit parameters is not sufficient to exploit the sub-percent level $\\Lambda$CDM\\ predictions in the nonlinear regime. On the other hand, observed deviations in $f(l)$ of $\\gtrsim 3\\%$ would falsify $\\Lambda$CDM\\ and $\\gtrsim 10\\%$ our most general quintessence class even considering these uncertainties. \n\n\\subsection{Supernova Light-Curve Fitting}\n\\label{sec:sn}\n\nThe analyses of the datasets outlined in \\S \\ref{sec:data} contribute further systematic uncertainties in our predictions, the largest of which arise from the fitting of SN light-curves. The two most widely used light-curve fitters are SALT2 and MLCS2k2. Assuming a flat cosmological model with a constant dark energy equation of state $w$, the SALT2 and MLCS2k2 methods yield $(\\Omega_{\\rm m},w)=(0.26 \\pm 0.03, -0.96 \\pm 0.13)$ and $(\\Omega_{\\rm m},w)=(0.31 \\pm 0.03, -0.76 \\pm 0.13)$, respectively \\cite{SDSS_SN}. Thus these two methods lead to a discrepancy in the dark energy equation of state of $\\Delta w \\sim 0.2$. This might be due to a possible mismatch in the UV spectra between low-redshift and intermediate-redshift SNe \\cite{Foley:2010mm} and efforts are underway to reduce the resulting error. We have chosen here to use SN data analyzed using the SALT2 technique, but we now briefly discuss what happens when we instead use the MLCS2k2 data.\\footnote{Note that Ref.~\\cite{Mortonson:2010mj} made the opposite choice, using the MLCS2k2 SN constraints for their main results.}\n\nEven for the flat $\\Lambda$CDM model class, the differences between SALT2 and MLCS2k2 estimates of $\\Omega_{\\rm m}$ and $w$ significantly alter our predictions for the shear power spectrum, as can be seen in the upper panel of Fig.~\\ref{fig:shear_ms}. In the lower panel we show the comparison for flat quintessence; in both cases, the CL regions shift by $\\sim 0.5$--$1\\sigma$. This shift to higher shear with MLCS2k2 is driven mainly by the preference for higher $\\Omega_{\\rm m}$ when using this method; this higher $\\Omega_{\\rm m}$ also increases the present day matter power spectrum normalization for a fixed amplitude of initial curvature fluctuations $A_s$. The corresponding effect for cluster abundance was found in Ref.~\\cite{Mortonson:2010mj}.\nWith improvements in the calibration of SN data and modeling of the \neffects of dust extinction, it is likely that these systematics can be reduced\nso that they will not be a limiting uncertainty for future predictions of \nshear statistics.\n\n\\begin{figure}[t]\n\\includegraphics[width=8cm]{shear_ms}\n\\caption{Shear power spectrum predictions for the CFHTLS source redshift distribution, where we show the bias resulting from the choice of SN light curve fitter. \nUpper panel: Flat $\\Lambda$CDM predictions for SALT2 (blue solid) and MLCS2k2 (grey hatched), plotted relative to the ML SALT2 flat $\\Lambda$CDM result. Once again the shaded regions correspond to 68\\% CL and the curves bound the 95\\% CL regions. \nLower panel: Flat quintessence predictions for SALT2 (blue solid) and MLCS2k2 (grey hatched), also plotted relative to the ML SALT2 flat $\\Lambda$CDM result.}\n\\label{fig:shear_ms}\n\\end{figure}\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nIn this paper we have provided robust statistical predictions for weak lensing observables in a variety of cosmological contexts. Given existing local distance measures of $H_0$, the Type Ia SNe distance-redshift relation, BAO distance measures, and the CMB temperature and polarization power spectra, we have constrained the expected cosmic shear power spectrum in the $\\Lambda$CDM model. We then generalized this model class to show how the predictions widen when curvature, EDE, and a generalized dark energy equation of state $w(z)$ are allowed. We further showed how these predictions are affected by uncertainties in the nonlinear matter power spectrum from warm dark matter and baryons, to find that the former is negligible whereas the latter is a comparable source of uncertainty to current statistical errors beyond $l\\sim 10^{3}$. In the near future baryonic effects will likely become the dominant source of error unless modeling improves.\n\nIn a similar fashion as in the clusters study of Ref.~\\cite{Mortonson:2010mj}, we find that any observation that claims to rule out the $\\Lambda$CDM model via a shear excess generically also rules out the entire quintessence paradigm, as the extra freedom allowed by the free function $w(z)\\ge -1$ can only serve to reduce the relative growth if we normalize to the CMB and constrain the distance-redshift relation to the CMB and SNe at opposite ends of the\nexpansion history.\nAdding EDE, warm dark matter, massive neutrinos, or baryonic AGN feedback tends to exacerbate the reduction, as they too only reduce growth and therefore suppress the cosmic shear power spectrum. Therefore a measured shear excess would require updating our models to include significant baryonic cooling, primordial non-Gaussianity, non-smooth dark energy, \nphantom dark energy, an interacting dark sector, or even a modification to the gravitational\nforce law.\n\nIn our analysis we have focused only on those uncertainties pertaining to the predictions for the shear statistics presented here, as opposed to uncertainties in the measurement of\nthe shear itself. The latter becomes important when comparing the data to these predictions. \nFor example in comparing our predictions to current data in Fig.~\\ref{fig:xip_comp}, we have\nemployed the statistical and systematic error estimates in the literature. Uncertainties relating to shape measurement, photometric redshifts, and intrinsic alignments feed into the data error covariance. Accounting for these effects, the flat $\\Lambda$CDM\\ model is consistent with the\ncurrent data.\n\nFuture imaging surveys, both from the ground (e.g.~DES and LSST) and from space ({\\it Euclid} and\/or {\\it WFIRST}) will provide the necessary number of galaxy shape measurements to enable precise tests of dark energy model classes, as long as the various shear systematics can be either eliminated or understood sufficiently well. Moreover, improved \nCMB constraints from {\\it Planck} will soon reduce the uncertainties in the \ncosmic shear predictions by a factor of a few, further enhancing the ability \nof weak lensing observations to detect deviations from the concordance cosmological model but also requiring more stringent control on astrophysical systematic errors in both the predictions and the measurements.\n\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\\begin{acknowledgments}\n\nWe thank Chris Hirata, Richard Massey, Catherine Heymans, and Tim Schrabback for assistance regarding weak lensing survey parameters and results and Scott Dodelson, Eduardo Rozo, Elise Jennings, David Weinberg, and Mark Wyman for useful conversations. RAV and WH acknowledge the support of the Kavli Institute for Cosmological Physics at the University of Chicago through grants NSF PHY-0114422 and NSF PHY-0551142 and an endowment from the Kavli Foundation and its founder Fred Kavli. MJM and TE acknowledge support from CCAPP at the Ohio State University. WH acknowledges additional support from\nDOE contract DE-FG02-90ER-40560 and the Packard Foundation.\n\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nGeometric simplicial complexes, such as Vietoris--Rips or \\v{C}ech, are one of the cornerstones of topological data analysis.\nOne can approximate the shape of a dataset $X$ by building a growing sequence of Vietoris--Rips complexes with $X$ as the underlying set, and then computing persistent homology.\nThe shape of the data, as measured by persistence, is reflective of important patterns within~\\cite{Carlsson2009}.\n\nThe popularity of Vietoris--Rips complexes relies on at least three facts.\nFirst, Vietoris--Rips filtrations and their persistent homology signatures are computable~\\cite{bauer2019ripser}.\nSecond, Vietoris--Rips persistent homology is stable~\\cite{chazal2009gromov,ChazalDeSilvaOudot2014}, meaning that the topological data analysis pipeline is robust to certain types of noise.\nThird, Vietoris--Rips complexes are topologically faithful at low scale parameters: one can use them to recover the homotopy types~\\cite{Latschev2001} or homology groups~\\cite{ChazalOudot2008} of an unknown underlying space, when given only a finite noisy sampling.\n\nAt higher scale parameters, we mostly do not know how Vietoris--Rips complexes behave.\nThis is despite the fact that one of the key insights of persistent homology is to allow the scale parameter to vary from small to large, tracking the lifetimes of features as the scale increases.\nOur practice is ahead of our theory in this regard: data science practitioners are building Vietoris--Rips complexes with scale parameters larger than those for which the reconstruction results of~\\cite{Latschev2001,ChazalOudot2008} apply.\n\nStability implies that as more and more data points are sampled from some ``true'' underlying space $M$, the Vietoris--Rips persistent homology of the dataset $X$ converges to the Vietoris--Rips persistent homology of $M$.\nThe simplest possible case is when the dataset $X$ is sampled from a manifold $M$, and so we cannot fully understand the Vietoris--Rips persistent homology of data without also understanding the Vietoris--Rips persistent homology of manifolds.\nHowever, not much is known about Vietoris--Rips complexes of manifolds, except at small scales~\\cite{Hausmann1995}.\nEven the Vietoris--Rips persistent homology of the $n$-sphere $\\ensuremath{\\mathbb{S}}^n$ is almost entirely unknown.\n\nTwo potential obstacles for understanding the homotopy types of Vietoris--Rips complexes of a manifold $M$ are:\n\\begin{enumerate}\n\t\\item the natural inclusion $M \\hookrightarrow \\vr{M}{r}$ is not continuous, and\n\t\\item we do not yet have a full Morse theory for Vietoris--Rips complexes of manifolds.\n\\end{enumerate}\n\nThere are by now several strategies for handling these two obstacles.\nOne strategy is to remain in the setting of Vietoris--Rips simplicial complexes.\nObstacle (1) is then unavoidable.\nRegarding obstacle (2), Bestvina--Brady Morse theory has only been successfully applied at low scale parameters, allowing Zaremsky~\\cite{zaremsky2019} to prove that $\\vr{\\ensuremath{\\mathbb{S}}^n}{r}$ recovers the homotopy type of $\\ensuremath{\\mathbb{S}}^n$ for $r$ small enough, but not to derive new homotopy types that appear as $r$ increases.\nSimplicial techniques have been considered for a long time, but even successes such as an understanding of the homotopy types of the Vietoris--Rips complexes of the circle at all scales~\\cite{AA-VRS1} are not accompanied by a broader Morse theory (though some techniques feel Morse-theoretic).\n\nA second strategy is very recent.\n\\cite{lim2020vietoris,okutan2019persistence} show that the Vietoris--Rips simplicial complex filtration is equivalent to thickenings of the Kuratowski embedding into $L^\\infty(M)$ or any other \\emph{injective} metric space: in particular, the two filtrations have the same persistent homology.\nThis overcomes obstacle (1): the inclusion of a metric space into a thickening of its Kuratowski embedding is continuous, and indeed an isometry onto its image.\nThis connection has created new opportunities, such as the Morse theoretic techniques employed by Katz~\\cite{katz1983filling,katz1989diameter,katz9filling,katz1991neighborhoods}.\nThis Morse theory allow one to prove the first new homotopy types that occur for Vietoris--Rips simplicial complexes of the circle and 2-sphere, but have not yet inspired progress for larger scales, or for spheres above dimension two.\n\nA third strategy is to consider Vietoris--Rips metric thickenings, which rely on optimal transport and Wasserstein-distances~\\cite{AAF}.\nWe refer to these spaces as the $\\infty$-metric thickenings, for reasons that will become clear in the following paragraph.\nSuch thickenings were invented in order to enable Morse-theoretic proofs of the homotopy types of Vietoris--Rips type spaces.\nThe first new homotopy type of the $\\infty$-Vietoris--Rips metric thickening of the $n$-sphere is known~\\cite{AAF}, but only for a single (non-persistent) scale parameter.\nIt was previously only conjectured that the $\\infty$-Vietoris--Rips metric thickenings have the same persistent homology as the more classical Vietoris--Rips simplicial complexes~\\cite[Conjecture~6.12]{AAF}; one of our contributions is to answer this conjecture in the affirmative.\n\\cite{MirthThesis} considers a Morse theory in Wasserstein space, which is inspired in part by applications to $\\infty$-Vietoris--Rips metric thickenings, but which does not apply as-is to these thickenings as the $\\infty$-diameter functional is not ``$\\lambda$-convex''~\\cite{santambrogio2017euclidean}.\n\nWe introduce a generalization: the $p$-Vietoris--Rips metric thickening for any $1\\le p\\le \\infty$.\nLet $X$ be an arbitrary metric space.\nFor $1\\leq p \\leq \\infty$, the $p$-Vietoris--Rips metric thickening at scale parameter $r> 0$ contains all probability measures on $X$ whose $p$-diameter is less than $r$.\nFor $p$ finite, the $p$-diameter of a probability measure $\\alpha$ on the metric space $X$ is defined as \n\\[\\mathrm{diam}_p(\\alpha):=\\left(\\iint_{X\\times X} d_X^p(x,x')\\,\\alpha(dx)\\,\\alpha(dx')\\right)^{1\/p},\\]\nand $\\mathrm{diam}_\\infty(\\alpha)$ is defined to be the diameter of the support of $\\alpha$.\n\n\nThe $p$-Vietoris--Rips metric thickenings at scale $r$ form a metric bifiltration of $X$ that is covariant in $r$ and contravariant in $p$.\nIndeed, we have an inclusion map $\\vrpp{p}{X}{r} \\hookrightarrow \\vrpp{p'}{X}{r'}$ for $r\\le r'$ and $p \\ge p'$; see Figure~\\ref{fig:vrpBifiltration}.\n\\[\n\\xymatrix{\n \\vrpp{p'}{X}{r} \\ar@{^{(}->}[r] & \\vrpp{p'}{X}{r} \\\\\n \\vrp{X}{r} \\ar@{^{(}->}[r]\\ar@{^{(}->}[u] & \\vrp{X}{r'}\\ar@{^{(}->}[u]\n\t}\n\\]\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.75\\textwidth]{vrp_bifiltration_v2.pdf}\n\\caption{The $p$-Vietoris--Rips bifiltration $\\vrp{X}{r}$, where $X$ is a metric space over 3 points, and where $\\vrpp{p}{X}{r}\\subseteq\\vrpp{p'}{X}{r'}$ for $r\\le r'$ and $p \\ge p'$.}\n\\label{fig:vrpBifiltration}\n\\end{figure}\n\nAs one of our main contributions, we prove that the $p$-Vietoris--Rips metric thickening is stable.\nThis means that if two totally bounded metric spaces $X$ and $Y$ are close in the Gromov--Hausdorff distance, then their filtrations $\\vrpf{X}$ and $\\vrpf{Y}$ are close in the homotopy interleaving distance.\nThis was previously unknown even in the case $p=\\infty$ (see~\\cite[Conjecture~6.14]{AAF}); we prove stability for all $1\\le p\\le \\infty$.\nAs a consequence, it follows that the (undecorated) persistent homology diagrams for the Vietoris--Rips simplicial complexes $\\vr{X}{r}$ and for the $p=\\infty$ Vietoris--Rips metric thickenings $\\vrppf{\\infty}{X}$ are identical.\nIn other words, the persistent homology barcodes for $\\vrf{X}$ and $\\vrppf{\\infty}{X}$ are identical up to replacing closed interval endpoints with open endpoints, or vice-versa.\nThis answers~\\cite[Conjecture~6.12]{AAF} in the affirmative.\nAnother consequence of stability is that the $p$-metric thickenings give the same persistence diagrams whether one considers all Radon probability measures, or instead the restricted setting of only measures with finite support.\n\nWe furthermore introduce the $p$-\\v{C}ech metric thickenings, again for $1\\le p \\le \\infty$, and prove analogues of all of the above results.\nThe $p$-\\v{C}ech metric thickening at scale parameter $r> 0$ contains all probability measures supported on $X$ whose $p$-radius is less than $r$.\n\nWe also deduce the complete spectrum of homotopy types of $2$-Vietoris--Rips and $2$-\\v{C}ech metric thickenings of the $n$-sphere, equipped with the Euclidean metric $\\ell_2$:\n$\\vrpp{2}{(\\ensuremath{\\mathbb{S}}^n,\\ell_2)}{r}$ first attains the homotopy type of $\\ensuremath{\\mathbb{S}}^n$, and then for all scales $r$ larger than $\\sqrt{2}$ the space is contractible.\nBy contrast, the Vietoris--Rips simplicial complexes or the $\\infty$-Vietoris--Rips metric thickenings of the $n$-sphere (with either the Euclidean or the geodesic metric) are only known for a bounded range of scales, including only a single change in homotopy type (\\cite[Section~5]{AAF}),\neven though infinitely many changes in homotopy type are conjectured (c.f.~\\cite[Question~8.1]{ABF2}).\nSee however,~\\cite[Corollary 7.18]{lim2020vietoris} for results for round spheres with the $\\ell_\\infty$ metric.\n\nOne of our main motivations for introducing the $p$-Vietoris--Rips and $p$-\\v{C}ech metric thickenings is to enable effective Morse theories on these types of spaces.\nThe $p$-variance for \\v{C}ech metric thickenings is a minimum of linear functionals, and therefore fits in the framework of Morse theory for min-type functions~\\cite{baryshnikov2014min,gershkovich1997morse,bryzgalova1978maximum,matov1982topological}.\nOn the Vietoris--Rips side, we remark that gradient flows of functionals on Wasserstein can be defined when the functional is ``$\\lambda$-convex''~\\cite{santambrogio2017euclidean,MirthThesis}.\nThough the $\\infty$-diameter is not $\\lambda$-convex, we hope that the $p$-diameter functional for $p<\\infty$ may be $\\lambda$-convex in certain settings.\n\n\n\\paragraph*{Organization.}\nIn Section~\\ref{sec:background} we describe background material and set notation.\nWe define the $p$-Vietoris--Rips and $p$-\\v{C}ech metric thickenings in Section~\\ref{sec:p-relaxation}, and consider their basic properties in Section~\\ref{sec:basic-properties}.\nIn Section~\\ref{sec:stability} we prove stability.\nWe consider Hausmann-type theorems in Section~\\ref{sec:hausmann}, and deduce the $2$-Vietoris--Rips metric thickenings of Euclidean spheres in Section~\\ref{sec:spheres}.\nIn Section~\\ref{app:spread} we bound the length of intervals in $p$-Vietoris--Rips and $p$-\\v{C}ech metric thickenings using a generalization of the spread of a metric space, called the $p$-spread.\n\nWe conclude the paper by providing some discussion in Section~\\ref{sec:conclusion}.\n\nIn Appendix~\\ref{app:Metrization_weak_topology} we explain how the $q$-Wasserstein distance metrizes the weak topology for $1\\le q<\\infty$.\nWe describe connections to min-type Morse theories in Appendix~\\ref{app:Morse}.\nIn Appendix~\\ref{app:finite-Cech} we show that $p$-\\v{C}ech thickenings of finite metric spaces are homotopy equivalent to simplicial complexes.\nWe prove the persistent homology diagrams of the $p$-Vietoris--Rips and $p$-\\v{C}ech metric thickenings of a family of discrete metric spaces in Appendix~\\ref{app:pd_delta_1_space}, and we describe the $0$-dimensional persistent homology of the $p$-Vietoris--Rips and $p$-\\v{C}ech metric thickenings of an arbitrary metric space in Appendix~\\ref{app:PH0}.\nWe consider crushings in Appendix~\\ref{app:crushings}.\nIn Appendix~\\ref{app:ambient}, we show that the main properties we prove for the (intrinsic) $p$-\\v{C}ech metric thickening also hold for the ambient $p$-\\v{C}ech metric thickening.\n\n\n\n\\section{Background}\n\\label{sec:background}\n\nThis section introduces background material and notation.\n\n\\paragraph{Metric spaces and the Gromov--Hausdorff distance.}\nLet $(X,d_X)$ be a metric space.\nFor any $x\\in X$, we let $B(x;r):=\\{y\\in X~|~d_X(x,y)0$ sufficiently small depending on the curvature of $M$, Hausmann produces a map $T\\colon \\vr{M}{r}\\to M$ from the simplicial complex to the manifold that is not canonical in the sense that it depends on a total order of all points in the manifold.\nAlso, the inclusion map $M\\hookrightarrow\\vr{M}{r}$ is not continuous, and therefore cannot be a homotopy inverse for $T$.\nNevertheless, Hausmann is able to prove $T$ is a homotopy equivalence without constructing an explicit inverse.\nBy contrast, in the context of metric thickenings, one can produce a canonical map $\\vrppfin{\\infty}{M}{r}\\to M$ by mapping a measure to its Fr\\'{e}chet mean (whenever $r$ is small enough so that measures of diameter less than $r$ have unique Fr\\'{e}chet means).\nThe (now continuous) inclusion $M\\hookrightarrow\\vrppfin{\\infty}{M}{r}$ can be shown to be a homotopy inverse via linear homotopies~\\cite[Theorem~4.2]{AAF}.\n\nOne of the main contributions of this paper is showing how these various spaces relate to each other, especially when it comes to persistent homology.\nThis allows one to work either simplicially, geometrically, or with measures --- whichever perspective is most convenient for the task at hand.\n\n\\paragraph{Homology and Persistent homology.}\nFor each integer $k\\geq 0$, let $H_k$ denote either the simplicial or the singular homology functor from the cateogry $\\mathrm{Top}$ of topological spaces to the category $\\mathrm{Vec}$ of vector spaces and linear transformations.\nWe use coefficients in a fixed field, so that homology groups are furthermore vector spaces.\nFor background on persistent homology, we refer the reader to~\\cite{EdelsbrunnerHarer,edelsbrunner2000topological,zomorodian2005computing}.\n\nIn applications of topology, such as topological data analysis~\\cite{Carlsson2009}, one often models a dataset not as a single space $X$, but instead as an increasing sequence of spaces.\nWe refer to an increasing sequence of spaces, i.e.\\ a functor from the poset $(\\ensuremath{\\mathbb{R}}, \\leq)$ to $\\mathrm{Top}$, as a \\emph{filtration}.\nIf $X$ is a metric space, then a common filtration is the Vietoris-Rips simplicial complex filtration $\\vrf{X}$.\nIn this paper we will introduce relaxed versions, the $p$-Vietoris--Rips metric thickening filtrations $\\vrpf{X}$ (see Section~\\ref{sec:p-relaxation}).\n\nBy applying homology (with coefficients in a field) to a filtration, we obtain a functor from the poset $(\\ensuremath{\\mathbb{R}}, \\leq)$ to $\\mathrm{Vec}$, i.e.\\ a \\emph{persistence module}.\nWe will use symbols like $V$, $W$ and so on to denote persistence modules.\nFollowing~\\cite{chazal2016structure}, a persistence module is \\emph{Q-tame} if for any $s0$, there is a point $z\\in X$ with\n\\[\\left(\\int_Xd_X^p(x, z)\\alpha(dx)\\right)^\\frac{1}{p} \\le \\mathrm{rad}_p(\\alpha)+\\varepsilon.\\]\nThen we have\n\\begin{align*}\n\\mathrm{diam}_p(\\alpha) & = \\Big(\\iint_{X\\times X} \\big(d_X(x,x')\\big)^p\\alpha(dx)\\,\\alpha(dx')\\Big)^{\\frac{1}{p}} \\\\\n& \\leq \\Big(\\iint_{X\\times X} \\big(d_X(x,z) + d_X(z, x')\\big)^p\\alpha(dx)\\,\\alpha(dx')\\Big)^{\\frac{1}{p}} \\\\\n& \\leq \\Big(\\iint_{X\\times X} \\big(d_X(x,z)\\big)^p\\alpha(dx)\\,\\alpha(dx')\\Big)^{\\frac{1}{p}} + \\Big(\\iint_{X\\times X} \\big(d_X(z,x')\\big)^p\\alpha(dx)\\,\\alpha(dx')\\Big)^{\\frac{1}{p}} \\\\\n& \\le 2\\,\\mathrm{rad}_p(\\alpha)+2\\varepsilon.\n\\end{align*}\nSince $\\varepsilon>0$ was arbitrary, this shows $\\mathrm{diam}_p(\\alpha)\\leq 2\\,\\mathrm{rad}_p(\\alpha)$.\n\\end{proof}\n\n\\begin{remark}\nThe tightness of the bound $\\mathrm{rad}_p(\\alpha)\\leq \\mathrm{diam}_p(\\alpha)$ can be seen from the calculation for the uniform measure on $Z_n$, the metric space on a set of size $n$ with all interpoint distances equal to $1$.\nTo see the (asymptotic) tightness of $\\mathrm{diam}_p(\\alpha)\\leq 2\\,\\mathrm{rad}_p(\\alpha)$, we consider the metric space $Z_n\\cup \\{O\\}$, where the newly introduced ``center\" $O$ has distance $1\/2$ to every other point.\nThen for the measure $\\alpha := \\sum_{z\\in Z_n} \\frac{1}{n}\\delta_z$, we have $\\mathrm{rad}_p(\\alpha) = 1\/2$ and $\\mathrm{diam}_p(\\alpha) = (\\frac{n-1}{n})^{\\frac{1}{p}}$.\nWe obtain asymptotic tightness by letting $n$ go to infinity.\n\\end{remark}\n\n\n\\subsection{The relaxed Vietoris--Rips and \\v{C}ech metric thickenings}\n\n\\begin{defn}[$p$-Vietoris--Rips filtration]\\label{defn:vrp}\nFor each $r>0$ and $p\\in[1,\\infty]$, let the \\emph{$p$-Vietoris--Rips metric thickening at scale $r$} be\n\\[\\vrp{X}{r}:=\\{\\alpha\\in\\ensuremath{\\mathcal{P}}_X~|~\\mathrm{diam}_p(\\alpha)0$, not only does $\\mathrm{VR}_p(X;r)$ become larger and larger as $p$ decreases, but also for $p=0$, $\\mathrm{VR}_0(X;r)$ contains \\emph{all} Radon probability measures on $X$ and thus has trivial reduced homology.\n \n\\end{remark}\n\n\\begin{defn}[$p$-\\v{C}ech filtration]\n\\label{defn:cechp}\nFor each $r>0$ and $p\\in[1,\\infty]$, let the \\emph{$p$-\\v{C}ech metric thickening at scale $r$} be\n\\[\\cechp{X}{r}:=\\{\\alpha\\in\\ensuremath{\\mathcal{P}}_X~|~\\mathrm{rad}_p(\\alpha) 0$, we have\n\\[\\vrp{X}{r}\\subseteq \\cechp{X}{r}\\subseteq\\vrp{X}{2r}.\\]\n\\end{prop}\n\n\\begin{proof}\nThis follows immediately from Proposition~\\ref{prop:basic-diamp}, which implies that for any $\\alpha\\in\\ensuremath{\\mathcal{P}}_X$,\n\\[\\mathrm{diam}_p(\\alpha) \\ge \\mathrm{rad}_p(\\alpha) \\ge \\tfrac{1}{2}\\,\\mathrm{diam}_p(\\alpha).\\]\n\\end{proof}\n\nIf $X$ and $Z$ are metric spaces with $X\\subseteq Z$, and if the metric on $Z$ is an extension of that on $X$, then following Gromov~\\cite[Section~1B]{Gromov93} we say that $Z$ is a \\emph{$r$-metric thickening of $X$} if for all $z\\in Z$, there is some $x\\in X$ with $d_Z(x,z)\\le r$.\n\n\\begin{prop}\n\tLet $X$ be a bounded metric space.\nWhen equipped with the $q$-Wasserstein metric for $1\\le q\\le p$, we have that $\\cechp{X}{r}$ and $\\vrp{X}{r}$ are each $r$-metric thickenings of $X$.\n\\end{prop}\n\n\\begin{proof}\nLet $\\alpha\\in\\cechp{X}{r}$.\nHence there exists some $x\\in X$ with $r > \\ensuremath{d_{\\mathrm{W},p}^X}(\\delta_x,\\alpha) \\ge \\ensuremath{d_{\\mathrm{W},q}^X}(\\delta_x,\\alpha)$, which shows that $\\cechp{X}{r}$ equipped with the $q$-Wasserstein metric is an $r$-metric thickening of $X$.\n\nThe Vietoris--Rips case follows immediately since $\\vrp{X}{r}\\subseteq \\cechp{X}{r}$ by Proposition~\\ref{prop:nesting}.\n\\end{proof}\n\n\n\n\\begin{lem}\nFor all $r>0$ and all $p,q\\in[1,\\infty]$ with $p\\geq q$, one has\n\\[\\vrpp{p}{X}{r}\\subseteq \\vrpp{q}{X}{r}\n\\quad\\mbox{and}\\quad\n\\cechpp{p}{X}{r} \\subseteq \\cechpp{q}{X}{r}.\\]\n\\end{lem}\n\n\\begin{proof}\nThis comes from applying the H\\\"{o}lder inequality on $\\mathrm{diam}_p$ and $\\mathrm{rad}_p$.\n\\end{proof}\n\n\n\n\\begin{example}\\label{ex:Delta-n}\nWe recall that $Z_{n+1}$ is the metric space consisting of $n+1$ points with all interpoint distances equal to 1.\nFor any natural number $n$, the Vietoris--Rips or \\v{C}ech simplicial complex filtrations of $Z_{n+1}$ do not produce any non-diagonal point in their persistence diagram except in homological dimension zero.\nHowever, for any $1\\left(\\frac{n}{n+1}\\right)^{\\frac{1}{p}}$, both $\\vrp{Z_{n+1}}{{r}}$ and $\\cechp{Z_{n+1}}{{r}}$ become the $n$-simplex $\\Delta_{n}$ which is contractible.\n\nFrom this we get the persistence diagrams\n\\[ \\dgm^\\ensuremath{\\mathrm{VR}}_{k,p}(Z_{n+1}) = \\dgm^\\ensuremath{\\mathrm{\\check{C}}}_{k,p}(Z_{n+1}) = \\begin{cases}\n\\left(0, \\left(\\tfrac{1}{2}\\right)^{\\tfrac{1}{p}}\\,\\right)^{\\otimes {n}}\\oplus (0, \\infty) & \\,\\text{if $k=0$}, \\\\\n\\left(\\left(\\tfrac{k}{k+1}\\right)^{\\tfrac{1}{p}}, \\left(\\tfrac{k+1}{k+2}\\right)^{\\frac{1}{p}}\\,\\right)^{\\otimes {{n}\\choose{k+1}}} & \\,\\text{if $1\\le k \\leq n-1$}, \\\\\n\\emptyset & \\,\\text{if $k \\ge n$}.\n\\end{cases}\n\\]\nNote that from the definition of $\\vrppf{\\infty}{Z_{n+1}}$, we have $\\dgm^\\ensuremath{\\mathrm{VR}}_{k,\\infty}=\\emptyset$ for $k\\geq 1$, and furthermore that $\\lim_{p\\uparrow \\infty}\\dgm^\\ensuremath{\\mathrm{VR}}_{k,p}(Z_{n+1}) = \\dgm^\\ensuremath{\\mathrm{VR}}_{k,\\infty}(Z_{n+1})$ for each $k\\geq 0$.\nAn analogous result holds for $\\cechpf{Z_{n+1}}$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{homotopy_types_Delta.pdf}\n\\caption{Homotopy types of both $\\vrp{Z_{n+1}}{\\boldsymbol{\\cdot}}$ and $\\cechp{Z_{n+1}}{\\boldsymbol{\\cdot}}$.}\n\\label{fig:homotopyTypesDelta}\n\\end{figure}\n\n\\end{example}\n\n\n\n\n\n\\subsection{A more general setting: controlled invariants}\nWe now generalize the relaxed $p$-Vietoris-Rips and $p$-\\v{C}ech filtrations.\n\n\\begin{defn}[Controlled invariants on $\\ensuremath{\\mathcal{P}}_X$]\\label{def:controlled-invariant}\nLet $\\mathfrak{i}$ be a functional that associates to each bounded metric space $X$ a function $\\mathfrak{i}^X\\colon \\ensuremath{\\mathcal{P}}_X\\to \\mathbb{R}$.\nWe say, for some $L>0$, that $\\mathfrak{i}$ is an \\emph{$L$-controlled} invariant if the following conditions are satisfied:\n\\begin{enumerate}\n\t\\item \\emph{Stability under pushforward}: For any map $f$ between finite metric spaces $X$ and $Y$ and any $\\alpha$ in $\\ensuremath{\\mathcal{P}}_X$,\n \\[\\mathfrak{i}^Y(f_\\sharp(\\alpha)) \\leq \\mathfrak{i}^X(\\alpha)+L\\cdot \\mathrm{dis}(f).\\]\n\t\\item \\emph{Stability with respect to $\\dWqq{\\infty}$}: For any bounded metric space $X$ and any $\\alpha, \\beta$ in $\\ensuremath{\\mathcal{P}}_X$,\n\t\\[|\\mathfrak{i}^X(\\alpha) - \\mathfrak{i}^X(\\beta)| \\leq 2L\\cdot \\dWqq{\\infty}(\\alpha, \\beta).\\]\n\\end{enumerate}\n\\end{defn}\n\nIn the next section, we will prove that both $\\mathrm{diam}_p$ and $\\mathrm{rad}_p$ induce controlled invariants.\n\nFor any controlled invariant $\\mathfrak{i}$, we will use the notation $\\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}$ to denote the sublevel set filtration induced by the pair $(\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X)$.\nThat is,\n\\[ \\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r}=(\\mathfrak{i}^X)^{-1}\\big((-\\infty,r)\\big)\\mbox{ for }r\\in \\ensuremath{\\mathbb{R}}\\]\nand\n\\[ \\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}=\\left\\{(\\mathfrak{i}^X)^{-1}\\big((-\\infty,r)\\big) \\subseteq (\\mathfrak{i}^X)^{-1}\\big((-\\infty,r')\\big)\\right\\}_{r\\leq r'}.\\]\nThis construction generalizes both the definition of $\\vrpf{X}$ and that of $\\cechpf{X}$.\n\nSimilarly, any controlled invariant $\\mathfrak{i}\\colon \\ensuremath{\\mathcal{P}}^\\fin_X\\to\\ensuremath{\\mathbb{R}}$ induces an analogous sublevel set filtration $\\filtf{\\ensuremath{\\mathcal{P}}^\\fin_X}{\\mathfrak{i}^X}$ of $\\ensuremath{\\mathcal{P}}^\\fin_X$.\nThis construction generalizes both the definition of $\\vrpffin{X}=\\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mathrm{diam}_p^X}$ and that of $\\cechpffin{X}=\\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mathrm{rad}_p^X}$.\n\n\nWe'll see later on in Corollary~\\ref{cor:cP_cPfin_0-interleaved} that the filtrations $\\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}$ and $\\filtf{\\ensuremath{\\mathcal{P}}^\\fin_X}{\\mathfrak{i}^X}$ yield persistence modules which are $0$-interleaved.\n\n\n\n\n\\section{Basic properties}\\label{sec:basic-properties}\n\nWe prove some basic properties.\nThese include the convexity of the Wasserstein distance, the fact that $\\mathrm{diam}_p$ and $\\mathrm{rad}_p$ are $1$-controlled, and the fact that for $X$ finite, the $p=\\infty$ metic thickenings $\\vrpp{\\infty}{X}{r}$ and $\\cechpp{\\infty}{X}{r}$ are homeomorphic to the simplicial complexes $\\vr{X}{r}$ and $\\cech{X}{r}$, respectively.\nWe begin with the property that nearby measurses have nearby integrals.\nThroughout this section, $X$ is a bounded metric space.\n\n\\begin{lem}\\label{lem:dwq_stability_wrt_Lipschitz}\nLet $\\varphi$ be an $L$-Lipschitz function on $X$.\nThen for any $\\alpha, \\beta$ in $\\ensuremath{\\mathcal{P}}_X$ and any $q\\in [1, \\infty]$, we have\n\\[\\left\\vert \\left(\\int_X|\\varphi(x)|^q\\, \\, \\alpha(dx)\\right)^{\\frac{1}{q}} -\\left(\\int_X|\\varphi(x)|^q\\, \\, \\beta(dx)\\right)^{\\frac{1}{q}} \\right\\vert\\leq L \\cdot \\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha, \\beta).\\]\n\\end{lem}\n\\begin{proof}\nLet $\\mu \\in \\ensuremath{\\mathrm{Cpl}}(\\alpha,\\beta)$ be any coupling.\nThen we have\n\\begin{align*}\n& \\left\\vert\\left( \\int_X|\\varphi(x)|^q\\, \\, \\alpha(dx)\\right)^{\\frac{1}{q}} -\\left(\\int_X|\\varphi(x)|^q\\, \\, \\beta(dx)\\right)^{\\frac{1}{q}} \\right\\vert \\\\\n=\\ & \\bigg\\vert \\left( \\iint_{X\\times X}|\\varphi(x)|^q\\, \\, \\mu(dx\\times dx') \\right)^{\\frac{1}{q}} -\\left( \\iint_{X\\times X}|\\varphi(x')|^q\\, \\, \\mu(dx\\times dx') \\right)^{\\frac{1}{q}}\\bigg\\vert \\\\\n\\leq\\ & \\left(\\iint_{X\\times X}|\\varphi(x) - \\varphi(x')|^q\\, \\, \\mu(dx\\times dx')\\right)^{\\frac{1}{q}} \\\\\n\\leq\\ & L\\left(\\iint_{X\\times X}d_X^q(x, x')\\, \\mu(dx\\times dx')\\right)^{\\frac{1}{q}}.\n\\end{align*}\nWe obtain the claim by infimizing the right hand side with respect to the coupling $\\mu$.\n\\end{proof}\n\n\nWe will often use the following convexity result of the Wasserstein distance; for a more general result, see~\\cite[Theorem 4.8]{villani2008optimal}.\n\n\\begin{lem}\\label{lem:bound-distance-convex-comb}\nIf $c_1, \\dots, c_n$ are non-negative real numbers satisfying $\\sum_{k=1}^n c_k = 1$ and if $\\alpha_1, \\dots, \\alpha_n,\\alpha'_1, \\dots, \\alpha'_n \\in \\ensuremath{\\mathcal{P}}_X$, then for all $q \\in [1, \\infty)$, we have\n\\[\n\\ensuremath{d_{\\mathrm{W},q}^X} \\Big( \\sum_{k=1}^n c_k \\alpha_k ,\\, \\sum_{k=1}^n c_k \\alpha'_k \\Big) \\leq \\left( \\sum_{k=1}^n c_k \\left(\\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha_k, \\alpha'_k) \\right)^q \\right)^\\frac{1}{q}\n\\]\nand\n\\[\n\\dWqq{\\infty} \\Big( \\sum_{k=1}^n\\,c_k \\alpha_k ,\\, \\sum_{k=1}^n c_k\\,\\alpha'_k \\Big) \\leq \\max_k\\,\\dWqq{\\infty}(\\alpha_k, \\alpha'_k).\\]\n\\end{lem}\n\n\n\\begin{proof}\nLet $\\varepsilon>0$ and $q \\in [1,\\infty)$.\nFor each $k$, suppose $\\mu_k$ is a coupling between $\\alpha_k$ and $\\alpha'_k$ such that\n\\[ \\iint_{X\\times X}d^q_X(x,x')\\,\\mu_k(dx\\times dx') < (\\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha_k, \\alpha'_k) + \\varepsilon)^q.\\]\nThen it can be checked that $\\sum_{k=1}^n c_k \\mu_k$ is a coupling between $\\sum_{k=1}^n c_k \\alpha_k$ and $\\sum_{k=1}^n c_k \\alpha'_k$.\nWe have\n\\begin{align*}\n\\ensuremath{d_{\\mathrm{W},q}^X} \\left( \\sum_{k=1}^n\\,c_k \\alpha_k ,\\, \\sum_{k=1}^n c_k\\,\\alpha'_k \\right) & \\leq \\left( \\iint_{X\\times X}d_X^q(x,x')\\,\\sum_{k=1}^n c_k \\mu_k(dx\\times dx') \\right)^\\frac{1}{q} \\\\\n& < \\left( \\sum_{k=1}^n c_k \\left(\\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha_k, \\alpha'_k) + \\varepsilon \\right)^q \\right)^\\frac{1}{q} \\\\\n& \\leq \\left( \\sum_{k=1}^n c_k \\left(\\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha_k, \\alpha'_k) \\right)^q \\right)^\\frac{1}{q} + \\varepsilon.\n\\end{align*}\nSince this holds for all $\\varepsilon>0$, the claimed inequality holds.\nThe case for $q = \\infty$ can be checked separately using the same matching.\n\\end{proof}\n\nWe define an invariant generalizing $\\mathrm{diam}_p$, and then prove that it is $1$-controlled.\n\n\\begin{defn}[$\\mi_{q, p}^X$ invariant]\\label{defn:iqp}\nLet $X$ be a bounded metric space.\nFor any $p, q \\in [1, \\infty]$, we define the \\emph{$\\mi_{q, p}$ invariant}, which associates to each bounded metric space $X$ a function $\\mi_{q, p}^X\\colon \\ensuremath{\\mathcal{P}}_X\\to \\ensuremath{\\mathbb{R}}$, \n\\[\\mi_{q, p}^X(\\alpha)=\\left(\\int_X\\big(\\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha,\\delta_x)\\big)^p\\alpha(dx)\\right)^{1\/p}.\\]\nNote that $\\mi_{q, p}^X$ recovers $\\mathrm{diam}_p^X$ when $p=q$.\n\\end{defn}\n\n\\begin{remark}\nIn the contruction of $\\mi_{q, p}^X$, one can get other interesting filtration functions by replacing the Wasserstein distance with other distances between measures.\nThese include, for example, the L\\'{e}vy--Prokhorov metric, the Fortet--Mourier metric, and the Kantorovich--Rubinstein metric.\nFor definitions of these metrics, see~\\cite[Section 3]{bogachev2018weak}.\n\\end{remark}\n\n\\begin{lem}\\label{lem:iqp_distortion_via_pushforward}\nLet $X$ and $Y$ be bounded metric spaces, and let $f \\colon X\\to Y$ be a map.\nThen for any element $\\alpha\\in \\ensuremath{\\mathcal{P}}_X$ and $p\\in[1,\\infty]$, we have\n\\[\\mi_{q, p}^Y(f_\\sharp(\\alpha)) \\leq \\mi_{q, p}^X(\\alpha)+\\mathrm{dis}(f).\\]\n\\end{lem}\n\n\\begin{proof}\nWe compute\n\\begin{align*}\n\\mi_{q, p}^Y(f_\\sharp(\\alpha)) & =\\left(\\int_Y\\big(d_{\\mathrm{W}, q}^Y(f_\\sharp(\\alpha),\\delta_y)\\big)^p\\,f_\\sharp(\\alpha)(dy)\\right)^{1\/p} \\\\\n& =\\left(\\int_X\\big(d_{\\mathrm{W}, q}^Y(f_\\sharp(\\alpha),\\delta_{f(x)})\\big)^p\\,\\alpha(dx)\\right)^{1\/p} \\\\\n& =\\left(\\int_X\\bigg(\\int_Yd_Y^q(y,f(x))\\,f_\\sharp(\\alpha)(dy)\\bigg)^\\frac{p}{q}\\,\\alpha(dx)\\right)^{1\/p} \\\\\n& =\\left(\\int_X\\bigg(\\int_Xd_Y^q(f(x'),f(x))\\,\\alpha(dx')\\bigg)^\\frac{p}{q}\\,\\alpha(dx)\\right)^{1\/p} \\\\\n& \\leq\\left(\\int_X\\bigg(\\int_X\\big(d_X(x',x)+ \\mathrm{dis}(f) \\big)^q\\,\\alpha(dx')\\bigg)^\\frac{p}{q}\\,\\alpha(dx)\\right)^{1\/p} \\\\\n& \\leq \\mi_{q, p}^X(\\alpha) + \\mathrm{dis}(f).\n\\end{align*}\n\\end{proof}\n\n\\begin{lem}\\label{lem:iqp_stability_wrt_dW}\nLet $X$ be a bounded metric space.\nThen for any $\\alpha, \\beta$ in $\\ensuremath{\\mathcal{P}}_X$, we have\n\\[|\\mathfrak{i}^X_{q,p}(\\alpha) - \\mi_{q, p}^X(\\beta)|\\leq \\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha, \\beta) + \\ensuremath{d_{\\mathrm{W},p}^X}(\\alpha, \\beta).\\]\nIn particular, $|\\mathfrak{i}^X_{q,p}(\\alpha) - \\mi_{q, p}^X(\\beta)|\\leq 2\\,\\dWqq{\\infty}(\\alpha, \\beta)$.\n\\end{lem}\n\n\\begin{proof}\nWe first note that\n\\begin{align*}\n\\mi_{q, p}^X(\\alpha) & =\\left(\\int_X\\big(\\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha,\\delta_x)\\big)^p\\alpha(dx)\\right)^{1\/p} \\\\\n& \\leq\\left(\\int_X\\big(\\ensuremath{d_{\\mathrm{W},q}^X}(\\beta ,\\delta_x) + \\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha, \\beta)\\big)^p\\alpha(dx)\\right)^{1\/p} \\\\\n& \\leq \\left(\\int_X\\big(\\ensuremath{d_{\\mathrm{W},q}^X}(\\beta ,\\delta_x) \\big)^p \\alpha(dx)\\right)^{1\/p} + \\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha, \\beta).\n\\end{align*}\nWe also have\n\\begin{align*}\n&\\left\\vert\\left(\\int_X\\big(\\ensuremath{d_{\\mathrm{W},q}^X}(\\beta ,\\delta_x) \\big)^p \\alpha(dx)\\right)^{1\/p} - \\mi_{q, p}(\\beta)\\right \\vert \\\\\n=& \\bigg\\vert\\left(\\int_X\\big(\\ensuremath{d_{\\mathrm{W},q}^X}(\\beta ,\\delta_x) \\big)^p \\alpha(dx)\\right)^{1\/p} - \\left(\\int_X\\big(\\ensuremath{d_{\\mathrm{W},q}^X}(\\beta ,\\delta_x) \\big)^p \\beta(dx)\\right)^{1\/p}\\bigg \\vert \\\\\n\\leq & \\ensuremath{d_{\\mathrm{W},p}^X}(\\alpha, \\beta),\n\\end{align*}\nwhere the last line follows from Lemma~\\ref{lem:dwq_stability_wrt_Lipschitz} since $\\ensuremath{d_{\\mathrm{W},q}^X}(\\beta, \\delta_x)$ is a $1$-Lipschitz function in $x$.\nTherefore,\n\\[\\mi_{q, p}(\\alpha) \\leq \\mi_{q, p}(\\beta) + \\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha, \\beta) + \\ensuremath{d_{\\mathrm{W},p}^X}(\\alpha, \\beta).\\]\nWe then get the result by swapping the roles of $\\alpha$ and $\\beta$.\n\\end{proof}\n\n\\begin{remark}\nThis establishes the continuity of $\\mi_{q, p}$ for $q, p \\in [1, \\infty)$.\nIf $p$ or $q$ equals infinity, then $\\mi_{q, p}$ is not necessarily continuous in the weak topology since $\\dWqq{\\infty}$ does not necessarily induce the same topology on $\\ensuremath{\\mathcal{P}}_X$ as $\\ensuremath{d_{\\mathrm{W},p}^X}$ does for $p<\\infty$.\nIndeed, if $X=[0,1]$ is the unit interval, then $\\frac{n-1}{n}\\delta_0+\\frac{1}{n}\\delta_1\\to\\delta_0$ in $\\ensuremath{\\mathcal{P}}_X$ as $n\\to\\infty$, even though $\\mathrm{diam}_\\infty(\\frac{n-1}{n}\\delta_0+\\frac{1}{n}\\delta_1)$ is equal to $1$ for all $n\\ge 1$, and hence does not converge to $0=\\mathrm{diam}(\\delta_0)$.\n\\end{remark}\n\nThe above two lemmas imply that $\\mi_{q, p}$ (and hence $\\mathrm{diam}_p=\\mathfrak{i}_{p.p}$) is a $1$-controlled invariant.\nWe next consider the analogous properties for $\\mathrm{rad}_p$.\n\n\\begin{lem}[$\\mathrm{rad}_p$ under a map with bounded distortion] \\label{lem:rad_p_dist}\nLet $X$ and $Y$ be bounded metric spaces, and let $f\\colon X\\to Y$ be a map.\nThen for any $\\alpha \\in \\ensuremath{\\mathcal{P}}_X$ and $p\\in[1,\\infty]$, we have\n\\[\\mathrm{rad}_p(f_\\sharp(\\alpha)) \\leq \\mathrm{rad}_p(\\alpha)+ \\mathrm{dis}(f).\\]\n\\end{lem}\n\n\\begin{proof}\nWe only give the proof for the case $p<\\infty$; the case $p=\\infty$ is similar.\n\\begin{align*}\n\\mathrm{rad}_p(f_\\sharp(\\alpha)) & = \\inf_{y\\in Y}\\left(\\int_{Y}d_Y^p(y,y')\\,f_\\sharp(\\alpha)(dy')\\right)^{\\frac{1}{p}} \\\\\n& \\leq \\inf_{y\\in f(X)}\\left(\\int_{Y}d_Y^p(y,y')\\,f_\\sharp(\\alpha)(dy')\\right)^{\\frac{1}{p}} \\\\\n& =\\inf_{x\\in X}\\left(\\int_{Y}d_Y^p(f(x),y')\\,f_\\sharp(\\alpha)(dy')\\right)^{\\frac{1}{p}} \\\\\n& =\\inf_{x\\in X}\\left(\\int_{X}d_Y^p(f(x),f(x'))\\,\\alpha(dx')\\right)^{\\frac{1}{p}} \\\\\n& \\leq \\inf_{x\\in X}\\left(\\int_{X}\\left(d_X(x, x') + \\mathrm{dis}(f)\\right)^{p}\\,\\alpha(dx')\\right)^{\\frac{1}{p}} \\\\\n& \\leq \\inf_{x\\in X}\\left(\\int_{X}d_X^p(x,x')\\,\\alpha(dx')\\right)^{\\frac{1}{p}} + \\mathrm{dis}(f) \\\\\n& =\\mathrm{rad}_p(\\alpha) + \\mathrm{dis}(f).\n\\end{align*}\n\\end{proof}\n\n\\begin{lem}[Stability of $\\mathrm{rad}_p$]\\label{lem:stability_of_rad_p}\nLet $X$ be a bounded metric space.\nFor any two probability measures $\\alpha,\\beta \\in \\ensuremath{\\mathcal{P}}_X$ and for every $p\\in[1,\\infty]$, we have\n\\[ |\\mathrm{rad}_p(\\alpha)-\\mathrm{rad}_p(\\beta)|\\leq \\ensuremath{d_{\\mathrm{W},p}^X}(\\alpha,\\beta).\\]\n\\end{lem}\n\n\\begin{proof}\nWe compute\n\\begin{align*}\n\\mathrm{rad}_p(\\alpha) & = \\inf_{x\\in X} \\ensuremath{d_{\\mathrm{W},p}^X}(\\alpha, \\delta_x) \\\\\n& \\leq \\inf_{x\\in X} \\left(\\ensuremath{d_{\\mathrm{W},p}^X}(\\beta, \\delta_x) + \\ensuremath{d_{\\mathrm{W},p}^X}(\\beta, \\alpha)\\right) \\\\\n&= \\inf_{x\\in X} \\ensuremath{d_{\\mathrm{W},p}^X}(\\beta, \\delta_x) + \\ensuremath{d_{\\mathrm{W},p}^X}(\\beta, \\alpha) \\\\\n& = \\mathrm{rad}_p(\\beta) + \\ensuremath{d_{\\mathrm{W},p}^X}(\\beta, \\alpha).\n\\end{align*}\n\\end{proof}\n\n\\begin{remark}\nLemma~\\ref{lem:stability_of_rad_p} establishes the continuity of $\\mathrm{rad}_p$ for $p \\in [1,\\infty)$, as these functions are 1-Lipschitz.\nSimilarly to $\\mathrm{diam}_\\infty$, we note that $\\mathrm{rad}_\\infty$ is not necessarily continuous because the metric topology given by $\\dWqq{\\infty}$ is not necessarily equal to the weak topology.\n\\end{remark}\n\nThe above two lemmas imply that $\\mathrm{rad}_p$ is a $1$-controlled invariant.\n\nWe end this section of basic properties by showing that $\\ensuremath{\\mathcal{P}}_X$ is homeomorphic to a simplex when $X$ is finite.\nHence for $X$ finite the $p=\\infty$ metic thickenings $\\vrpp{\\infty}{X}{r}$ and $\\cechpp{\\infty}{X}{r}$ are homeomorphic to the simplicial complexes $\\vr{X}{r}$ and $\\cech{X}{r}$, respectively (see also \\cite[Corollary~6.4]{AAF}).\n\n\\begin{lem}\n\\label{lem:fin-prob-simplex}\nIf $X$ is a finite metric space with $n$ points, then $\\ensuremath{\\mathcal{P}}_X$ is homeomorphic to the standard $(n-1)$-simplex.\n\\end{lem}\n\n\\begin{proof}\nLet $X = \\{x_1, \\dots, x_n\\}$.\nThe space $\\ensuremath{\\mathcal{P}}_X$ of probability measures is in bijection with the standard $n-1$ simplex $\\Delta_{n-1} = \\{ (y_1, \\dots, y_n) \\in \\ensuremath{\\mathbb{R}}^n \\mid \\sum_{i=1}^n y_i = 1,\\, y_i \\geq 0 \\text{ for all $i$} \\}$ via the function $f\\colon \\ensuremath{\\mathcal{P}}_X \\to \\Delta_{n-1}$ defined by $f \\left( \\sum_{i=1}^n a_i \\delta_{x_i} \\right) = (a_1, \\dots, a_n)$.\nSuppose we have a sequence $\\{\\alpha_k\\}$ in $\\ensuremath{\\mathcal{P}}_X$ given by $\\alpha_k = \\sum_{i=1}^n a_{k,i} \\delta_{x_i}$.\nBy the definition of weak convergence, $\\{ \\alpha_k \\}$ converges to $\\alpha = \\sum_{i=1}^n a_i \\delta_{x_i}$ in $\\ensuremath{\\mathcal{P}}_X$ if and only if $\\int_X \\phi(x) \\alpha_k(dx)$ converges to $\\int_X \\phi(x) \\alpha(dx)$ for all bounded and continuous functions $\\phi\\colon X \\to \\ensuremath{\\mathbb{R}}$.\nThese integrals are equal to $\\sum_{i=1}^n a_{k,i} \\phi (x_i)$ and $\\sum_{i=1}^n a_i \\phi (x_i)$ respectively, so $\\{ \\alpha_k \\}$ converges to $\\alpha$ if and only if $\\lim_{k \\to \\infty} a_{k,i} = a_i$ for each $i$.\nTherefore, $\\{ \\alpha_k \\}$ converges to $\\alpha$ in $\\ensuremath{\\mathcal{P}}_X$ if and only if $\\{ f(\\alpha_k) \\}$ converges to $f(\\alpha)$ in $\\Delta_{n-1}$.\n\\end{proof}\n\nIn general, the $p=\\infty$ metric thickenings $\\vrpp{\\infty}{X}{r}$ and $\\cechpp{\\infty}{X}{r}$ are in bijection as sets with the geometric realizations of the usual Vietoris--Rips and \\v{C}ech simplicial complexes on $X$.\nTherefore, Lemma~\\ref{lem:fin-prob-simplex} implies the following.\n\n\\begin{lem}\\label{lem:finite_infty-thickenings_homeomorphic_to_complexes}\nFor a finite metric space $X$ and any $r>0$, we have homeomorphisms $\\vrpp{\\infty}{X}{r} \\cong \\vr{X}{r}$ and $\\cechpp{\\infty}{X}{r} \\cong \\cech{X}{r}$.\n\\end{lem}\n\n\n\n\\section{Stability}\\label{sec:stability}\n\nIn this section we establish the stability of all of the filtrations we have introduced in this paper.\nIn Section~\\ref{ssec:statement-of-stability} we state our main results, Theorem~\\ref{thm:general_stability} and its immediate consequence, Theorem~\\ref{thm:main}.\nWe give two applications of stability.\nFirst, Section~\\ref{sec:tameness} applies the stability theorem in order to show that various persistence modules of interest are Q-tame, and therefore have persistence diagrams.\nNext, in Section~\\ref{sec:comparability} we apply stability to show the close relationship between $\\infty$-metic thickenings and simplicial complexes, generalizing and answering in the affirmative~\\cite[Conjecture~6.12]{AAF} which states that these filtrations have identical persistence diagrams.\nWe give the proof of the stability theorem, Theorem~\\ref{thm:general_stability}, in Section~\\ref{ssec:proof-stability}.\n\n\n\n\\subsection{Statement of the stability theorem}\n\\label{ssec:statement-of-stability}\n\nLet $X$ and $Y$ be totally bounded metric spaces, let $p\\in [1,\\infty]$, and let $k\\ge 0$.\nWe will see, for example, that we have the following stability bound:\n\\begin{equation}\n\\label{eq:main}\n\\ensuremath{d_{\\mathrm{I}}}\\big(H_k\\circ\\vrpf{X},H_k\\circ\\vrpf{Y}\\big) \\leq 2\\,d_\\mathrm{GH}(X,Y).\n\\end{equation}\nThis implies that if two metric spaces are close in the Gromov--Hausdorff distance, then their resulting filtrations and persistence modules are also close.\n\n\n\n\\begin{example}\n\\label{ex:tight}\nIn this example we consider the metric space $Z_{n+1}$ from Example~\\ref{ex:Delta-n}, which has $n+1$ points all at interpoint distance 1 apart.\nLet $\\zeta_{p,k}$ be the interleaving distance between persistent homology modules\n\\[\\zeta_{p,k}:=\\ensuremath{d_{\\mathrm{I}}}\\big(H_k\\circ\\vrpf{Z_{n+1}},H_k\\circ\\vrpf{Z_{m+1}}\\big) \\quad \\mbox{for }n\\neq m.\\]\nNotice that from Example~\\ref{ex:Delta-n}, when $n\\neq m$ and $p$ is finite, we have\n\\[\\zeta_{p,0} = \\frac{1}{2} \\left(\\frac{1}{2}\\right)^{\\frac{1}{p}} \n\\quad \\mbox{and} \\quad \n\\zeta_{p,1} = \\frac{1}{2}\\left(\\left(\\frac{2}{3}\\right)^{\\frac{1}{p}}- \\left(\\frac{1}{2}\\right)^{\\frac{1}{p}}\\right).\\]\nHowever, when $p=\\infty$, $\\zeta_{\\infty,0} = \\frac{1}{2}$ and $\\zeta_{\\infty,k} =0$ for all $k\\geq 1$.\n\t\nFrom these calculations we can make the following observations:\n\t\n\\begin{enumerate}\n \\item[(i)] For $p$ infinite, the only value of $k$ for which $\\zeta_{\\infty,k}\\neq 0$ is $k=0$.\n\t\\item[(ii)] For $p$ finite, we have $\\zeta_{p,1}> 0$.\n\t\\item[(iii)] $\\sup_k\\zeta_{\\infty,k} = \\frac{1}{2} > \\frac{1}{2} \\left(\\frac{1}{2}\\right)^{\\frac{1}{p}} = \\sup_{k} \\zeta_{p,k} $ for $p$ finite.\n\\end{enumerate}\n\t\nFrom items (i) and (ii) above we can see that whereas persistence diagrams for $k\\geq 1$ of $p=\\infty$ thickenings do not contain discriminative information for the $Z_{n}$ spaces, in contrast, the analogous quantities for $p$ finite do absorb useful information.\n\t\nSince it is known (cf.~\\cite[Example 4.1]{dgh-props}) that $d_\\mathrm{GH}(Z_{n+1}, Z_{m+1})=\\frac{1}{2}$ whenever $n\\neq m$, item (iii) above suggests that the lower bound for Gromov--Hausdorff given by Equation~\\eqref{eq:main} may not be tight for $p$ finite.\nThis phenomenon can actually be explained by the more general theorem below, which identifies a certain pseudo-metric between filtrations which mediates between the two terms appearing in Equation~\\eqref{eq:main}.\n\n\\end{example}\n\n\n\n\\begin{thm}\\label{thm:general_stability}\nLet $\\mathfrak{i}$ be an $L$-controlled invariant.\nThen for any two totally bounded metric spaces $X$ and $Y$ and any integer $k\\geq 0$, we have\n\\begin{align*}\n\\ensuremath{d_{\\mathrm{I}}}\\big(H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X},H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}_Y}{\\mathfrak{i}^Y}\\big)&\\leq d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X), (\\ensuremath{\\mathcal{P}}_Y, \\mathfrak{i}^Y)\\big)\\leq 2L\\cdotd_\\mathrm{GH}(X, Y) \\\\\n\\ensuremath{d_{\\mathrm{I}}}\\big(H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}^\\fin_X}{\\mathfrak{i}^X},H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}^\\fin_Y}{\\mathfrak{i}^Y}\\big)&\\leq d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}^\\fin_X, \\mathfrak{i}^X), (\\ensuremath{\\mathcal{P}}^\\fin_Y, \\mathfrak{i}^Y)\\big)\\leq 2L\\cdotd_\\mathrm{GH}(X, Y).\n\\end{align*}\n\\end{thm}\nNote that there are instances when the quantity in the middle vanishes yet the spaces $X$ and $Y$ are non-isometric; see Appendix~\\ref{app:crushings} for results about this in terms of the notion of \\emph{crushing} considered by Hausmann~\\cite{Hausmann1995} and~\\cite{memoli2021quantitative,AAF}.\n\n\\begin{cor}\\label{cor:iqp_stability}\nFor any two totally bounded metric spaces $X$ and $Y$, for any $q, p \\in [1, \\infty]$, and for any integer $k \\geq 0$, we have\n\\begin{align*}\n\\ensuremath{d_{\\mathrm{I}}}\\big(H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mi_{q, p}^X},H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}_Y}{\\mi_{q, p}^Y}\\big) &\\leq d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}_X,\\,\\mi_{q, p}^X), (\\ensuremath{\\mathcal{P}}_Y,\\,\\mi_{q, p}^Y)\\big) \\leq 2\\,d_\\mathrm{GH}(X, Y) \\\\\n\\ensuremath{d_{\\mathrm{I}}}\\big(H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}^\\fin_X}{\\mi_{q, p}^X},H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}^\\fin_Y}{\\mi_{q, p}^Y}\\big) &\\leq d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}^\\fin_X,\\,\\mi_{q, p}^X), (\\ensuremath{\\mathcal{P}}^\\fin_Y,\\,\\mi_{q, p}^Y)\\big) \\leq 2\\,d_\\mathrm{GH}(X, Y).\n\\end{align*}\n\\end{cor}\n\n\\begin{proof}\nFrom Lemma~\\ref{lem:iqp_distortion_via_pushforward} and Lemma~\\ref{lem:iqp_stability_wrt_dW} we know $\\mi_{q, p}$ is a $1$-controlled invariant, and then we apply Theorem~\\ref{thm:general_stability}.\n\\end{proof}\n\n\\begin{thm}\\label{thm:main}\nLet $X$ and $Y$ be totally bounded metric spaces, let $p\\in[1,\\infty]$, and let $k \\geq 0$ be an integer.\nThen the $k$-th persistent homology of $\\vrpf{X}$ and $\\vrpf{Y}$ are $\\varepsilon$-interleaved for any $\\varepsilon\\geq d_\\mathrm{GH}(X, Y)$, and similarly for $\\cechpf{X}$ and $\\cechpf{Y}$:\n\\begin{align*}\n\\ensuremath{d_{\\mathrm{I}}}\\big(H_k\\circ\\vrpf{X},H_k\\circ\\vrpf{Y}\\big) &\\leq d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}_X, \\mathrm{diam}_p^X), (\\ensuremath{\\mathcal{P}}_Y, \\mathrm{diam}_p^Y)\\big) \\leq 2\\,d_\\mathrm{GH}(X,Y) \\\\\n\\ensuremath{d_{\\mathrm{I}}}\\big(H_k\\circ\\vrpffin{X},H_k\\circ\\vrpffin{Y}\\big) &\\leq d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}^\\fin_X, \\mathrm{diam}_p^X), (\\ensuremath{\\mathcal{P}}^\\fin_Y, \\mathrm{diam}_p^Y)\\big) \\leq 2\\,d_\\mathrm{GH}(X,Y) \\\\\n\\ensuremath{d_{\\mathrm{I}}}\\big(H_k\\circ\\cechpf{X},H_k\\circ\\cechpf{Y}\\big) &\\leq d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}_X, \\mathrm{rad}_p^X), (\\ensuremath{\\mathcal{P}}_Y, \\mathrm{rad}_p^Y)\\big) \\leq 2\\,d_\\mathrm{GH}(X,Y) \\\\\n\\ensuremath{d_{\\mathrm{I}}}\\big(H_k\\circ\\cechpffin{X},H_k\\circ\\cechpffin{Y}\\big) &\\leq d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}^\\fin_X, \\mathrm{rad}_p^X), (\\ensuremath{\\mathcal{P}}^\\fin_Y, \\mathrm{rad}_p^Y)\\big) \\leq 2\\,d_\\mathrm{GH}(X,Y).\n\\end{align*}\n\\end{thm}\n\n\\begin{proof}\nThe $p$-Vietoris--Rips case follows from Corollary~\\ref{cor:iqp_stability} by letting $q=p$, in which case $\\mathfrak{i}_{p,p}^X=\\mathrm{diam}_p^X$ (see Definition~\\ref{defn:iqp}).\nThe $p$-\\v{C}ech case follows from Theorem~\\ref{thm:general_stability} since $\\mathrm{rad}_p$ is a $1$-controlled invariant by Lemmas~\\ref{lem:rad_p_dist} and~\\ref{lem:stability_of_rad_p}.\n\\end{proof}\n\nIn the corollary below, we use Theorem~\\ref{thm:general_stability} to show that a controlled invariant on all Radon probability measures produces a filtration at interleaving distance zero from that same invariant restricted to only finitely supported measures.\n\n\\begin{cor}\n\\label{cor:cP_cPfin_0-interleaved}\nLet $X$ be a totally bounded metric space, let $k \\geq 0$ be an integer, and let $\\mathfrak{i}$ be an $L$-controlled invariant.\nThen\n\\[\\ensuremath{d_{\\mathrm{I}}}(H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}, H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}^\\fin_X}{\\mathfrak{i}^X}) = d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X), (\\ensuremath{\\mathcal{P}}^\\fin_X, \\mathfrak{i}^X)\\big) = 0.\\]\n\\end{cor}\n\n\\begin{proof}\nFor any $\\varepsilon>0$, let $U$ be a $\\varepsilon$-net in $X$.\nThen the stability theorem, Theorem~\\ref{thm:general_stability}, shows\n\\begin{align*}\nd_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X), (\\ensuremath{\\mathcal{P}}_U), \\mathfrak{i}^U)\\big) &\\leq 2L\\cdot d_\\mathrm{GH}(X, U) \\leq 2L \\varepsilon \\\\\nd_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}^\\fin_X, \\mathfrak{i}^X), (\\ensuremath{\\mathcal{P}}^\\fin_U, \\mathfrak{i}^U)\\big) &\\leq 2L\\cdot d_\\mathrm{GH}(X, U) \\leq 2L \\varepsilon\n\\end{align*}\nNote $(\\ensuremath{\\mathcal{P}}_U, \\mathfrak{i}^U) = (\\ensuremath{\\mathcal{P}}^\\fin_U, \\mathfrak{i}^U)$ since $U$ is finite.\nSince $d_\\mathrm{HT}$ satisfies the triangle inequality (Proposition~\\ref{prop:dht_pseudometric}), we have\n\\[d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X), (\\ensuremath{\\mathcal{P}}^\\fin_X, \\mathfrak{i}^X)\\big)\\leq 4L\\varepsilon.\\]\nBy letting $\\varepsilon$ go to zero, we see $d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X), (\\ensuremath{\\mathcal{P}}^\\fin_X, \\mathfrak{i}^X)\\big) = 0$.\nIt then follows from Lemma~\\ref{lem:dht_bound_interleaving} that $\\ensuremath{d_{\\mathrm{I}}}(H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}, H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}^\\fin_X}{\\mathfrak{i}^X})=0$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\subsection{Consequence of stability: Tameness}\n\\label{sec:tameness}\n\nWe next determine conditions under which a persistence module $H_k\\circ\\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}$ associated with a controlled invariant $\\mathfrak{i}$ will be Q-tame.\nIn particular, we apply the results to the Vietoris--Rips and \\v{C}ech metric thickenings.\n\n\\begin{cor}\\label{cor:controlled_invariant_tameness}\nLet $\\mathfrak{i}$ be a controlled invariant and let $k \\geq 0$ be an integer.\nSuppose the persistence module $H_k\\circ\\filtf{\\ensuremath{\\mathcal{P}}_V}{\\mathfrak{i}^V}$ (resp.\\ $H_k\\circ\\filtf{\\ensuremath{\\mathcal{P}}^\\fin_V}{\\mathfrak{i}^V}$) is $Q$-tame for any finite metric space $V$.\nThen for any totally bounded metric space $X$, the persistence module $H_k\\circ\\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}$ (resp.\\ $H_k\\circ\\filtf{\\ensuremath{\\mathcal{P}}^\\fin_X}{\\mathfrak{i}^X}$) is $Q$-tame.\n\\end{cor}\n\n\n\\begin{proof}\nSince $X$ is totally bounded, Theorem~\\ref{thm:general_stability} implies that the persistence module $H_k\\circ\\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}$ can be approximated arbitrarily well in the interleaving distance by the Q-tame persistence modules on finite $\\varepsilon$-nets $V_\\varepsilon$ as $\\varepsilon$ goes to zero.\nThen the result follows from Lemma~\\ref{lem:qtame_approximation}.\n\\end{proof}\n\nNow, we give a sufficient condition for the Q-tameness of the persistence module $H_k\\circ\\filtf{\\ensuremath{\\mathcal{P}}_V}{\\mathfrak{i}^V}$ over a finite metric space $V$.\n\n\\begin{cor}\\label{cor:qtame_cts_finite}\nSuppose $\\mathfrak{i}$ is a controlled invariant such that for any finite metric space $V$, $\\mathfrak{i}^V$ is a continuous function on $\\ensuremath{\\mathcal{P}}_V$.\nThen for any totally bounded metric space $X$, the persistence modules $H_k\\circ\\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}$ and $H_k\\circ\\filtf{\\ensuremath{\\mathcal{P}}^\\fin_X}{\\mathfrak{i}^X}$ are Q-tame for any integer $k\\geq 0$.\n\\end{cor}\n\n\\begin{proof}\nAccording to Corollary~\\ref{cor:controlled_invariant_tameness}, it suffices to check Q-tameness over finite metric spaces.\nIf $V$ is a finite metric space, then we may identify $\\ensuremath{\\mathcal{P}}_V$ with a simplex by Lemma~\\ref{lem:fin-prob-simplex}.\nThen $\\filtf{\\ensuremath{\\mathcal{P}}_V}{\\mathfrak{i}^V}$ is a sublevel set filtration on a simplex, and since $\\mathfrak{i}^V$ is continuous,\nTheorem 2.22 of~\\cite{chazal2016structure} shows $H_k\\circ\\filtf{\\ensuremath{\\mathcal{P}}_V}{\\mathfrak{i}^V}$ is Q-tame.\n\\end{proof}\n\nHere we summarize the $Q$-tameness results related to relaxed Vietoris--Rips and \\v{C}ech metric thickenings.\n\n\\begin{cor}\\label{cor:totally_bounded_implies_q-tame}\nLet $X$ be a totally bounded metric space, let $p\\in [1, \\infty]$, and let $k \\geq 0$ be an integer.\nThe persistence modules $H_k\\circ \\vrpf{X}$, $H_k\\circ \\vrpffin{X}$, $H_k\\circ \\cechpf{X}$, and $H_k\\circ \\cechpffin{X}$ are Q-tame.\n\\end{cor}\n\n\\begin{proof}\nFor $H_k\\circ \\vrpf{X}$, when $p$ is finite, Lemma~\\ref{lem:iqp_stability_wrt_dW} implies $\\mathrm{diam}_p$ is a continuous function over any $\\ensuremath{\\mathcal{P}}_X$ where $X$ is a bounded metric space.\nThen we get the result by applying Corollary~\\ref{cor:qtame_cts_finite}.\nWhen $p= \\infty$, Lemma~\\ref{lem:finite_infty-thickenings_homeomorphic_to_complexes} shows the persistent homology associated with $\\mathrm{diam}_\\infty$ is Q-tame over finite metric spaces.\nWe then get the result by applying Corollary~\\ref{cor:controlled_invariant_tameness}.\n\nWe obtain the case of $H_k\\circ \\vrpffin{X}$ from $H_k\\circ \\vrpf{X}$ by using the interleaving distance result in Corollary~\\ref{cor:cP_cPfin_0-interleaved} along with the Q-tame approximation result in Lemma~\\ref{lem:qtame_approximation}.\n\nFor $H_k\\circ \\cechpf{X}$, when $p$ is finite, Lemma~\\ref{lem:stability_of_rad_p} implies $\\mathrm{rad}_p$ is a continuous function over any $\\ensuremath{\\mathcal{P}}_X$ where $X$ is a bounded metric space.\nThe rest is similar to the Vietoris--Rips case.\n\\end{proof}\n\n\\begin{remark}\nThe proof that $H_k \\circ \\cechpf{X}$ is Q-tame can be made more direct at one step.\nFor $U$ finite, to see that $H_k \\circ \\cechpf{U}$ is Q-tame, one could appeal to Theorem~\\ref{thm_pCech_homotopy_equiv_simplicial_complexes} in Appendix~\\ref{app:finite-Cech} instead of Theorem~2.22 of~\\cite{chazal2016structure}.\n\\end{remark}\n\nThe Q-tame persistence modules given by this theorem allow us to discuss persistence diagrams, using the results of~\\cite{chazal2016structure}.\n\n\\begin{cor}\n\\label{cor:p-bottleneck-stability}\nLet $X$ be a totally bounded metric space, let $p\\in [1, \\infty]$, and let $k \\geq 0$ be an integer. \nThen $H_k\\circ \\vrpf{X}$ and $H_k\\circ \\vrpffin{X}$ have the same persistence diagram, denoted $\\dgm^\\ensuremath{\\mathrm{VR}}_{k,p}(X)$.\nSimilarly, $H_k\\circ \\cechpf{X}$ and $H_k\\circ \\cechpffin{X}$ have the same persistence diagram, denoted $\\dgm^\\ensuremath{\\mathrm{\\check{C}}}_{k,p}(X)$\n\\end{cor}\n\n\\begin{proof}\nPersistence diagrams are well-defined for Q-tame persistence modules, so for any totally bounded metric space $X$, any $p \\in [1,\\infty]$, and any $k \\geq 0$, by Corollary~\\ref{cor:totally_bounded_implies_q-tame} we have persistence diagrams associated to $H_k\\circ \\vrpf{X}$ and $H_k\\circ \\vrpffin{X}$.\nFrom Corollary~\\ref{cor:cP_cPfin_0-interleaved} we know that the interleaving distance between $H_k\\circ \\vrpf{X}$ and $H_k\\circ \\vrpffin{X}$ is zero, and so the Isometry Theorem~\\cite[Theorem 4.11]{chazal2016structure} implies that these persistence modules have the same (undecorated) persistence diagram, denoted $\\dgm^\\ensuremath{\\mathrm{VR}}_{k,p}(X)$.\nThe same proof also works for \\v{C}ech metric thickenings.\n\\end{proof}\n\n\n\nCombining the Isometry Theorem (\\cite[Theorem 4.11]{chazal2016structure}) with Theorem~\\ref{thm:main} and Corollary~\\ref{cor:totally_bounded_implies_q-tame}, we obtain the following.\n\n\\begin{cor}\nIf $X$ and $Y$ are totally bounded metric spaces, then for any $p \\in [1, \\infty]$ and any integer $k \\geq 0$, we have\n\\begin{align*}\n\\ensuremath{d_{\\mathrm{B}}}\\left( \\dgm^\\ensuremath{\\mathrm{VR}}_{k,p}(X), \\dgm^\\ensuremath{\\mathrm{VR}}_{k,p}(Y) \\right) &\\leq 2\\,d_\\mathrm{GH}(X,Y) \\\\\n\\ensuremath{d_{\\mathrm{B}}}\\left( \\dgm^\\ensuremath{\\mathrm{\\check{C}}}_{k,p}(X), \\dgm^\\ensuremath{\\mathrm{\\check{C}}}_{k,p}(Y) \\right) &\\leq 2\\,d_\\mathrm{GH}(X,Y).\n\\end{align*}\n\\end{cor}\n\n\n\n\\subsection{Consequence of stability: Connecting $\\infty$-metric thickenings and simplicial complexes}\n\\label{sec:comparability}\n\nWe show how the $\\infty$-Vietoris--Rips and $\\infty$-\\v{C}ech metric thickenings recover the persistent homology of the Vietoris--Rips and \\v{C}ech simplicial complexes.\nOur Corollary~\\ref{cor:infty-simplicial-dgms} answers~\\cite[Conjecture~6.12]{AAF} in the affirmative.\n\nWe recall that $\\vrp{X}{r}$ denotes the $p$-Vietoris--Rips metric thickening, that $\\vrpfin{X}{r}$ denotes the $p$-Vietoris--Rips metric thickening for measures of finite support, and that $\\vr{X}{r}$ denotes the Vietoris--Rips simplicial complex.\nTheorem~\\ref{thm:main} shows, for any $p\\in[1,\\infty]$, any $\\delta>0$, and any finite $\\delta$-net $U_\\delta$ of a totally bounded metric space $X$, that\n\\[\nd_\\mathrm{I}\\big(H_k\\circ\\vrpffin{X},H_k\\circ\\vrpffin{U_\\delta}\\big)\\leq 2\\delta.\n\\]\nFor $p=\\infty$, by Lemma~\\ref{lem:finite_infty-thickenings_homeomorphic_to_complexes}, we have $H_k\\circ\\vrppf{\\infty}{U_\\delta} \\cong H_k\\circ\\vrf{U_\\delta}$, so from the above we have\n\\[\nd_\\mathrm{I}\\big(H_k\\circ\\vrppffin{\\infty}{X} , H_k\\circ\\vrf{U_\\delta}\\big)\\leq 2\\delta.\n\\]\nNow, by the triangle inequality for the interleaving distance, by the inequality above, and by the Gromov--Hausdorff stability of $X\\mapsto H_k\\circ\\vrf{X}$~\\cite{ChazalDeSilvaOudot2014,chazal2009gromov}, we have\n\\begin{align*}\n& d_\\mathrm{I}(H_k\\circ\\vrppffin{\\infty}{X},H_k\\circ\\vrf{X}) \\\\\n\\leq\\ & d_\\mathrm{I}(H_k\\circ\\vrppffin{\\infty}{X},H_k\\circ\\vrf{U_\\delta}) + d_\\mathrm{I}(H_k\\circ\\vrf{U_\\delta},H_k\\circ\\vrf{X}) \\\\\n\\leq\\ & 4\\delta.\n\\end{align*}\n\nSince this holds for any $\\delta>0$, we find $d_\\mathrm{I}(H_k\\circ\\vrppffin{\\infty}{X},H_k\\circ\\vrf{X}) = 0$.\nThis implies that the bottleneck distance between persistence diagrams is $0$, and the (undecorated) diagrams are in fact equal (see for instance~\\cite[Theorem 4.20]{chazal2016structure}).\nWe can apply Corollary~\\ref{cor:cP_cPfin_0-interleaved} to get the same result for $\\vrppf{\\infty}{X}$ (measures with infinite support), and the same proof works in the \\v{C}ech case.\nWe state this as the following theorem.\n\n\n\\begin{cor}\n\\label{cor:infty-simplicial-dgms}\nFor any totally bounded metric space $X$ and any integer $k \\geq 0$, we have\n\\begin{align*}\nd_\\mathrm{I}(H_k\\circ\\vrppffin{\\infty}{X},\\ & H_k\\circ\\vrf{X}) = 0 \\\\\nd_\\mathrm{I}(H_k\\circ\\vrppf{\\infty}{X},\\ & H_k\\circ\\vrf{X}) = 0 \\\\\n\\dgm^\\ensuremath{\\mathrm{VR}}_{k,\\infty}(X) &= \\dgm^\\ensuremath{\\mathrm{VR}}_{k}(X) \\\\\nd_\\mathrm{I}(H_k\\circ\\cechppffin{\\infty}{X},\\ & H_k\\circ\\cechf{X}) = 0 \\\\\nd_\\mathrm{I}(H_k\\circ\\cechppf{\\infty}{X},\\ & H_k\\circ\\cechf{X}) = 0 \\\\\n\\dgm^\\ensuremath{\\mathrm{\\check{C}}}_{k,\\infty}(X) &= \\dgm^\\ensuremath{\\mathrm{\\check{C}}}_{k}(X).\n\\end{align*}\n\\end{cor}\n\n\n\n\\begin{remark}\nWhereas the $\\infty$-metric thickenings $\\vrppf{\\infty}{X}$ and the simplicial complexes $\\vrf{X}$ yield persistence modules with an interleaving distance of 0, we are in the interesting landscape where the metric thickenings $\\vrpf{X}$ may yield something new and different for $p<\\infty$.\nFor example, in Section~\\ref{sec:spheres} we explore the new persistence modules that arise for $p$-metric thickenings of Euclidean spheres $(\\ensuremath{\\mathbb{S}}^n,\\ell_2)$ in the case $p=2$.\n\\end{remark}\n\n\\begin{question}\nIs the simplicial complex $\\vr{X}{r}$ always homotopy equivalent to $\\vrppfin{\\infty}{X}{r}$ or $\\vrpp{\\infty}{X}{r}$?\nCompare with~\\cite[Remark~3.3]{AAF}.\n\\end{question}\n\n\\begin{question}\nFrom~\\cite[Theorem~5.2]{lim2020vietoris} we know that for $X$ compact, bars in the persistent homology of the simplicial complex filtration $\\vrf{X}$ are of the form $(a,b]$ or $(a,\\infty)$.\nIs the same true for $\\vrppf{\\infty}{X}$?\nNote that we are using the $<$ convention, i.e.\\ a simplex is included in $\\vr{X}{r}$ when its diameter is strictly less than $r$, and a measure is included in $\\vrpp{\\infty}{X}{r}$ when the diameter of its support is strictly less than $r$.\n\\end{question}\n\n\n\n\n\n\\subsection{The proof of stability}\n\\label{ssec:proof-stability}\n\nThe main idea behind the proof of our stability result, Theorem~\\ref{thm:general_stability}, is to approximate the metric space $X$ by a finite approximating submetric space (a net) $U$.\nThe advantage is that it is easier to construct a continuous map with domain $\\ensuremath{\\mathcal{P}}_U$ than one with domain $\\ensuremath{\\mathcal{P}}_X$, and then we can construct a continuous map from $\\ensuremath{\\mathcal{P}}_X$ to $\\ensuremath{\\mathcal{P}}_U$ by using a partition of unity.\n\nWe begin with some necessary lemmas.\n\n\\begin{lem}\\label{lem:Ptau_cts_via_partition}\nLet $U$ be a finite $\\delta$-net of a bounded metric space $X$, and let $\\{\\zeta_u^U\\}_{u\\in U}$ be a continuous partition of unity subordinate to the open covering $\\bigcup\\limits_{u\\in U} B(u;\\delta)$ of $X$.\n\nFor any $r>0$, we have the continuous map $\\Phi_U \\colon \\ensuremath{\\mathcal{P}}_X \\to \\ensuremath{\\mathcal{P}}_U$ defined by\n\\[\\alpha \\mapsto \\sum_{u\\in U} \\int_X\\zeta_u^U(x)\\,\\alpha(dx) \\cdot\\delta_{u}.\\]\nFor any $q\\in [1, \\infty]$, we have $\\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha, \\Phi_U(\\alpha))< \\delta$.\n\\end{lem}\n\n\\begin{proof}\nThe map $\\Phi_U$ is well-defined as\n\\[\\sum_{u\\in U} \\int_X\\zeta_u^U(x)\\,\\alpha(dx) =\\int_X\\,\\sum_{u\\in U} \\,\\zeta_u^U(x)\\,\\alpha(dx) = \\int_X\\, \\alpha(dx) = 1.\\]\nFor the continuity, since the weak topology on $\\ensuremath{\\mathcal{P}}_X$ and $\\ensuremath{\\mathcal{P}}_U$ can be metrized, it suffices to show for a weakly convergent sequence $\\alpha_n\\in \\ensuremath{\\mathcal{P}}_X$, that the image $\\Phi_U(\\alpha_n)$ is also weakly convergent.\nAs $U$ is a finite metric space, it suffices to show that for any fixed $u_0\\in U$, the sequence of real numbers $\\big(\\Phi_U(\\alpha_n)(\\{u_0\\})\\big)_n$ is itself convergent.\nNote that\n\\[ \\Phi_U(\\alpha_n)(\\{u_0\\}) = \\int_X\\zeta_{u_0}^U(x)\\,\\alpha_n(dx).\\]\nSince $\\zeta_{u_0}^U$ is a bounded continuous function on $X$, we obtain the desired convergence through the weak convergence of $\\alpha_n$.\n\nLastly, we must show that $\\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha, \\Phi_U(\\alpha))< \\delta$.\nFor any $u\\in U$, we use the notation $w_u^\\alpha := \\int_X \\zeta_u^U(x)\\alpha(dx)$, so that $\\Phi_U(\\alpha)=\\sum_{u\\in U} w_u^\\alpha \\cdot\\,\\delta_u$ and\n\\[ \\sum_{u\\in U} w_u^\\alpha = \\sum_{u\\in U} \\int_X \\zeta_u^U(x)\\,\\alpha(dx) = \\int_X \\sum_u \\zeta_u^U(x)\\,\\alpha(dx) = \\int_X \\alpha(dx) = 1.\\]\nLet $\\zeta_u^U\\,\\alpha$ denote the measure such that $\\zeta_u^U\\,\\alpha(B) = \\int_B\\zeta_u^U(x)\\alpha(dx)$ for any measurable set $B\\subseteq X$.\nWe have\n\\begin{equation*}\n\\alpha = \\sum_{u\\in U} \\zeta_u^U\\,\\alpha = \\sum_{\\substack{u\\in U, \\\\ w^{\\alpha}_u\\neq 0}} w_u^\\alpha \\cdot\\,\\left(\\tfrac{1}{w^{\\alpha}_u}\\,\\zeta_u^U\\,\\alpha \\right).\n\\end{equation*}\nFrom this we get,\n\\begin{align*}\n\\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha, \\Phi_U(\\alpha)) & = \\ensuremath{d_{\\mathrm{W},q}^X}\\left(\\sum_{\\substack{u\\in U, \\\\ w^{\\alpha}_u\\neq 0}} w_u^\\alpha \\cdot\\,\\left(\\tfrac{1}{w^{\\alpha}_u}\\,\\zeta_u^U\\,\\alpha \\right), \\sum_{\\substack{u\\in U,\\\\ w^{\\alpha}_u\\neq 0}} w_u^\\alpha \\cdot\\,\\delta_u\\right) \\\\\n& \\leq \\left( \\sum_{\\substack{u\\in U, \\\\ w^{\\alpha}_u\\neq 0}} w_u^\\alpha\\cdot\\left( \\ensuremath{d_{\\mathrm{W},q}^X}\\left(\\tfrac{1}{w^{\\alpha}_u}\\,\\zeta_u^U\\,\\alpha ,\\, \\delta_u\\right) \\right)^q \\right)^\\frac{1}{q}\\\\\n& < \\delta.\n\\end{align*}\nThe first inequality is by Lemma~\\ref{lem:bound-distance-convex-comb}.\nThe last inequality comes from the fact that each $\\tfrac{1}{w^{\\alpha}_u}\\,\\zeta_u^U\\,\\alpha$ is a probability measure supported in $B(u;\\delta)$, and thus $\\ensuremath{d_{\\mathrm{W},q}^X}\\left(\\tfrac{1}{w^{\\alpha}_u}\\,\\zeta_u^U\\,\\alpha ,\\, \\delta_u\\right) < \\delta$.\n\\end{proof}\n\nFor any two totally bounded metric spaces $X$ and $Y$, through partitions of unity we can build continuous maps between $\\ensuremath{\\mathcal{P}}_X$ and $\\ensuremath{\\mathcal{P}}_Y$, as follows.\nLet $X$ and $Y$ be two totally bounded metric spaces, let $\\eta > 2\\,d_\\mathrm{GH}(X, Y)$, and let $\\delta>0$.\nWe fix finite $\\delta$-nets $U\\subseteq X$ of $X$ and $V\\subseteq Y$ of $Y$.\nBy the triangle inequality, we have $d_\\mathrm{GH}(U, V)<\\frac{\\eta}{2}+2\\delta$.\nSo there exist maps $\\varphi\\colon U\\to V$ and $\\psi\\colon V\\to U$ with\n\\[\\max\\big(\\mathrm{dis}(\\varphi),\\mathrm{dis}(\\psi),\\mathrm{codis}(\\varphi,\\psi)\\big)\\leq \\eta +4\\delta.\\]\nWe then define the maps $\\widehat{\\Phi}\\colon \\ensuremath{\\mathcal{P}}_X\\to \\ensuremath{\\mathcal{P}}_Y$ and $\\widehat{\\Psi}\\colon \\ensuremath{\\mathcal{P}}_Y\\to \\ensuremath{\\mathcal{P}}_X$ to be\n\\[\\widehat{\\Phi}:=\\iota_V\\circ \\varphi_\\sharp \\circ \\Phi_U \\quad\\text{and}\\quad \\widehat{\\Psi} :=\\iota_U \\circ \\psi_\\sharp\\circ \\Phi_V,\\]\nwhere $\\iota_U \\colon \\ensuremath{\\mathcal{P}}_U\\hookrightarrow \\ensuremath{\\mathcal{P}}_X$ and $\\iota_V \\colon \\ensuremath{\\mathcal{P}}_V\\hookrightarrow \\ensuremath{\\mathcal{P}}_Y$ are inclusions and $\\Phi_U$ and $\\Phi_V$ are the maps defined in Lemma~\\ref{lem:Ptau_cts_via_partition}.\nThen both $\\widehat{\\Phi}$ and $\\widehat{\\Psi}$ are continuous maps by Lemma~\\ref{lem:Ptau_cts_via_partition} and by the continuity of pushforwards of continuous maps.\nThe use of the finite $\\delta$-nets $U$ and $V$ is important because they can be compared by continuous maps $\\varphi$ and $\\psi$, which yield continuous maps $\\varphi_\\sharp$ and $\\psi_\\sharp$.\nThe following lemma shows that $\\widehat{\\Phi}$ and $\\widehat{\\Psi}$ are homotopy equivalences.\n\n\\begin{lem}\\label{lem:homotopy_scaffording}\nWith the above notation, we have homotopy equivalences $\\widehat{\\Psi}\\circ \\widehat{\\Phi} \\simeq \\mathrm{id}_{\\ensuremath{\\mathcal{P}}_X}$ and $\\widehat{\\Phi}\\circ \\widehat{\\Psi} \\simeq \\mathrm{id}_{\\ensuremath{\\mathcal{P}}_Y}$ via the linear families $H^X_t\\colon \\ensuremath{\\mathcal{P}}_X\\times [0, 1]\\to \\ensuremath{\\mathcal{P}}_X$ and $H^Y_t\\colon \\ensuremath{\\mathcal{P}}_Y\\times [0, 1]\\to \\ensuremath{\\mathcal{P}}_Y$ given by\n\\[H^X_t(\\alpha) := (1-t)\\,\\widehat{\\Psi}\\circ \\widehat{\\Phi}(\\alpha)+ t\\,\\alpha \\quad\\text{and}\\quad H^Y_t(\\beta) := (1-t)\\,\\widehat{\\Phi}\\circ \\widehat{\\Psi}(\\beta)+ t\\,\\beta.\\]\nMoreover, for any $q\\in [1, \\infty]$ and any $t\\in [0, 1]$, we have $\\ensuremath{d_{\\mathrm{W},q}^X}(H_t^X(\\alpha), \\alpha) < \\eta + 6\\delta$ and $d_{\\mathrm{W}, q}^Y(H_t^Y(\\beta), \\beta) < \\eta + 6\\delta.$\n\\end{lem}\n\n\\begin{proof}\nThe homotopies $H^X_t$ and $H^Y_t$ are continuous by Proposition~\\ref{prop:linear_homotopies}, since $\\widehat{\\Phi}$ and $\\widehat{\\Psi}$ are continuous.\nWe will only present the estimate $\\ensuremath{d_{\\mathrm{W},q}^X}(H_t^X(\\alpha), \\alpha) < \\eta + 6\\delta$, as the other inequality can be proved in a similar way.\nWe first calculate the expression for $\\widehat{\\Psi} \\circ\\widehat{\\Phi} (\\alpha)$, obtaining\n\\begin{align*}\n\\widehat{\\Psi} \\circ\\widehat{\\Phi} (\\alpha)\n& = \\psi_\\sharp\\left(\\Phi_V\\left(\\varphi_\\sharp\\left(\\Phi_U(\\alpha)\\right)\\right)\\right) \\\\\n& = \\psi_\\sharp\\left(\\Phi_V\\left(\\varphi_\\sharp\\left(\\sum_{u\\in U} \\int_X\\zeta^U_u(x)\\,\\alpha(dx)\\cdot\\delta_{u}\\right)\\right)\\right) \\\\\n& = \\psi_\\sharp\\left(\\Phi_V\\left(\\sum_{u\\in U} \\int_X\\zeta^U_u(x)\\,\\alpha(dx)\\cdot\\delta_{\\varphi(u)}\\right)\\right) \\\\\n& =\\psi_\\sharp\\left(\\sum_{v\\in V}\\int_Y \\zeta^V_v(y)\\, \\left(\\sum_{u\\in U} \\int_X\\zeta^U_u(x)\\,\\alpha(dx)\\cdot\\delta_{\\varphi(u)}\\right)(dy) \\cdot \\delta_v\\right) \\\\\n& = \\psi_\\sharp\\left(\\sum_{v\\in V}\\bigg( \\sum_{u\\in U} \\int_X\\zeta^U_u(x)\\,\\alpha(dx)\\cdot \\int_Y\\zeta_v^V(y)\\cdot \\delta_{\\varphi(u)}(dy)\\bigg)\\cdot \\delta_v\\right) \\\\\n& = \\psi_\\sharp\\left(\\sum_{v\\in V}\\bigg( \\sum_{u\\in U} \\int_X\\zeta^U_u(x)\\,\\alpha(dx)\\cdot \\zeta^V_v({\\varphi(u)})\\bigg)\\cdot \\delta_v\\right) \\\\\n& =\\sum_{v\\in V}\\bigg( \\sum_{u\\in U} \\int_X\\zeta^U_u(x)\\,\\alpha(dx)\\cdot \\zeta^V_v({\\varphi(u)})\\bigg)\\cdot \\delta_{\\psi(v)} \\\\\n& =\\sum_{v\\in V}\\bigg( \\sum_{u\\in U} w^{\\alpha}_u\\cdot \\zeta^V_v({\\varphi(u)})\\bigg)\\cdot \\delta_{\\psi(v)},\n\\end{align*}\nwhere we again use the notation $w^{\\alpha}_u:=\\int_X \\zeta_u^U(x)\\alpha(dx)$. \nWe have\n\\[H_t^X(\\alpha) = (1-t)\\sum_{v\\in V}\\sum_{u\\in U} \\bigg( w^{\\alpha}_u\\cdot\\zeta^V_v({\\varphi(u)})\\bigg)\\cdot \\delta_{\\psi(v)} + t\\, \\alpha.\\]\nThe coefficients in front of measures in the previous formula sums to one as we can see from the following computation:\n\\[(1-t)\\sum_{v\\in V}\\sum_{u\\in U} \\bigg( w^{\\alpha}_u\\cdot\\zeta^V_v({\\varphi(u)})\\bigg) + t = (1-t)\\sum_{u\\in U} w^{\\alpha}_u + t = 1.\\]\nNote that\n\\begin{align*}\n\\alpha & =(1-t) \\sum_{u\\in U} \\zeta_u^U\\,\\alpha + t\\,\\alpha \\\\\n& = (1-t)\\sum_{v\\in V}\\sum_{u\\in U} \\,\\zeta^V_v({\\varphi(u)})\\cdot\\,(\\zeta_u^U\\,\\alpha ) + t\\, \\alpha \\\\\n& =(1-t)\\sum_{v\\in V}\\sum_{\\substack{u\\in U, \\\\ w^{\\alpha}_u\\neq 0}} \\,\\bigg(w_u^\\alpha\\cdot \\zeta^V_v({\\varphi(u)})\\bigg)\\cdot\\,\\left(\\tfrac{1}{w^{\\alpha}_u}\\,\\zeta_u^U\\,\\alpha \\right) + t\\, \\alpha. \\end{align*}\nBased on these observations, we then apply the inequality from Lemma~\\ref{lem:bound-distance-convex-comb} to obtain\n\\begin{align*}\n& \\ensuremath{d_{\\mathrm{W},q}^X}(H_t^X(\\alpha), \\alpha) \\\\\n=\\,& \\ensuremath{d_{\\mathrm{W},q}^X}\\bigg((1-t)\\sum_{v\\in V}\\sum_{\\substack{u\\in U, \\\\ w^{\\alpha}_u\\neq 0}} \\bigg(w_u^\\alpha\\cdot \\zeta^V_v({\\varphi(u)})\\bigg)\\cdot \\delta_{\\psi(v)} + t\\, \\alpha,\\, \\\\\n& \\quad\\quad\\quad \\quad\\quad\\quad(1-t)\\sum_{v\\in V}\\sum_{\\substack{u\\in U, \\\\ w^{\\alpha}_u\\neq 0}} \\,\\bigg(w_u^\\alpha\\cdot \\zeta^V_v({\\varphi(u)})\\bigg)\\cdot\\,\\left(\\tfrac{1}{w^{\\alpha}_u}\\,\\zeta_u^U\\,\\alpha \\right) + t\\, \\alpha\\bigg) \\\\\n\\leq\\, & \\left( (1-t)\\sum_{v\\in V} \\sum_{\\substack{u\\in U, \\\\ w^{\\alpha}_u\\neq 0}} \\bigg(w_u^\\alpha\\cdot \\zeta^V_v({\\varphi(u)})\\bigg)\\cdot \\bigg(\\ensuremath{d_{\\mathrm{W},q}^X}\\big( \\delta_{\\psi(v)} ,\\,\\tfrac{1}{w^{\\alpha}_u}\\,\\zeta_u^U\\,\\alpha\\big)\\bigg)^p\\right)^\\frac{1}{p} \\\\\n\\leq\\, & \\left( (1-t)\\sum_{v\\in V} \\sum_{\\substack{u\\in U, \\\\ w^{\\alpha}_u\\neq 0}} \\bigg(w_u^\\alpha\\cdot \\zeta^V_v({\\varphi(u)})\\bigg)\\cdot \\bigg(\\ensuremath{d_{\\mathrm{W},q}^X}\\big( \\delta_{\\psi(v)} ,\\,\\delta_u\\big)+\\ensuremath{d_{\\mathrm{W},q}^X}\\big( \\delta_{u} ,\\,\\tfrac{1}{w^{\\alpha}_u}\\,\\zeta_u^U\\,\\alpha\\big)\\bigg)^p\\right)^\\frac{1}{p}.\n\\end{align*}\nFor each non-zero term in the summand of the last expression, we have $d_Y(v,\\,\\varphi(u)) < \\delta$ which implies $d_X(\\psi(v), u)< 5\\delta + \\eta$ via the codistortion assumption.\nTherefore, $\\ensuremath{d_{\\mathrm{W},q}^X}\\big( \\delta_{\\psi(v)} ,\\,\\delta_u\\big)< 5\\delta + \\eta$ as well.\nMoreover, since $\\frac{1}{w^{\\alpha}_u}\\,\\zeta_u^U\\,\\alpha$ is a probability measure supported in the $\\delta$-ball centered at the point $u$, we have $\\ensuremath{d_{\\mathrm{W},q}^X}\\big( \\delta_{u} ,\\,\\frac{1}{w^{\\alpha}_u}\\,\\zeta_u^U\\,\\alpha\\big)<\\delta$.\nTherefore, we have $\\ensuremath{d_{\\mathrm{W},q}^X}(H_t^X(\\alpha), \\alpha)<6\\delta + \\eta$.\n\\end{proof}\n\n\\begin{remark}\nThe above lemma still holds if we replace $\\ensuremath{\\mathcal{P}}_X$ by $\\ensuremath{\\mathcal{P}}^\\fin_X$ and $\\ensuremath{\\mathcal{P}}_Y$ by $\\ensuremath{\\mathcal{P}}^\\fin_Y$ as all related maps naturally restrict to the set of finitely supported measures, $\\ensuremath{\\mathcal{P}}^\\fin_X$.\n\\end{remark}\n\nWe are now ready to prove our main result, Theorem~\\ref{thm:general_stability}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:general_stability}]\nFor $\\mathfrak{i}$ an $L$-controlled invariant, for totally bounded metric spaces $X$ and $Y$, and for any integer $k\\geq 0$, we must show\n\\begin{align*}\n\\ensuremath{d_{\\mathrm{I}}}\\big(H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X},H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}_Y}{\\mathfrak{i}^Y}\\big)&\\leq d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X), (\\ensuremath{\\mathcal{P}}_Y, \\mathfrak{i}^Y)\\big)\\leq 2L\\cdotd_\\mathrm{GH}(X, Y) \\\\\n\\ensuremath{d_{\\mathrm{I}}}\\big(H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}^\\fin_X}{\\mathfrak{i}^X},H_k\\circ \\filtf{\\ensuremath{\\mathcal{P}}^\\fin_Y}{\\mathfrak{i}^Y}\\big)&\\leq d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}^\\fin_X, \\mathfrak{i}^X), (\\ensuremath{\\mathcal{P}}^\\fin_Y, \\mathfrak{i}^Y)\\big) \\leq 2L\\cdotd_\\mathrm{GH}(X, Y).\n\\end{align*}\nWe prove only the first line above, as the finitely supported case in the second line has a nearly identical proof.\nThe inequality involving $\\ensuremath{d_{\\mathrm{I}}}$ and $d_\\mathrm{HT}$ follows from Lemma~\\ref{lem:dht_bound_interleaving}.\nHence it suffices to prove the inequality $d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X), (\\ensuremath{\\mathcal{P}}_Y, \\mathfrak{i}^Y)\\big)\\leq 2L\\cdotd_\\mathrm{GH}(X, Y)$ involving $d_\\mathrm{HT}$ and $d_\\mathrm{GH}$.\n\nFollowing the construction in Lemma~\\ref{lem:homotopy_scaffording}, let $\\eta > 2\\cdot d_\\mathrm{GH}(X, Y)$ and $\\delta>0$.\nWe fix finite $\\delta$-nets $U\\subset X$ of $X$ and $V\\subset Y$ of $Y$.\nBy the triangle inequality, we have $d_\\mathrm{GH}(U, V)<\\frac{\\eta}{2}+2\\delta$, and so there exist maps $\\varphi:U\\to V$ and $\\psi\\colon V\\to U$ with\n\\[\\max\\big(\\mathrm{dis}(\\varphi),\\mathrm{dis}(\\psi),\\mathrm{codis}(\\varphi,\\psi)\\big)\\leq \\eta +4\\delta.\\]\nFor any $\\delta>0$, we will show that $(\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X)$ and $(\\ensuremath{\\mathcal{P}}_Y, \\mathfrak{i}^Y)$ are $(\\eta+6\\delta)\\, L$--homotopy equivalent.\nBy the definition of a $\\delta$-homotopy (Definition~\\ref{defn:delta-homotopy}), it suffices to show that\n\\begin{itemize}\n \\item the map $\\widehat{\\Phi}$ is a $(\\eta+ 6\\delta)\\, L$--map from $(\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X)$ to $(\\ensuremath{\\mathcal{P}}_Y, \\mathfrak{i}^Y)$,\n \\item the map $\\widehat{\\Psi}$ is a $(\\eta+ 6\\delta)\\, L$--map from $(\\ensuremath{\\mathcal{P}}_Y, \\mathfrak{i}^Y)$ to $(\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X)$,\n \\item the map $\\widehat{\\Psi}\\circ\\widehat{\\Phi}\\colon \\ensuremath{\\mathcal{P}}_X\\to \\ensuremath{\\mathcal{P}}_X$ is $(2\\eta + 12\\delta)\\, L$--homotopic to $\\mathrm{id}_{\\ensuremath{\\mathcal{P}}_X}$ with respect to $( \\mathfrak{i}^X, \\mathfrak{i}^X)$,\n \\item the map $\\widehat{\\Phi}\\circ\\widehat{\\Psi}\\colon \\ensuremath{\\mathcal{P}}_X\\to \\ensuremath{\\mathcal{P}}_X$ is $(2\\eta + 12\\delta)\\, L$--homotopic to $\\mathrm{id}_{\\ensuremath{\\mathcal{P}}_Y}$ with respect to $( \\mathfrak{i}^Y, \\mathfrak{i}^Y)$.\n\\end{itemize}\nWe will only present the proof for the first and third items; the other two can be proved similarly.\nWe have\n\\[ \\mathfrak{i}^Y(\\widehat{\\Phi}(\\alpha)) = \\mathfrak{i}^Y(\\varphi_\\sharp\\circ\\Phi_U(\\alpha))\n\\leq \\mathfrak{i}^X(\\Phi_U(\\alpha)) + (\\eta + 4\\delta)\\, L\n\\leq \\mathfrak{i}^X(\\alpha) + (\\eta + 6\\delta)\\, L, \\]\nwhere the first inequality is from the stability of the invariant $\\mathfrak{i}$ under pushforward and from the bound on the distortion of $\\varphi$, and where the second inequality is from the stability of the invariant with respect to Wasserstein distance and from the bound $\\ensuremath{d_{\\mathrm{W},q}^X}(\\alpha, \\Phi_U(\\alpha))< \\delta$ in Lemma~\\ref{lem:Ptau_cts_via_partition}.\nThis proves the first item.\n\nFor the third item, we use the homotopy $H_t^X$ in Lemma~\\ref{lem:homotopy_scaffording}.\nThen it suffices to show for any fixed $t\\in [0, 1]$, the map $H_t^X$ is a $(2\\eta + 12\\delta)\\, L$--map from $(\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X)$ to $(\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X)$, that is,\n\\[ \\mathfrak{i}^X(H_t^X(\\alpha)) \\leq \\mathfrak{i}^X(\\alpha) + (2\\eta + 12\\delta)\\, L.\\]\nThis comes from the inequality \\(d_{\\mathrm{W}, \\infty}^X(H_t^X(\\alpha), \\alpha)< \\eta + 6\\delta\\) from Lemma~\\ref{lem:homotopy_scaffording}, and the from the stability assumption of $\\mathfrak{i}$ with respect to Wasserstein-distance.\n\nNow, since $(\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X)$ and $(\\ensuremath{\\mathcal{P}}_Y, \\mathfrak{i}^Y)$ are $(\\eta+6\\delta)\\, L$--homotopy equivalent for any $\\eta > 2\\cdot d_\\mathrm{GH}(X, Y)$ and $\\delta>0$, it follows from the definition of the $d_\\mathrm{HT}$ distance (Definition~\\ref{defn:dHT}) that $d_\\mathrm{HT}\\big((\\ensuremath{\\mathcal{P}}_X, \\mathfrak{i}^X), (\\ensuremath{\\mathcal{P}}_Y, \\mathfrak{i}^Y)\\big)\\leq 2L\\cdotd_\\mathrm{GH}(X, Y)$.\nThis completes the proof.\n\\end{proof}\n\n\n\n\\section{A Hausmann type theorem for $2$-Vietoris-Rips and $2$-\\v{C}ech thickenings of Euclidean submanifolds}\n\\label{sec:hausmann}\n\n\n\nHausmann's theorem~\\cite{Hausmann1995} states that if $X$ is a Riemannian manifold and if the scale $r>0$ is sufficiently small (depending on the curvature $X$), then the Vietoris--Rips simplicial complex $\\vr{X}{r}$ is homotopy equivalent to $X$.\nA short proof using an advanced version of the nerve lemma is given in~\\cite{virk2021rips}.\nVersions of Hausmann's theorem have been proven for $\\infty$-metric thickenings in~\\cite{AAF}, where $X$ is equipped with the Riemannian metric, and in~\\cite{AM}, where $X$ is equipped with a Euclidean metric.\nIn this section we prove a Hausmann type theorem for the $2$-Vieteoris--Rips and $2$-\\v{C}ech metric thickenings.\nOur result is closest to that in~\\cite{AM} (now with $p=2$ instead of $p=\\infty$): we work with any Euclidean subset $X$ of positive reach.\nThis includes (for example) any embedded $C^k$ submanifold of $\\ensuremath{\\mathbb{R}}^n$ for $k\\ge 2$, with or without boundary~\\cite{Thale}.\n\nWe begin with a definition and some lemmas that will be needed in the proof.\n\nLet $\\alpha$ be a measure in $\\ensuremath{\\mathcal{P}}_1(\\ensuremath{\\mathbb{R}}^n)$, the collection of all Radon measures with finite first moment.\nFor any coordinate function $x_i$, where $1\\leq i\\leq n$, we have\n\\[\n \\int_{\\ensuremath{\\mathbb{R}}^n} |x_i|\\, \\alpha(dx) \\leq \\int_{\\ensuremath{\\mathbb{R}}^n} \\norm{x}\\,\\alpha(dx)<\\infty.\n\\]\nTherefore, the \\emph{Euclidean mean} map $m\\colon \\ensuremath{\\mathcal{P}}_1(\\ensuremath{\\mathbb{R}}^n)\\to \\ensuremath{\\mathbb{R}}^n$ given by the following vector-valued integral is well-defined:\n\\[\n m(\\alpha):= \\int_{\\ensuremath{\\mathbb{R}}^n} x\\, \\alpha(dx).\n\\]\n\n\\begin{lem}\nLet $X$ be a metric space and let $f\\colon X\\to \\ensuremath{\\mathbb{R}}^n$ be a bounded continuous function.\nThen the induced map $m\\circ f_\\sharp \\colon \\ensuremath{\\mathcal{P}}_X \\to \\ensuremath{\\mathbb{R}}^n$ is continuous.\n\\end{lem}\n\n\\begin{proof}\nFor a sequence $\\alpha_n$ that weakly converges to $\\alpha$, we have the following vector-valued integral:\n\\[\nm\\circ f_\\sharp(\\alpha_n) = \\int_{\\mathbb{R}^n}x\\, f_\\sharp(\\alpha_n)(dx) = \\int_X f(x)\\,\\alpha_n(dx).\\]\nAs $f$ is bounded and continuous, the above limit converges to $m\\circ f_\\sharp(\\alpha)$ as $\\alpha_n$ converges to $\\alpha$.\nTherefore, the map $m\\circ f_\\sharp$ is continuous.\n\\end{proof}\n\n\n\\begin{lem}\\label{lem:dHBoundViaDiamp}\nLet $\\alpha$ be a probability measure in $\\ensuremath{\\mathcal{P}}_1(\\ensuremath{\\mathbb{R}}^n)$.\nThen for $p\\in [1, \\infty]$, there is some $z \\in \\mathrm{supp}(\\alpha)$ with\n\\[\\norm{m(\\alpha) - z} \\leq \\mathrm{diam}_p(\\alpha).\\]\n\\end{lem}\n\n\\begin{proof}\nAs $\\mathrm{diam}_1(\\alpha)\\leq \\mathrm{diam}_p(\\alpha)$ for all $p\\in[1,\\infty]$, it suffices to show $ \\norm{m(\\alpha) - z} \\leq \\mathrm{diam}_1(\\alpha)$.\nWe consider the following formula:\n\\begin{align*}\n\\int_{\\ensuremath{\\mathbb{R}}^n}\\norm{m(\\alpha) - z} \\alpha(dz) & = \\int_{\\ensuremath{\\mathbb{R}}^n} \\norm{\\int_{\\ensuremath{\\mathbb{R}}^n} (x - z) \\alpha(dx)} \\alpha(dz)\\\\\n& \\leq \\int_{\\ensuremath{\\mathbb{R}}^n}\\int_{\\ensuremath{\\mathbb{R}}^n}\\norm{ (x - z)} \\alpha(dx)\\alpha(dz)\\\\\n& = \\mathrm{diam}_1(\\alpha).\n\\end{align*}\nSo there must be some $z\\in \\mathrm{supp}(\\alpha)$ with $\\norm{m(\\alpha) - z} \\leq \\mathrm{diam}_1(\\alpha)$.\n\\end{proof}\n\n\\begin{lem}\\label{lem:Frechet_2_decomposition}\nFor any probability measure $\\alpha\\in \\ensuremath{\\mathcal{P}}_1(\\ensuremath{\\mathbb{R}}^n)$ and for any $x\\in \\ensuremath{\\mathbb{R}}^n$, we can write the associated squared $2$-Fr\\'{e}chet function $F_{\\alpha, 2}^2(x)$ as\n\\begin{equation*}\nF_{\\alpha, 2}^2(x) = \\norm{x - m(\\alpha)}^2 +F_{\\alpha, 2}^2(m(\\alpha)).\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nBy the linearity of the inner product, we have\n\\begin{align*}\nF_{\\alpha, 2}^2(x) & := \\int_{\\ensuremath{\\mathbb{R}}^n}\\norm{x - y}^2 \\alpha(dy) \\\\\n& =\\int_{\\ensuremath{\\mathbb{R}}^n}\\Big(\\norm{x}^2 - 2\\langle x, y\\rangle + \\norm{y}^2\\Big)\\alpha(dy) \\\\\n& =\\int_{\\ensuremath{\\mathbb{R}}^n}\\Big(\\norm{x}^2 - 2\\langle x, y\\rangle +\\norm{m(\\alpha)}^2 - \\norm{m(\\alpha)}^2 + \\norm{y}^2\\Big)\\alpha(dy) \\\\\n& = \\norm{x}^2 - 2 \\langle x, m(\\alpha)\\rangle+ \\norm{m(\\alpha)}^2 + \\int_{\\ensuremath{\\mathbb{R}}^n}\\Big(-\\norm{m(\\alpha)}^2 + \\norm{y}^2\\Big)\\alpha(dy) \\\\\n& = \\norm{x - m(\\alpha)}^2 + \\int_{\\ensuremath{\\mathbb{R}}^n}\\Big(\\norm{m(\\alpha)}^2 - 2\\langle m(\\alpha), y\\rangle + \\norm{y}^2\\Big)\\alpha(dy) \\\\\n& = \\norm{x - m(\\alpha)}^2 + \\int_{\\ensuremath{\\mathbb{R}}^n}\\norm{m(\\alpha) - y}\\alpha(dy) \\\\\n& =\\norm{x - m(\\alpha)}^2 +F_{\\alpha, 2}^2(m(\\alpha)).\n\\end{align*}\n\\end{proof}\n\n\n\nRecall the \\emph{medial axis} of $X$ is defined as the closure\n\\[\\mathrm{med}(X) = \\overline{\\{y\\in \\ensuremath{\\mathbb{R}}^n~|~\\exists\\, x_1 \\neq x_2 \\in X\\,\\text{with } \\norm{y - x_1} = \\norm{y - x_2} = \\inf_{x\\in X}\\norm{y-x}\\}}.\\]\nThe \\emph{reach} $\\tau$ of $X$ is the closest distance $\\tau := \\inf_{x\\in X,y\\in\\mathrm{med}(X)}\\norm{x-y}$ between points in $X$ and $\\mathrm{med}(X)$.\nLet $U_\\tau(X)$ be the $\\tau$-neighborhood of $X$ in $\\ensuremath{\\mathbb{R}}^n$, that is, the set of points $y\\in\\ensuremath{\\mathbb{R}}^n$ such that there is some $x\\in X$ with $\\norm{y - x}<\\tau$.\nThe definition of reach implies that for any point $y$ in the open neighborhood $U_\\tau(X)$, there is a unique closest point $x\\in X$.\nThe associated nearest projection map $\\pi\\colon U_\\tau(X)\\to X$ is continuous; see~\\cite[Theorem~4.8(8)]{federer1959curvature} or~\\cite[Lemma~3.7]{AM}.\n\n\n\\begin{lem}\n\\label{lem:boundedness_of_rad_and_diam}\nLet $X$ be a bounded subset of $\\ensuremath{\\mathbb{R}}^n$ with reach $\\tau(X)>0$.\nLet $\\alpha \\in \\ensuremath{\\mathcal{P}}_X$ have its Euclidean mean $m(\\alpha)$ in the neighborhood $U_\\tau(X)$.\nThen along the linear interpolation family\n\\[\\alpha_t := (1-t)\\alpha + t\\,\\delta_{\\phi(\\alpha)},\\]\nwhere $\\phi$ is the composition $\\pi\\circ m\\colon \\ensuremath{\\mathcal{P}}_X\\to X$, both $\\mathrm{diam}_2$ and $\\mathrm{rad}_2$ obtain their maximum \nat $t=0$.\n\\end{lem}\n\n\\begin{proof}\nAccording to Lemma~\\ref{lem:Frechet_2_decomposition}, for any $x\\in X$ we have\n\\[F_{\\alpha, 2}^2(x) = \\norm{x - m(\\alpha)}^2 +F_{\\alpha, 2}^2(m(\\alpha)).\\]\nAs $m(\\alpha)$ is inside $U_\\tau(X)$, the first term $\\norm{x - m(\\alpha)}$ is minimized over $x\\in X$ at $x=\\phi(\\alpha)$, and hence so is $F_{\\alpha, 2}(x)$.\nTherefore, we have $\\mathrm{rad}_2(\\alpha) = \\inf_{x\\in X} F_{\\alpha, 2}(x) = F_{\\alpha, 2}(\\phi(\\alpha))$.\nSo for $\\mathrm{rad}_2(\\alpha_t)$, we have\n\\begin{align*}\n(\\mathrm{rad}_2(\\alpha_t))^2 &\\leq F_{\\alpha_t,2}^2(\\phi(\\alpha)) \\\\\n&= \\int_{\\ensuremath{\\mathbb{R}}^n} \\norm{x - \\phi(\\alpha)}^2 \\alpha_t(dx) \\\\\n&=(1-t)\\int_{\\ensuremath{\\mathbb{R}}^n} \\norm{x - \\phi(\\alpha)}^2 \\alpha(dx) + t \\int_{\\ensuremath{\\mathbb{R}}^n} \\norm{x - \\phi(\\alpha)}^2\\delta_{\\phi(\\alpha)}(dx) \\\\\n&= (1-t)\\int_{\\ensuremath{\\mathbb{R}}^n} \\norm{x - \\phi(\\alpha)}^2\\alpha(dx) \\\\\n&= (1-t) F_{\\alpha,2}^2(\\phi(\\alpha)) \\\\\n&\\leq (\\mathrm{rad}_2(\\alpha))^2.\n\\end{align*}\n\nWe now consider $\\mathrm{diam}_2$.\nRecall that $F_{\\alpha, 2}(\\phi(\\alpha))\\leq F_{\\alpha, 2}(x)$ for all $x\\in X$.\nFrom this, we have $(\\mathrm{diam}_2(\\alpha))^2 = \\int_X F^2_{\\alpha, 2}(x)\\,\\alpha(dx) \\ge F_{\\alpha, 2}^2(\\phi(\\alpha))$.\nTherefore\n\\begin{align*}\n\\big(\\mathrm{diam}_2(\\alpha_t)\\big)^2 & = \\iint_{\\ensuremath{\\mathbb{R}}^n\\times \\ensuremath{\\mathbb{R}}^n}\\norm{x - y}^2\\alpha_t(dx)\\ \\alpha_t(dy) \\\\\n& = (1-t)^2\\iint_{\\ensuremath{\\mathbb{R}}^n\\times \\ensuremath{\\mathbb{R}}^n}\\norm{x - y}^2 \\alpha(dx)\\,\\alpha(dy) \\\\\n& \\quad \\quad + 2t(1-t)\\iint_{\\ensuremath{\\mathbb{R}}^n\\times \\ensuremath{\\mathbb{R}}^n} \\norm{x - y}^2 \\alpha(dx)\\,\\delta_{\\phi(\\alpha)}(dy) \\\\\n& \\quad \\quad + t^2\\iint_{\\ensuremath{\\mathbb{R}}^n\\times \\ensuremath{\\mathbb{R}}^n}\\norm{x - y}^2\\delta_{\\phi(\\alpha)}(dx)\\,\\delta_{\\phi(\\alpha)}(dy) \\\\\n& = (1-t)^2 \\big(\\mathrm{diam}_2(\\alpha)\\big)^2 + 2t(1-t)\\, F_{\\alpha, 2}^2(\\phi(\\alpha)) \\\\\n& \\leq (1-t)^2\\big(\\mathrm{diam}_2(\\alpha)\\big)^2 +2t(1-t)\\,\\big(\\mathrm{diam}_2(\\alpha)\\big)^2 \\\\\n& = (1-t^2)\\big(\\mathrm{diam}_2(\\alpha)\\big)^2 \\\\\n& \\leq\\big(\\mathrm{diam}_2(\\alpha)\\big)^2.\n\\end{align*}\n\\end{proof}\n\n\\begin{thm}\\label{thm:vr2_Hausmann}\nLet $X$ be a bounded subset of $\\ensuremath{\\mathbb{R}}^n$ with positive reach $\\tau$.\nThen for all $0\\sqrt{2}$, the spaces $\\vrpp{2}{(\\ensuremath{\\mathbb{S}}^n,\\ell_2)}{r}$ and $\\cechpp{2}{(\\ensuremath{\\mathbb{S}}^n,\\ell_2)}{r}$ are contractible.\nBy restricting to finitely supported measures, we get the same result for $\\vrppfin{2}{(\\ensuremath{\\mathbb{S}}^n,\\ell_2)}{r}$ and $\\cechppfin{2}{(\\ensuremath{\\mathbb{S}}^n,\\ell_2)}{r}$.\n\\end{thm}\n\n\\begin{remark}\nA similar phenomenon is found for the Vietoris--Rips filtration of $(\\ensuremath{\\mathbb{S}}^n,\\ell_\\infty)$ where $\\vr{(\\ensuremath{\\mathbb{S}}^n,\\ell_\\infty)}{r}$ has the homotopy type of $\\ensuremath{\\mathbb{S}}^n$ when $r\\leq \\frac{2}{\\sqrt{n+1}}$ and becomes contractible when $r>\\frac{2}{\\sqrt{n+1}}$; see~\\cite[Section 7.3]{lim2020vietoris}.\nIn contrast, the homotopy types of the Vietoris--Rips and \\v{C}ech filtrations of geodesic circle include all possible odd-dimensional spheres~\\cite{AA-VRS1}.\nThe first new homotopy type of the $\\infty$-Vietoris--Rips metric thickening of the $n$-sphere (with either the geodesic or the Euclidean metric) is known~\\cite[Theorem~5.4]{AAF}, but only for a single (non-persistent) scale parameter, and only when using the convention $\\mathrm{diam}_\\infty(\\alpha)\\le r$ instead of $\\mathrm{diam}_\\infty(\\alpha) < r$.\n\\end{remark}\n\nWe note that Theorem~\\ref{thm:vr2_Hausmann} only implies that $\\vrpp{2}{(\\ensuremath{\\mathbb{S}}^n,\\ell_2)}{r}\\simeq \\ensuremath{\\mathbb{S}}^n$ for $r\\le\\tau(\\ensuremath{\\mathbb{S}}^n)=1$, whereas Theorem~\\ref{thm:vp2Euclidean} extends this range to $r\\le\\sqrt{2}$.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:vp2Euclidean}]\nFrom Lemma~\\ref{lem:diam2rad2Sphere} we know that $\\sqrt{2}$ is the maximal possible $2$-diameter on $\\ensuremath{\\mathcal{P}}_{(\\ensuremath{\\mathbb{S}}^n,\\ell_2)}$.\nTherefore, when $r> \\sqrt{2}$, the spaces $\\vrpp{2}{(\\ensuremath{\\mathbb{S}}^n,\\ell_2)}{r} = \\ensuremath{\\mathcal{P}}_{(\\ensuremath{\\mathbb{S}}^n,\\ell_2)}$ and $\\vrppfin{2}{(\\ensuremath{\\mathbb{S}}^n,\\ell_2)}{r} = \\ensuremath{\\mathcal{P}}^\\fin_{(\\ensuremath{\\mathbb{S}}^n,\\ell_2)}$ are both convex and hence contractible.\n\nWhen $00$ and any $\\xi> \\mathrm{spread}_\\infty(X) $, the space $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r}$ can be contracted inside of $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r+2L\\xi}$.\nIn particular, any homology class of $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r}$ will vanish in $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r+2L\\xi}$.\n\\end{prop}\n\n\\begin{proof}\nAs $\\xi > \\mathrm{spread}_\\infty(X)$, there is some $\\alpha_\\xi\\in \\ensuremath{\\mathcal{P}}^\\fin_X$ such that for all $\\alpha\\in \\ensuremath{\\mathcal{P}}_X$ we have $\\dWqq{\\infty}(\\alpha_\\xi, \\alpha)< \\xi$.\nConsider the linear homotopy\n\\[h_t\\colon [0,1]\\times \\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r}\\to \\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r+2L\\xi}\\]\ndefined by\n\\[(t, \\alpha)\\mapsto (1-t)\\,\\alpha + t\\, \\alpha_{\\xi}.\\]\nThis gives a homotopy between the inclusion from $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r}$ to $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r+2L\\xi}$ and a constant map, and therefore implies our conclusion so long as the homotopy is well-defined.\nIt then suffices to show $\\mathfrak{i}^X(h_t(\\alpha)) < r+ 2L\\xi$.\nThis comes from the stability condition of the invariant with respect to Wasserstein distances and from Lemma~\\ref{lem:bound-distance-convex-comb}:\n\\begin{equation*}\n\\mathfrak{i}^X(h_t(\\alpha)) \\leq \\mathfrak{i}^X(\\alpha) + 2L\\cdot\\dWqq{\\infty}(h_t(\\alpha), \\alpha) \\leq \\mathfrak{i}^X(\\alpha) + 2L\\cdot\\dWqq{\\infty}(\\alpha_\\xi, \\alpha) < r + 2L\\xi.\n\\end{equation*}\n\\end{proof}\n\n\\begin{remark}\nIt is not difficult to see that for the $\\mi_{q, p}$ invariants we can improve the bound $\\xi>\\mathrm{spread}_\\infty(X)$ to $\\xi>\\max\\{\\mathrm{spread}_p(X),\\mathrm{spread}_q(X)\\}$.\n\\end{remark}\n\nWe also have the following stronger contractibility conclusion for $\\vrp{X}{\\boldsymbol{\\cdot}}$ and $\\cechp{X}{\\boldsymbol{\\cdot}}$.\n\n\\begin{thm}\nLet $X$ be a bounded metric space.\nFor any $p\\in [1, \\infty]$ and any $r>\\mathrm{spread}_p(X)$, both $\\vrp{X}{r}$ and $\\cechp{X}{r}$ are contractible.\n\\end{thm}\n\n\\begin{proof}\nAs $r > \\mathrm{spread}_p(X)$, there is some $\\alpha_r\\in \\ensuremath{\\mathcal{P}}^\\fin_X$ such that for all $\\alpha\\in \\ensuremath{\\mathcal{P}}_X$ we have $\\ensuremath{d_{\\mathrm{W},p}^X}(\\alpha_r, \\alpha)< r$.\nIn particular, this implies $\\mathrm{diam}_p(\\alpha_r) = (\\int_X F_{\\alpha_r,p}^p(x)\\,\\alpha_r(dx))^{1\/p}< r$ and $\\mathrm{rad}_p(\\alpha_r) = \\inf_{x\\in X}F_{\\alpha_r,p}(x) < r$.\nNow, let $\\mathfrak{i}$ be either $\\mathrm{diam}_p$ or $\\mathrm{rad}_p$.\nConsider the linear homotopy\n\\[h_t\\colon [0,1]\\times \\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r}\\to \\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r} \\]\ndefined by\n\\[(t, \\alpha)\\mapsto (1-t)\\,\\alpha + t\\, \\alpha_{r}.\\]\nIt then suffices to show for all $t\\in [0, 1]$, we have $\\mathrm{diam}_p(h_t(\\alpha)) < r$ and $\\mathrm{rad}_p(h_t(\\alpha))< r $, so that this linear homotopy from the identity map on $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r}$ to the constant map to $\\alpha_r$ is well-defined.\n\nIn the $\\mathrm{diam}_p$ case, we have\n\\begin{align*}\n\\mathrm{diam}_p^p(h_t(\\alpha)) & =(1-t)^2\\iint_{X\\times X}d^p_X(x, x') \\alpha(dx)\\,\\alpha(dx') + 2t\\,(1-t)\\iint_{X\\times X} d^p_X(x, x') \\alpha_r(dx)\\,\\alpha(dx') \\\\\n& \\quad \\quad \\quad + t^2\\iint_{X\\times X}d^p_X(x, x') \\alpha_r(dx)\\,\\alpha_r(dx') \\\\\n& = (1-t)^2 \\mathrm{diam}_p^p(\\alpha) + 2t\\,(1-t)\\int_X \\left(\\ensuremath{d_{\\mathrm{W},p}^X}(\\alpha_r, \\delta_{x'})\\right)^p\\,\\alpha(dx') \\\\\n& \\quad \\quad \\quad + t^2\\int_X\\left(\\ensuremath{d_{\\mathrm{W},p}^X}(\\alpha_r, \\delta_{x'})\\right)^p\\,\\alpha_r(dx') \\\\\n& < (1-t)^2 r^p+ 2t\\,(1-t)r^p +t^2r^p \\\\\n& =r^p.\n\\end{align*}\n\nIn the $\\mathrm{rad}_p$ case, as $\\mathrm{rad}_p(\\alpha) \\max(\\mathfrak{i}^X)$, we have $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r}=\\ensuremath{\\mathcal{P}}_X$, and therefore $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r}$ is contractible.\nInspired by the definition of the spread, we have the following result.\n\n\\begin{lem}\\label{lem:bound_barcode_via_metric_spread}\nLet $X$ be a metric space, let $\\mathfrak{i}$ be an $L$-controlled invariant, let $U$ be a finite subset of $X$.\nThen for any $\\xi > \\max\\left( d_\\mathrm{H}(U, X),\\, \\frac{\\max(\\mathfrak{i}^U) - r}{2L}\\right)$, the space $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r}$ is contractible in $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r+2L\\xi}$.\nIn particular, any homology class of $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r}$ will vanish in $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r+2L\\xi}$.\n\\end{lem}\n\n\\begin{proof}\nAs $\\xi > d_\\mathrm{H}(U, X)$, the balls $\\{B(u; \\xi)\\}_{u\\in U}$ form a open covering of $X$.\nWe choose a partition of unity subordinate to the covering and build the map $\\Phi_U \\colon \\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r} \\to \\ensuremath{\\mathcal{P}}_U$ as in Lemma~\\ref{lem:Ptau_cts_via_partition}.\nSince $r + 2L\\,\\xi> \\max(\\mathfrak{i}^U)$, we know $\\ensuremath{\\mathcal{P}}_U = \\filt{\\ensuremath{\\mathcal{P}}_U}{\\mathfrak{i}^U}{r+2L\\xi}$.\nSince the inclusion map $\\iota_U \\colon U\\to X$ is of zero distortion, we have $\\mathfrak{i}^X((\\iota_U)_\\sharp(\\beta))\\leq \\mathfrak{i}^U(\\beta) \\leq \\max(\\mathfrak{i}^U)$ for all $\\beta \\in \\ensuremath{\\mathcal{P}}_U$.\nThis implies that the contractible set $\\ensuremath{\\mathcal{P}}_U$ is inside $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r+2L\\xi}$, and so we have the following diagram.\n\\begin{equation*}\n\\xymatrix{\n\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r} \\ar[rrd]_{\\Phi_U} \\ar[rr]^{\\nu^X_{r, r + 2L\\,\\xi}}\n&& \\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r+2L\\xi}\\\\\n&&\\ensuremath{\\mathcal{P}}_U\\ar@{^{(}->}[u]_{\\iota_U}}\n\\end{equation*}\nAs the image of $\\iota_U\\circ \\Phi_U$ maps $\\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r}$ into a contractible subset $\\ensuremath{\\mathcal{P}}_U\\subset \\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r+2L\\xi}$, it is homotopy equivalent to a constant map.\nTo obtain this conclusion, it thus suffices to show that $\\iota_U\\circ \\Phi_U$ is homotopy equivalent to the inclusion given by the structure map $\\nu^X_{r, r+ 2L\\, \\xi}$.\nLet's consider the linear homotopy\n\\[h_t\\colon [0,1]\\times \\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r}\\to \\filt{\\ensuremath{\\mathcal{P}}_X}{\\mathfrak{i}^X}{r+2L\\xi}\\]\ndefined by\n\\[(t, \\alpha)\\mapsto (1-t)\\,\\alpha + t\\, \\Phi_U(\\alpha).\\]\nFrom the stability property of $\\mathfrak{i}$, Lemma~\\ref{lem:bound-distance-convex-comb}, and the estimate in Lemma~\\ref{lem:Ptau_cts_via_partition}, we have\n\\begin{align*}\n \\mathfrak{i}^X(h_t(\\alpha)) & \\leq \\mathfrak{i}^X(\\alpha) + 2L\\cdot \\dWqq{\\infty}(\\alpha, h_t(\\alpha))\\\\\n & \\leq \\mathfrak{i}^X(\\alpha) + 2L\\cdot \\dWqq{\\infty}(\\alpha, \\Phi_U(\\alpha)) \\\\\n & \\leq \\mathfrak{i}^X(\\alpha) +2L\\,\\xi \\\\\n & < r + 2L\\,\\xi.\\end{align*}\nThis shows that the homotopy $h_t$ from $\\iota_U\\circ \\Phi_U$ to $\\nu^X_{r, r+ 2L\\, \\xi}$ is well-defined.\n\\end{proof}\n\n\\begin{remark}\nFor the 1-controlled $\\mathfrak{i}_{q, p}$ invariant on a bounded metric space $X$, the maximum of $\\mathfrak{i}_{q, p}^X$ is bounded by $\\mathrm{diam}(X)$.\nTherefore, for any finite subset $U$ we have\n\\[\n \\max\\left(d_\\mathrm{H}(U, X), \\mathrm{diam}(U)\\right) \\geq \\max\\left( d_\\mathrm{H}(U, X),\\, \\frac{\\max(\\mathfrak{i}_{q, p}^U) - r}{2}\\right).\n\\]\nHence, the previous lemma implies that the lifetime of features in $\\filtf{\\ensuremath{\\mathcal{P}}_X}{\\mi_{q, p}^X}$ is bounded by $2\\,\\mathrm{spread}(X)$.\nIn the $\\mathrm{diam}_p$ case, the factor $2$ can be removed, as we show next, and therefore matches the result in~\\cite[Proposition~9.6]{lim2020vietoris} for Vietoris--Rips simplicial complexes.\n\\end{remark}\n\n\\begin{prop}\nFor any $r>0$, $p\\in[1, \\infty]$, and $\\xi> \\mathrm{spread}(X)$, the space $\\vrp{X}{r}$ is contractible in $\\vrp{X}{r+\\xi}$.\nIn particular, any homology class on $\\vrp{X}{r}$ will vanish in $\\vrp{X}{r+\\xi}$.\n\\end{prop}\n\n\\begin{proof}\nAs $\\xi > \\mathrm{spread}(X)$, there exists some $U\\subset X$ such that $\\mathrm{diam}(U)< \\xi$ and $d_\\mathrm{H}(U, X)< \\xi$.\nAs the maximum of $\\mathrm{diam}_p^U$ on $\\ensuremath{\\mathcal{P}}_U$ is bounded by $\\mathrm{diam}(U)$, the contractible subset $\\ensuremath{\\mathcal{P}}_U$ lies inside $\\vrp{X}{r+\\xi}$, and therefore $\\iota_U\\circ \\Phi_U$ is homotopy equivalent to a constant map.\nLet $h_t$ be the linear homotopy used in Lemma~\\ref{lem:bound_barcode_via_metric_spread}.\nWhat is left to get a homotopy equivalence between $\\iota_U\\circ \\Phi_U$ and $\\nu^X_{r, r+ \\xi}$ is to show $\\mathrm{diam}_p(h_t(\\alpha)) < r + \\xi$.\nThis comes from the following calculation:\n\\begin{align*}\n&\\mathrm{diam}_p^p(h_t(\\alpha)) \\\\\n=\\, &(1-t)^2\\iint_{X\\times X}d^p(x, x') \\alpha(dx)\\,\\alpha(dx') + 2t\\,(1-t)\\iint_{X\\times X} d^p(x, x') \\alpha(dx)\\,\\Phi_U(\\alpha)(dx')\\\\\n&\\quad \\quad \\quad + t^2\\iint_{X\\times X}d^p(x, x') \\Phi_U(\\alpha)(dx)\\,\\Phi_U(\\alpha)(dx')\\\\\n=\\, & (1-t)^2 \\mathrm{diam}_p^p(\\alpha) + 2t\\,(1-t)\\int_X \\left(\\ensuremath{d_{\\mathrm{W},p}^X}(\\alpha, \\delta_{x'})\\right)^p\\,\\Phi_U(\\alpha)(dx')+ t^2\\mathrm{diam}_p^p(\\Phi_U(\\alpha))\\\\\n\\leq\\, & (1-t)^2 \\mathrm{diam}_p^p(\\alpha) + 2t\\,(1-t)\\int_X \\left(\\ensuremath{d_{\\mathrm{W},p}^X}(\\alpha, \\Phi_U(\\alpha)) + \\ensuremath{d_{\\mathrm{W},p}^X}(\\Phi_U(\\alpha), \\delta_{x'})\\right)^p\\,\\Phi_U(\\alpha)(dx')\\\\\n&\\quad \\quad \\quad + t^2\\mathrm{diam}_p^p(\\Phi_U(\\alpha))\\\\\n\\leq\\, & (1-t)^2 \\mathrm{diam}_p^p(\\alpha) + 2t\\,(1-t)\\big(\\xi + \\mathrm{diam}_p(\\Phi_U(\\alpha))\\big)^p + t^2\\mathrm{diam}_p^p(\\Phi_U(\\alpha))\\\\\n<\\, & (1-t)^2 r^p+ 2t(1-t)(r+ \\xi)^p +t^2\\xi^p\\\\\n<\\, & (r+ \\xi)^p.\n\\end{align*}\n\\end{proof}\n\nThe following result shows the lifetime of features of $\\cechp{X}{\\boldsymbol{\\cdot}}$ is also bounded by the metric spread of $X$, $\\mathrm{spread}(X)$.\n\n\\begin{prop}\nFor any $r>0$, $p\\in[1, \\infty]$, and $\\xi> \\mathrm{spread}(X)$, the space $\\cechp{X}{r}$ is contractible in $\\cechp{X}{r+\\xi}$.\nIn particular, any homology class on $\\cechp{X}{r}$ will vanish in $\\cechp{X}{r+\\xi}$.\n\\end{prop}\n\n\\begin{proof}\nAs $\\xi > \\mathrm{spread}(X)$, there exists some $U\\subset X$ such that $\\mathrm{diam}(U)< \\xi$ and $d_\\mathrm{H}(U, X)< \\xi$.\nThe maximum of $\\mathrm{rad}_p^U$ on $\\ensuremath{\\mathcal{P}}_U$ is bounded by $\\mathrm{diam}(U)$, and the contractible subset $\\ensuremath{\\mathcal{P}}_U$ lies inside $\\cechp{X}{r+\\xi}$.\nThis shows $\\iota_U\\circ \\Phi_U$ is homotopy equivalent to a constant map.\nLet $h_t = (1-t)\\,\\alpha + t\\, \\Phi_U(\\alpha)$ be the linear homotopy used in Lemma~\\ref{lem:bound_barcode_via_metric_spread}.\nWhat is left is to show is $\\mathrm{rad}_p(h_t(\\alpha)) < r + \\xi$.\nBy Lemma~\\ref{lem:stability_of_rad_p} and Lemma~\\ref{lem:bound-distance-convex-comb}, we have\n\\begin{align*}\n\\mathrm{rad}_p(h_t(\\alpha)) & \\leq \\mathrm{rad}_p(\\alpha) + \\dWqq{p}(\\alpha, h_t(\\alpha)) \\\\\n&\\leq r + t^\\frac{1}{p} \\dWqq{p}(\\alpha, \\Phi_U(\\alpha)) \\\\\n&\\leq r + \\xi.\n\\end{align*}\n\\end{proof}\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nFiltrations, i.e.\\ increasing sequences of spaces, play a foundational role in applied and computational topology, as they are the input to persistent homology.\nTo produce a filtration from a metric space $X$, one often considers a Vietoris--Rips or \\v{C}ech simplicial complex, with $X$ as its vertex set, as the scale parameter increases.\nSince a point in the geometric realization of a simplicial complex is a convex combination of the vertices of the simplex $[x_0,x_1,\\ldots,x_k]$ in which it lies, each such point can alternatively be identified with a probability measure: a convex combination of Dirac delta masses $\\delta_{x_0}$, $\\delta_{x_1}$, \\ldots, $\\delta_{x_k}$.\nWe can therefore re-interpret the Vietoris--Rips and \\v{C}ech simplicial complex filtrations instead as filtrations in the space of probability measures, which are referred to as the Vietoris--Rips and \\v{C}ech \\emph{metric thickenings}.\nIn~\\cite{AAF} it is argued that the metric thickenings have nicer properties for some purposes: for example, the inclusion from metric space $X$ into the metric thickening is always an isometry onto its image, whereas an inclusion from metric space $X$ into the vertex set of a simplicial complex is not even continuous unless $X$ is discrete.\nIn this paper, we prove that these two perspectives are compatible: the $\\infty$-Vietoris--Rips (resp.\\ \\v{C}ech) metric thickening filtration has the same persistent homology as the Vietoris--Rips (resp.\\ \\v{C}ech) simplicial complex filtration when $X$ is totally bounded.\nTherefore, when analyzing these filtrations, one can choose to apply either simplicial techniques (simplicial homology, simplicial collapses, discrete Morse theory) or measure-theoretic techniques (optimal transport, Karcher or Fr\\'{e}chet means), whichever is more convenient for the task at hand.\n\nThe measure-theoretic perspective motivates new filtrations to build on top of a metric space $X$.\nThough the Vietoris--Rips simplicial complex filtration is closely related (at interleaving distance zero) to the metric thickening filtration obtained by looking at sublevelsets in the space of probability measures of the $\\infty$-diameter functional, one can instead consider sublevelsets of the $p$-diameter functional for any $1\\le p\\le \\infty$.\nThe same is true upon replacing Vietoris--Rips with \\v{C}ech and replacing $p$-diameter with $p$-radius.\nThese relaxed $p$-Vietoris--Rips and $p$-\\v{C}ech metric thickenings enjoy the same stability results underlying the use of persistent homology: nearby metric spaces produce nearby persistence modules.\nThe generalization to $p<\\infty$ is a useful one: though determining the homotopy types of $\\infty$-Vietoris--Rips thickenings of $n$-spheres is a hard open problem, we give a complete description of the homotopy types of $2$-Vietoris--Rips thickenings of $n$-spheres for all $n$.\nWe also prove a Hausmann-type theorem in the case $p=2$, and ask if the $p<\\infty$ metric thickenings may be amenable to study using tools from Morse theory.\n\nMore generally, one can consider sublevelsets of any $L$-controlled function on the space of probability measures on $X$.\nWe prove stability in this much more general context.\nThis allows one to consider metric thickenings that are tuned to a particular task, perhaps incorporating other geometric notions besides just proximity, such as curvature, centrality, eccentricity, etc.\nOne can design an $L$-controlled functional to highlight specific features that may be useful for a particular data science task.\n\nWe hope these contributions inspire more work on metric thickenings and their relaxations.\nWe end with some open questions.\n\n\\begin{enumerate}\n\n\\item Is there an analogue to the Hausmann-type Theorem~\\ref{thm:vr2_Hausmann} which holds for $p\\in(2,\\infty)$? The case $p=\\infty$ was tackled in~\\cite{AM}.\nIn a similar spirit, it seems interesting to explore whether analogous theorems hold when the ambient space is a more general Hadamard space instead of $\\ensuremath{\\mathbb{R}}^d$.\n\n\\item Can one prove Latschev-type theorems~\\cite{Latschev2001} for $p$-metric thickenings?\n\n\\item For $p\\neq 2$, what are the homotopy types of $p$-Vietoris--Rips and $p$-\\v{C}ech thickenings of spheres at all scales?\nIs the homotopy connectivity a non-decreasing function of the scale, and if so, how quickly does the homotopy connectivity increase?\n\n\\item What are the homotopy types of $p$-Vietoris--Rips and $p$-\\v{C}ech metric thickenings of other manifolds, such as ellipses (see~\\cite{AAR}), ellipsoids, tori, and projective spaces~\\cite{katz1983filling,katz1983filling,katz1991rational,AdamsHeimPeterson}?\n\n\\item What versions of Morse theory~\\cite{milnor1963morse} can be developed in order to analyze the homotopy types of $p$-metric thickenings of manifolds as the scale increases?\nSee Appendix~\\ref{app:Morse} for some initial ideas in the case of $p$-\\v{C}ech thickenings.\nFor homogeneous spaces such as spheres, versions of Morse--Bott theory~\\cite{bott1954nondegenerate,bott1982lectures,bott1988morse} may be needed.\n\n\\item For $X$ finite, is $\\vrp{X}{r}$ always homotopy equivalent to a subcomplex of the complete simplex on the vertex set $X$?\nSee Appendix~\\ref{app:finite-Cech} for a proof of the \\v{C}ech case.\n\n\\item For $X$ finite with $n+1$ points, the space $\\ensuremath{\\mathcal{P}}_X$ is a $n$-simplex in $\\ensuremath{\\mathbb{R}}^{n+1}$ where coordinates are the weights of a measure at each point.\nIn this case, $\\mathrm{diam}_p$ is a quadratic polynomial on $\\ensuremath{\\mathbb{R}}^{n+1}$ and $\\mathrm{rad}_p$ is the minimum of $n+1$ linear equations.\nTherefore, both $\\vrp{X}{r}$ and $\\cechp{X}{r}$ are semi-algebraic sets in $\\ensuremath{\\mathbb{R}}^{n+1}$.\nCan one use linear programming along with the results of Appendix~\\ref{app:finite-Cech} to calculate the homology groups of $\\cech{X}{r}$, and the work on quadratic semi-algebraic sets~\\cite{basu2008computing, basu2008computing2, burgisser2018computing} to calculate the homology groups of $\\vrp{X}{r}$?\n\n\\end{enumerate}\n\n\n\n\\subsection*{Acknowledgements}\n\nHA would like to thank Florian Frick and \\v{Z}iga Virk and FM would like to thank Sunhyuk Lim for helpful conversations. \n\nWe acknowledge funding from these sources: NSF-DMS-1723003, NSF-CCF-1740761 and NSF-CCF-1839358.\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nHeavy quarkonium is a meson formed by two heavy quarks whose mass $m_Q$ is much bigger than $\\Lambda_{QCD}$. They are quite different to other mesons. The first difference is that a lot of the physics relevant to study this system happens at a energy scale $m_Q$ where perturbation theory is applicable. For example the typical size of the more deeply bounded quarkonium states is of order $1\/(m_Q v)$, where $v$ is the velocity of the heavy quarks around the center of mass, while for other mesons the size is of order $1\/\\Lambda_{QCD}$. A second difference is that heavy quarkonium is a non-relativistic system, meaning that $v\\ll 1$. For this reason it is expectable that at some accuracy it can be described by a Schr\\\"{o}dinger equation with some potential, hence we can say that it is the QCD analogous to the hydrogen atom. On the other hand this small $v$ can spoil naive perturbation theory in a way that will be discussed later.\n\nThe original idea of using quarkonium suppression as a probe of deconfinement in heavy ion collisions is found in \\cite{Matsui:1986dk}. Since then this phenomena has been studied experimentally in many facilities, as for example SPS, RHIC and LHC. In the future the CBM experiment will be able to probe the situation when the quark-gluon plasma has a large chemical potential. Even though quarkonium suppression is a well established experimental fact an understanding of the responsible mechanism and a quantitative description of experimental data is still missing. \n\nThe dissociation mechanism that was originally considered in \\cite{Matsui:1986dk} was color screening. In order to illustrate this mechanism we can make use of what happens in the perturbative limit of QCD. There the potential between two infinitely heavy color charges in the vacuum will have the form of a Coulomb potential $V(r)=-\\frac{\\alpha}{r}$. Then the two heavy charges will feel an attractive force. If instead of the vacuum we are in a medium in which color screening takes place then we will have instead of a Coulomb a Yukawa potential $V(r)=-\\frac{\\alpha}{r}e^{-m_D r}$ where $m_D$ is a Debye mass that will encode the information about the medium. If the two charges are separated by a distance that is much bigger than $1\/m_D$ they are not going to see each other so in an effective way they will behave like individual particles. \n\nHowever another mechanism for dissociation might also exists. The heavy quarkonium state will have some probability to be hit by some of the particles in the medium and this hit might induce a decay. This means that the quarkonium state might still exist but with a very short mean life. Indeed in a perturbative computation in \\cite{Laine:2006ns} it was shown that the correlator of the heavy quark part of the electromagnetic current fulfils a Schr\\\"{o}dinger equation with a potential that has an imaginary part. This imaginary part was bigger than the real part for the situation in which screening is important and hence it indicates that in the perturbative limit the decay width makes the different quarkonium states dissociate before screening becomes relevant.\n\nIn the previous discussion we have been talking about potentials and the Schr\\\"{o}dinger equation without talking about how they can be obtained from QCD. The question can be rephrased in the following way. Is there a controlled expansion in which we can obtain a Schr\\\"{o}dinger equation as a leading order approximation and we can estimate the error done by making such approximation? A way to answer this question that has been very successful in the vacuum is by using effective field theories (EFTs). This program for finite temperature $T$ and zero chemical potential $\\mu$ has been started in the last years \\cite{Escobedo:2008sy,Brambilla:2008cx}. Using these techniques the potential of \\cite{Laine:2006ns} has been confirmed for $T\\gg m_D\\sim 1\/r$. The case $1\/r\\gg T\\gg E\\gg m_D$ was studied in \\cite{Brambilla:2010vq} where corrections to decay width that can not be encoded in a potential were found. This result was later compared with lattice computations \\cite{Aarts:2011sm} yielding similar results. The case $T\\sim 1\/r$ was studied in muonic hydrogen \\cite{Escobedo:2010tu}. In this work we are going to focus on the recent investigations that have been done to translate the previous results in terms of cross sections \\cite{Brambilla:2011sg,Brambilla:2013dpa}. Opposite to what happens to the decay width these cross sections do not depend on the nature of the medium as long as it is isotropic and homogeneous, so they can be directly applied to mediums with a finite chemical potential as the one that is expected to be formed in CBM.\n\nThis work is organized as follows. In section \\ref{sec:nreft} we review non-relativistic EFTs, in section \\ref{sec:gluo} we discuss the process responsible of the decay of quarkonium in the medium at low densities and temperatures that is called gluo-dissociation, in section \\ref{sec:ine} we discuss the process that dominates at high densities and temperatures, inelastic parton scattering. Finally, section \\ref{sec:concl} is devoted to the conclusions.\n\\section{Non-relativistic EFTs for heavy quarkonium}\n\\label{sec:nreft}\nThe general idea of EFTs was introduced in \\cite{Weinberg:1978kz}. An EFT is a field theory which gives exactly the same result as a more general one but on a limited kinematic regime. They are useful for studying problems in which different energy scales are important. In most QCD processes $\\Lambda_{QCD}$ is an important scale but in heavy quarkonium we also have another important scale $m_Q$ that fulfils $m_Q\\gg \\Lambda_{QCD}$. Moreover, because heavy quarkonium is a non-relativistic system, there appears other energy scales as the inverse of the typical radius $1\/r\\sim m_Q v$ and $E\\sim m_Q v^2$ the binding energy. \n\nNon-relativistic QCD (NRQCD) \\cite{Caswell:1985ui} is an EFT in which modes with a virtuality of order $m_Q^2$ are integrated out. The fact that heavy quarks are heavy ensures that NRQCD can be derived from QCD using perturbation theory. In this EFT the heavy quark instead of being represented by a bispinor field is represented by two spinor fields, one for the heavy quark and another for the heavy anti-quark. Approximate symmetries as the spin symmetry can already be seen by looking at the NRQCD lagrangian. \n\nPotential NRQCD (pNRQCD) \\cite{Pineda:1997bj} is an EFT in which modes with a virtuality of order $1\/r^2$ are integrated out. In this case pNRQCD can be derived from NRQCD using perturbation theory only for states whose size is smaller than $1\/\\Lambda_{QCD}$. This is expected to happen for the more deeply bound states as $\\Upsilon(1S)$ with a Bohr radius of order $1\/a_0\\sim 1355 \\,\\texttt{MeV}$ or $J\/\\Psi$ with $1\/a_0\\sim 770\\,\\texttt{MeV}$ (both values are taken from \\cite{pinedathesis}). In the static limit agreement between high orders computations in pNRQCD and the lattice can be found up to distances of order $0.2 \\,\\textit{fm}$ and in fact it can be used to make a competitive determination of $\\alpha_s$ \\cite{Bazavov:2012ka}. Another important issue when doing computations is how $\\Lambda_{QCD}$ is related with the binding energy $E$. Through this work we are going to assume that $E\\gg \\Lambda_{QCD}$ when computations at this scale are needed. This might be reasonable for $\\Upsilon(1S)$ and $J\/\\Psi$ where $E\\sim 350\\,\\texttt{MeV}$ \\cite{pinedathesis}.\n\nIn pNRQCD when $1\/r\\gg \\Lambda_{QCD}$ there exist two kind of fields that represent heavy quark and heavy antiquarks pairs, a singlet field and an octet field. At leading order they both fulfil a Schr\\\"{o}dinger equation with a potential that encodes physics that is integrated out when going from NRQCD to pNRQCD. \n\nThe main advantage of using EFTs is that one can define a power counting such that the size of each contribution can be predicted just looking at the Lagrangian. As an example let us look in the computation of the binding energy of a given state in perturbation theory. If this is done directly in QCD we will find an expansion of the following type \n\\begin{equation}\nE=m_Q\\alpha_s\\sum_{n=0}^\\infty\\alpha_s^nA_n(v) \n\\end{equation}\nwhere $A_n$ are coefficients that in general are going to depend on ratios of the different scale that appear in the problem and therefore on $v$. There is nothing that forbids that a given $A_n$ goes to infinity as $v\\to 0$. If this happens we lost control of our perturbative expansion. This is indeed the case of some diagrams that appear in heavy quarkonium physics as for example the box diagram that is enhanced by a $m_Q r$ power. If the same computation is done using pNRQCD the series can be reorganized as\n\\begin{equation}\nE=m_Q\\alpha_s v^2\\sum_{n,m}\\alpha_s^n v^m B_{n,m} \n\\end{equation}\nwhere now we know that $B(n,m)$ will be of order $1$. \n\nWith the aim of having a way to correctly define the potential and the Schr\\\"{o}dinger equation and to have a well defined power counting non-relativistic EFTs were extended to finite temperature. In general the presence of a medium introduces another set of energy scales in the problem and the physics is going to be different depending on the relation with the scales that define the bound state in the vacuum. In a perturbative medium, as the one we are going to consider, the induced scales are the following. There is a scale $p_{Hard}$ which corresponds to the typical energy of the partons in the medium. This energy scale might be integrated out to yield an EFT called Hard Dense Loop (HDL) (Hard Thermal Loop for $\\mu=0$) \\cite{Braaten:1989mz}. Another important scale is the so-called Debye mass $m_D$ that corresponds to the inverse of the distance in which screening effects become important \n\\begin{equation}\nm_D^2=\\frac{N_c g^2 T^2}{3}+\\frac{g^2 T_F N_F}{3}\\left(T^2+\\frac{3\\mu^2}{\\pi^2}\\right)\n\\end{equation}\n\nDepending on the relation between $p_{Hard}$ and the rest of scales the dominant mechanism responsible for the decay width is going to be different. It happens that the key relation is that of $m_D$ with the binding energy. If $E\\gg m_D$ the dominant mechanism is gluo-dissociation while if $m_D\\gg E$ it is inelastic parton scattering. This conclusion can be easily obtained using pNRQCD power counting \\cite{Brambilla:2008cx,Escobedo:2010tu}. In the next two sections we are going to discuss these two processes in detail.\n\\section{Gluo-dissociation}\n\\label{sec:gluo}\nWhen $E\\gg m_D$ the decay width is dominated by the imaginary part of the leading order singlet self-energy. In the case of zero chemical potential this was computed for $T\\gg E$ in \\cite{Brambilla:2010vq}\n\\begin{equation}\n\\delta\\Gamma_n=\\frac{1}{3}N_C^2C_F\\alpha_{\\mathrm{s}}^3T-\\frac{16}{3m}C_F\\alpha_{\\mathrm{s}} TE_n+\\frac{4}{3}N_CC_F\\alpha_{\\mathrm{s}}^2T\\frac{2}{mn^2a_0} \n\\end{equation}\nand for $T\\sim E$ in \\cite{Escobedo:2008sy}\n\\begin{equation}\n\\delta\\Gamma_n=\\frac{4}{3}\\alpha_{\\mathrm{s}} C_FT\\langle n|r_i\\frac{|E_n-h_o|^3}{e^{\\beta |E_n-h_o|}-1}r_i|n\\rangle \n\\end{equation}\n\nAnother way to compute this decay width that has been used in the past is to compute the cross section for this process at $T=0$ and later convolute it with a partonic distribution function. There are two important questions to discuss. If this convolution is justified by perturbative thermal field theory and how the cross section obtained in pNRQCD relates with the one commonly used in phenomenological analyses. The cross section for gluo-dissociation was computed in \\cite{Bhanot:1979vb}. They used an OPE based on the multipole expansion. This expansion is also present in pNRQCD Lagrangian so in this sense the result has to be similar. The difference comes from the fact that they use the large $N_c$ limit while this approximation was not needed to compute the decay width in pNRQCD.\n\nIn order to extract the cross section from our previous computation of the decay width we use the cutting rules that were generalized to a thermal medium in \\cite{Kobes:1986za}. For the case of gluo-dissociation in fact the naively expected convolution is obtained \n\\begin{equation}\n\\Gamma=\\int\\frac{\\,d^3k}{(2\\pi)^3}f(k)\\sigma(k) \\nonumber\n\\end{equation}\nwith a cross section for the $1S$ state\n\\begin{equation}\n\\sigma_{1S}(k)=\\frac{\\alpha_{\\mathrm{s}} C_F}{3} 2^{10} \\pi^2 \\rho (\\rho +2)^2 \\frac{E_1^{4}}{mk^5}\n\\left(t(k)^2+\\rho ^2 \\right)\\frac{\\exp\\left(\\frac{4 \\rho}{t(k)} \\arctan\n\\left(t(k)\\right)\\right)}{ e^{\\frac{2 \\pi \\rho}{t(k)} }-1}\\,,\n\\label{crossEFT}\n\\end{equation}\nwhere $t(k)\\equiv\\sqrt{k\/\\vert E_1\\vert-1}$ and $\\rho=\\frac{1}{N_c^2-1}$. This result was also found in \\cite{Brezinski:2011ju} using different techniques. More details about gluo-dissociation in pNRQCD can be found in \\cite{Brambilla:2011sg}.\n\\section{Inelastic parton scattering}\n\\label{sec:ine}\nThis process dominates the decay width for $m_D\\gg E$. Here we are going to focus in the regime $p_{Hard}\\gg 1\/r\\sim m_D$ that is the more relevant one for dissociation. The process we are studying corresponds to $HQ+p\\to Q+\\bar{Q}+p$ which means that a heavy quarkonium $HQ$ is hit by a parton $p$ and goes into two heavy quarks $Q$ plus a parton. An approximation that is used in phenomenological analyses \\cite{Zhao:2010nk} is $\\sigma(HQ+p\\to Q+\\bar{Q}+p)=2\\sigma(Q+p\\to Q+p)$ (quasi-free approximation). In the right hand side the cross section computed in \\cite{Combridge:1978kx} is used. This approach neglects the interference terms that arise in the interaction of the heavy quark and the heavy antiquark. In the case in which the thermal corrections to the potential can be considered a perturbation one can use the cutting rules to extract a cross section that will not make this approximation. Doing this we notice two things. First that the naive convolution formula can not be applied in this case because we have a parton in the final state\n\\begin{equation}\n\\Gamma=\\int\\frac{\\,d^3k}{(2\\pi)^3}f(k)(1+f(k))\\sigma(k,m_D) \n\\end{equation}\nand second that $\\sigma(k,m_D)$ depends now on the medium through $m_D$ that acts as a thermal mass. For $1S$ states we find \n\\begin{equation}\n\\sigma_{1S}(k,m_D)=8\\pi C_F\\alpha_{\\mathrm{s}}^2N_Fa_0^2\\left(-\\frac{3}{2}+2\\log\\left(\\frac{2}{x}\\right)+\\log\\left(\\frac{y^2}{1+y^2}\\right)-\\frac{1}{y^2}\\log(1+y^2)\\right)\n\\end{equation}\nwhere $a_0$ is the Coulombic Bohr radius, $x=m_Da_0$ and $y=ka_0$. In conclusion, the EFT computation takes into account correctly the effects of Pauli blocking and Bose enhancement and improves on the quasi-free approximation. More details about inelastic parton scattering in EFT can be found in \\cite{Brambilla:2013dpa}\n\\section{Conclusions}\n\\label{sec:concl}\nEFTs have been used to compute a wide range of temperature regimes in the case of small chemical potential. This research has shown the importance of the imaginary part of the potential and the decay width for dissociation. In this work we discussed the physical process behind them in different regimes. For gluo-dissociation we found that the pNRQCD improved the previously known cross-section by going beyond the large $N_c$ approximation. For inelastic parton scattering it can be shown that the imaginary part of the potential improves over the quasi-free approximation in taking into account the interference terms.\n\\section*{Acknowledgements}\nI acknowledge Nora Brambilla, Jacopo Ghiglieri and Antonio Vairo for collaboration in the two papers in which this work is based. I also acknowledge financial support from the DFG project BR4058\/1-1.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{s:intro}\n\nAbout 80 years of experimental observations and theoretical arguments have pointed out \nthat a large fraction of the Universe is composed by Dark Matter particles\n\\footnote{For completeness, it is worth recalling that some efforts to find alternative explanations to Dark Matter have been proposed\nsuch as $MOdified$ $Gravity$ $Theory$ (MOG) and $MOdified$ $Newtonian$ \n$Dynamics$ (MOND); they hypothesize that the theory of gravity is incomplete and that a new gravitational theory might explain\nthe experimental observations.\nMOND modifies the law of motion for very small accelerations, while MOG modifies the Einstein's theory of\ngravitation to account for an hypothetical fifth fundamental force in addition to the gravitational, electromagnetic,\nstrong and weak ones. However, e.g.: i) there is no general underlying principle; ii)\nthey are generally unable to account for all small and large scale \nobservations; iii) they fail to reproduce accurately the Bullet Cluster; iv) \ngenerally they require some amount of DM particles as seeds for the structure formation.}.\n\nThe presently running DAMA\/LIBRA ($\\simeq$ 250 kg of full sensitive target-mass)\n\\cite{perflibra,modlibra,bot11,pmts,mu,review,papep,diu2014,norole} experiment,\nas well as the former DAMA\\-\/NaI ($\\simeq$ 100 kg of full sensitive target-mass) \\cite{allDM1,allDM,Nim98,RNC,ijma,chan,allRare},\nhas the main aim to investigate the presence of DM particles in the galactic halo by exploiting\nthe model independent DM annual modulation signature (originally suggested in Ref.~\\cite{Freese}).\n\nAs a consequence of the Earth's revolution around the Sun,\nwhich is moving in the Galaxy with respect to the Local Standard of\nRest towards the star Vega near\nthe constellation of Hercules, the Earth should be crossed\nby a larger flux of DM particles around $\\simeq$ 2 June\nand by a smaller one around $\\simeq$ 2 December.\nIn the former case the Earth orbital velocity is summed to the one of the\nsolar system with respect to the Galaxy, while in the latter\nthe two velocities are subtracted\\footnote{Thus,\nthe DM annual modulation signature has a different origin and peculiarities\nthan the seasons on the Earth and than effects\ncorrelated with seasons (consider the expected value of the\nphase as well as other requirements listed below).}.\nThis DM annual modulation signature is very distinctive since the effect\ninduced by DM particles must simultaneously satisfy\nall the following requirements: the rate must contain a component\nmodulated according to a cosine function (1) with\none year period (2) and a phase that peaks roughly\n$\\simeq$ 2 June (3); this modulation must only be found in a\nwell-defined low energy \nrange, where DM particle induced\nevents can be present (4); it must apply only to those events\nin which just one detector of many (9 in DAMA\/NaI and 25 in DAMA\/LIBRA) actually ``fires'' ({\\it single-hit}\n events), since the DM particle multi-interaction probability\nis negligible (5);\nthe modulation amplitude in the region\nof maximal sensitivity must be $\\simeq$ 7\\% for usually adopted\nhalo distributions (6), but it can be larger (even up to $\\simeq$ 30\\%)\nin case of some\npossible scenarios such as e.g. those in Ref.~\\cite{Wei01,Fre04}.\nThus, this signature is model independent and very effective; moreover,\nthe developed highly radio-pure NaI(Tl) target-detectors \\cite{perflibra} and the adopted procedures\nassure sensitivity to a wide range\nof DM candidates (both inducing nuclear recoils and\/or electromagnetic \nradiation), interaction types and astrophysical scenarios.\n\nIn particular, the experimental observable in DAMA experiments is the modulated component of the signal in \nNaI(Tl) target and not the constant part \nof it as in other approaches as those by CDMS, Xenon, etc., \nwhere in addition e.g.: i) different target materials are used; ii) the sensitivity is mainly restricted to candidates inducing just nuclear recoils;\niii) many (by the fact largely uncertain)\nselections\/subtractions of detectors and of data and (highly uncertain) extrapolations of detectors' features are applied. \n\nThe DM annual modulation signature might be mimicked only by systematic effects or side reactions\nable to account for the whole observed modulation amplitude and\nto simultaneously satisfy all the requirements given above.\nNo one is available or suggested by anyone over more than a decade \n\\cite{perflibra,modlibra,mu,review,diu2014,RNC,norole}. \n\nIt is also worth noting that the DM annual modulation signature acts itself as a strong background reduction as pointed out since the early paper by Ref. \\cite{Freese},\nand especially when all the above peculiarities can be \nexperimentally verified in suitable dedicated set-ups as it is the case of the DAMA experiments.\n\n\n\\section{The DAMA results}\\label{s:res}\n\nThe total exposure of DAMA\/LIBRA--phase1 is:\n1.04 ton $\\times$ yr in seven annual cycles;\nwhen including also that of the first generation DAMA\/NaI experiment it is\n$1.33$ ton $\\times$ yr, corresponding to 14 annual cycles.\nThe variance of the cosine during the DAMA\/LIBRA--phase1 data taking is 0.518,\nshowing that the set-up has been operational evenly throughout the years\n\\cite{modlibra,review}.\n\n\nMany independent data analyses have been carried out \\cite{modlibra,review}\nand all of them confirm the presence of a peculiar annual modulation in the {\\it single-hit} scintillation events in the \n2-6 keV energy interval,\nwhich -- in agreement with the requirements of the DM signature -- is absent in other parts of the energy spectrum and \nin the {\\it multiple-hit} scintillation events in the same 2-6 keV energy interval (this latter condition \ncorrespond to have ``switched off the beam\" of DM particles). All the analyses and details can be found in the literature given \nabove. In particular, Fig.~\\ref{fg:res} shows the time behaviour of the experimental\nresidual rates of the {\\it single-hit} scintillation\nevents for DAMA\/NaI \\cite{RNC} and DAMA\/LIBRA--phase1 \\cite{modlibra,review} cumulatively in the (2--6) keV energy interval.\nThe data points present the experimental errors as vertical bars and the associated \ntime bin width as horizontal bars.\nThe superimposed curve is the cosinusoidal function $A \\cos \\omega(t-t_0)$\nwith a period $T = \\frac{2\\pi}{\\omega} = 1$ yr, a phase $t_0 = 152.5$ day (June 2$^{nd}$) and\nmodulation amplitude, $A$, equal to the central value obtained by best fit on the data points.\nThe dashed vertical lines\ncorrespond to the maximum expected for the DM signal, while\nthe dotted vertical lines correspond to the expected minimum. The major upgrades are also pointed out.\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[width=0.95\\textwidth] {fig1.eps}\n\\end{center}\n\\vspace{-.4cm}\n\\caption{Experimental residual rate of the {\\it single-hit} scintillation events\nmeasured by DAMA\/NaI and DAMA\/LIBRA--phase1 in the (2--6) keV energy interval\nas a function of the time.\nThe data points present the experimental errors as vertical bars and the associated\ntime bin width as horizontal bars; see text. As always in DAMA results, the given rate is already corrected for the overall efficiency.\nThe major upgrades of the experiment are also pointed out.}\n\\label{fg:res}\n\\end{figure*}\n\nIn order to continuously monitor the running conditions, several pieces of information are acquired with the production data\nand quantitatively analysed. In particular, all the time behaviours\nof the running parameters, acquired with the production data,\nhave been investigated: the modulation amplitudes obtained for each\nannual cycle when fitting the time behaviours of the parameters including a cosine\nmodulation with the same phase and period as for DM particles are well compatible with zero.\nIn particular, \nno modulation has been found in any\npossible source of systematics or side reactions; thus, cautious upper limits (90\\% C.L.)\non possible contributions to the DAMA\/LIBRA measured modulation amplitude\nhave been derived (see e.g. \\cite{modlibra}).\nIt is worth noting that they do not quantitatively account for the\nmeasured modulation amplitudes, and are not able to simultaneously satisfy all the many requirements of the signature.\nSimilar analyses have also been carried out for\nthe DAMA\/NaI data\\cite{RNC}.\n\n\\begin{table*}[!ht]\n\\caption{Summary of the contributions to the total neutron flux at LNGS; the value, the relative modulation amplitude,\nand the phase of each component is reported.\nIt is also reported the counting rate in DAMA\/LIBRA for {\\it single-hit}\nevents, in the ($2-6$) keV energy region induced by neutrons, muons and solar neutrinos, detailed for each component.\nThe modulation amplitudes, $A_k$, are reported as well, while the last column shows the relative contribution\nto the annual modulation amplitude observed by DAMA, $S_m^{exp} \\simeq 0.0112$ cpd\/kg\/keV \\cite{modlibra}.\nAs can be seen, they are all negligible and they cannot give any significant contribution to\nthe observed modulation amplitude. In addition, neutrons, muons and solar neutrinos are not a competing background\nwhen the DM annual modulation signature is investigated since in no case they can mimic this signature. For details\nsee Ref. \\cite{norole} and references therein.}\n\\label{table:tab12}\n\\vspace{0.2cm}\n\\resizebox{0.98\\textwidth}{!}{\n\\begin{tabular}{|ll|ccc|cc|c|}\n\\hline\n\\multicolumn{2}{|c|} {Source} & $\\Phi^{(n)}_{0,k}$ & $\\eta_k$ & $t_k$ & $ R_{0,k}$ & $A_k = R_{0,k} \\eta_k $ & $A_k\/S_m^{exp}$ \\\\\n & & (neutrons cm$^{-2}$ s$^{-1}$) & & & (cpd\/kg\/keV) & (cpd\/kg\/keV) & \\\\\n\\hline\n & thermal n & $1.08 \\times 10^{-6}$ & $\\simeq 0$ & -- & $<8 \\times 10^{-6}$ & $\\ll 8 \\times 10^{-7}$ & $\\ll 7 \\times 10^{-5}$ \\\\\n & ($10^{-2}-10^{-1}$ eV) & & however $\\ll 0.1$ & & & & \\\\\n SLOW & & & & & & & \\\\\n neutrons & epithermal n & $2 \\times 10^{-6}$ & $\\simeq 0$ & -- & $<3 \\times 10^{-3}$ & $\\ll 3 \\times 10^{-4}$ & $\\ll 0.03$ \\\\\n & (eV-keV) & & however $\\ll 0.1$ & & & & \\\\\n\\hline\n & fission, $(\\alpha,n) \\rightarrow$ n & $\\simeq 0.9 \\times 10^{-7}$ & $\\simeq 0$ & -- & $< 6 \\times 10^{-4}$ & $\\ll 6 \\times 10^{-5}$ & $\\ll 5 \\times \n10^{-3}$ \\\\\n & (1-10 MeV) & & however $\\ll 0.1$ & & & & \\\\\n & & & & & & & \\\\\n & $\\mu \\rightarrow $ n from rock & $\\simeq 3 \\times 10^{-9}$ & 0.0129 & end of & $\\ll 7 \\times 10^{-4}$ & $\\ll 9 \\times 10^{-6}$ & $\\ll 8 \\times \n10^{-4}$ \\\\\n FAST & ($> 10$ MeV) & & & June & & & \\\\\n neutrons & & & & & & & \\\\\n & $\\mu \\rightarrow $ n from Pb shield & $\\simeq 6 \\times 10^{-9}$ & 0.0129 & end of & $\\ll 1.4 \\times 10^{-3}$ & $\\ll 2 \\times 10^{-5}$ & $\\ll 1.6 \n\\times 10^{-3}$ \\\\\n & ($> 10$ MeV) & & & June & & & \\\\\n & & & & & & & \\\\\n & $\\nu \\rightarrow $ n & $\\simeq 3 \\times 10^{-10}$ & 0.03342$^*$ & Jan. 4th$^*$ & $\\ll 7 \\times 10^{-5}$ & $\\ll 2 \\times 10^{-6}$ & $\\ll 2 \\times 10^{-4}$ \n\\\\\n & (few MeV) & & & & & & \\\\\n\\hline\n\\multicolumn{2}{|c|} {direct $\\mu$} & $\\Phi^{(\\mu)}_{0} \\simeq 20$ $\\mu$ m$^{-2}$d$^{-1}$ & 0.0129 & end of & $\\simeq 10^{-7}$ & $\\simeq 10^{-9}$ & $\\simeq 10^{-7}$ \\\\\n & & & & June & & & \\\\\n\\multicolumn{2}{|c|} {direct $\\nu$} & $\\Phi^{(\\nu)}_{0} \\simeq 6 \\times 10^{10}$ $\\nu$ cm$^{-2}$s$^{-1}$ & 0.03342$^*$ & Jan. 4th$^*$ & $\\simeq 10^{-5}$ \n& $3 \\times 10^{-7}$ & $3 \\times 10^{-5}$ \\\\\n\\hline\n\\end{tabular}}\n\\vspace{0.3cm}\n\\footnotesize{\\\\$^*$ The annual modulation of solar neutrino is due to the different Sun-Earth distance along the year; so the relative modulation amplitude is twice the\neccentricity of the Earth orbit and the phase is given by the perihelion.}\n\\end{table*}\n\nNo other experimental result has been verified over so long time\nso accurately and with various significant upgrades of the set-ups.\n\nFor completeness I mention that sometimes naive statements\nwere put forwards as the fact that\nin nature several phenomena may show some kind of periodicity.\nThe point is whether they could\nmimic the annual modulation signature in DAMA\/LIBRA (and former DAMA\/NaI), i.e.~whether they\ncould quantitatively account for the observed\nmodulation amplitude and also simultaneously\nsatisfy all the requirements of the DM annual modulation signature. The same is also for side reactions.\nThis has already been deeply investigated in Ref.~\\cite{perflibra,modlibra} and references\ntherein;\nthe arguments and the quantitative conclusions, presented there, also\napply to the entire DAMA\/LIBRA--phase1 data. Additional arguments can be found\nin Ref.~\\cite{mu,review,diu2014,norole}.\nIn particular, Ref. \\cite{norole} further outlines in a simple and intuitive way why neutrons (of whatever origin), muons and \nsolar neutrinos cannot give any significant contribution to the DAMA annual modulation results and -- in addition -- can never mimic the DM annual modulation signature since some of its specific requirements fail. \nTable \\ref{table:tab12} summarizes the safety upper limits on the contributions (if any) to the observed modulation amplitude\ndue to the total neutron flux at LNGS,\neither from $(\\alpha$,n) reactions, from fissions and from muons' and solar-neutrinos' interactions in the rocks and in the lead\naround the experimental set-up; the direct contributions of muons and solar neutrinos are also reported there.\nAs seen in Table \\ref{table:tab12}, they are all negligible and they cannot give any significant contribution to\nthe observed modulation amplitude; in addition, neutrons, muons and solar neutrinos are not a competing background\nwhen the DM annual modulation signature is investigated since they cannot mimic this signature. For details\nsee Ref. \\cite{norole} and references therein.\n\nIn conclusion, DAMA gives a model-independent evidence -- at 9.3$\\sigma$ C.L. over 14 independent annual cycles --\nfor the presence of DM particles in the galactic halo.\n \n\\subsection{On comparisons}\\label{s:sub1}\n\nNo direct model independent comparison is possible in the field when different target materials \nand\/or approaches are used; the same is for the strongly model dependent indirect searches\\footnote{It should be noted that \nthe rising behaviour of the positron flux reported in Ref. \\cite{pamela,ams2} does not give any intrinsic evidence for production \ndue to DM annihilation; \nthis may arise only when a particular model of the competing background is assumed as e.g. the \nGALPROP code. But other more complete models \nexist which do not support any significant excess evidence. Moreover, an interpretation in terms of DM particle annihilation would require the assumption of: \ni) a very large boost factor ($\\sim$ 400) of the density; ii) to boost the annihilation cross section through an assumed new interaction type; iii) to adjust the propagation\nparameters; iv) to consider extra-source (subhalos, IMBHs); v) to consider only a leptophilic candidate to justify the absence of any excess in the \nantiproton spectrum. \nFinally, other well known sources can account for a similar\npositron fraction: pulsars, supernova\nexplosions near the Earth, SNR.}.\n\nIn order to perform corollary investigations on the nature of the DM particles, model-dependent\nanalyses are necessary\\footnote{For completeness, it is worth recalling that it does not exist any approach to investigate the nature \nof the candidate in the direct and indirect DM searches, which can offer this information independently on \nassumed astrophysical, nuclear and particle Physics scenarios. On the other hand, searches for new particles beyond the\nStandard Model of particle Physics at accelerators cannot\ncredit by themselves that a certain particle is in the halo as a solution or the only solution for DM particles, and -- in addition --\nDM candidates and scenarios (even for the neutralino) exist which cannot be investigated there.}. \nThus, many theoretical and experimental parameters and \nmodels are possible (see e.g. in \\cite{modlibra,review,norm,Wall14}) and many hypotheses must also be exploited, while specific experimental and theoretical assumptions are generally \nadopted in the field assuming a single arbitrary scenario without accounting neither for existing uncertainties nor for \nalternative possible scenarios, interaction types, etc.\n\nThe obtained DAMA 9.3 $\\sigma$ C.L. model independent evidence\nis compatible with a wide set of scenarios regarding the nature of the DM candidate\nand related astrophysical, nuclear and particle Physics. For examples\nsome scenarios and parameters are discussed e.g. in\nRef.~\\cite{allDM1,allDM,RNC,modlibra,review,norm,Wall14}.\nFurther large literature is available on the topics (see for example in the bibliography of Ref. \\cite{review}).\nBy the fact, both the negative results and all the possible positive hints\nare largely compatible with the DAMA model-independent DM annual\nmodulation results in various scenarios considering also the existing experimental and\ntheoretical uncertainties; the same holds for the strongly model dependent indirect approaches.\n\nIt is also worthwhile to further recall that these DAMA experiments are not only sensitive to DM particles with spin-independent coupling \ninducing just nuclear recoils, but also to other couplings and to other DM candidates \nas those giving rise to part or all the signal in electromagnetic form. Finally, scenarios exist in which \nother kind of targets\/approaches are disfavoured or even blind. \n\n\n\\section{DAMA\/LIBRA--phase2 and perspectives}\\label{s:ph2}\n\nAn important upgrade has started at end of 2010 replacing all the PMTs with new ones having higher Quantum Efficiency;\ndetails on the developments and on the reached performances in the operative conditions\nare reported in Ref. \\cite{pmts}. They have allowed us to lower the software energy threshold of the experiment \nto 1 keV and to improve also other features as e.g. the energy resolution \\cite{pmts}.\n\nSince the fulfillment of this upgrade and after some optimization periods, DAMA\/LIBRA--phase2\nis continuously running in order e.g.:\n(1) to increase the experimental sensitivity thanks to the lower software energy threshold; \n(2) to improve the corollary investigation on the nature of the\nDM particle and related astrophysical, nuclear and particle physics arguments;\n(3) to investigate other signal features and second order effects. This requires long and\n dedicated work for reliable collection and analysis of very large\nexposures.\n\nIn the future DAMA\/LIBRA will also continue its study on several other rare\nprocesses as also the former DAMA\/NaI apparatus did.\n\nFinally, further future improvements of the DAMA\/LIBRA set-up to increase the sensitivity (possible DAMA\/LIBRA-phase3) and the \ndevelopments towards the possible DAMA\/1ton (1 ton full sensitive mass on the contrary of other kind of detectors), we proposed in 1996, \nare considered at some extent. For the first case developments of new further radiopurer PMTs with high quantum efficiency are starting,\nwhile in the second case it would be necessary to overcome the present problems regarding: i) the supplying, \nselection and purifications of a large number of high quality NaI and, mainly, TlI powders; ii) the \navailability of equipments and competence for reliable measurements of small trace contaminants in ppt or lower region; \niii) the creation \nof updated protocols for growing, handling \nand maintaining the crystals; iv) the availability of large Kyropoulos equipments \nwith suitable platinum crucibles; v) etc.. At present, due to the change of rules for provisions of strategical materials, \nthe large costs and the lost of some equipments and competence also at industry level, a \nsatisfactory development appears quite difficult.\n\n\n\\section{Conclusions}\\label{s:concl}\n\nThe data of DAMA\/LIBRA--phase1 have further confirmed the presence of a\npeculiar annual modulation of the {\\it single-hit} events in the (2--6) keV energy region\nsatisfying all the many requirements of the DM annual modulation signature;\nthe cumulative\nexposure by the former DAMA\/NaI and DAMA\/LIBRA--phase1 is\n1.33 ton $\\times$ yr (orders of magnitude larger than those typically released in the field).\n\nAs required by the DM annual modulation signature:\n1) the {\\it single-hit} events show a clear cosine-like modulation as expected for the DM signal;\n2) the measured period is equal to $(0.998\\pm 0.002)$ yr well compatible with the 1 yr period as expected for the DM signal;\n3) the measured phase $(144\\pm 7)$ days is compatible with $\\simeq$ 152.5 days as expected for the DM signal;\n4) the modulation is present only in the low energy (2--6) keV interval and not in other higher energy regions, consistently with expectation for the DM signal;\n5) the modulation is present only in the {\\it single-hit} events, while it is absent in the {\\it multiple-hit} ones as expected for the DM signal;\n6) the measured modulation amplitude in NaI(Tl) of the {\\it single-hit} events in the (2--6) keV energy\n interval is: $(0.0112 \\pm 0.0012)$ cpd\/kg\/keV (9.3 $\\sigma$ C.L.).\nNo systematic or side processes able to simultaneously satisfy all the many\npeculiarities of the signature and to account for the whole measured modulation \namplitude is available.\n\nDAMA\/LIBRA--phase2 is continuously running in its new configuration with a lower software energy threshold aiming to improve\nthe knowledge on corollary aspects regarding the signal and on second order effects as discussed e.g. in Ref.~\\cite{review,diu2014}.\n\nFew comments on model--dependent comparisons have also been addressed here.\n\n\n\\section*{Acknowledgments}\n\nIt is a pleasure to thank all my DAMA collaborators who effectively dedicated their efforts to this experimental activity\nand the colleagues in this Workshop for the interesting topics we have discussed, for the question \nsection, and for the pleasant scientific environment.\n\n\\section*{Appendix: Questions \\& Answers}\n\nThis section shortly summarizes some of the topics extensively discussed at the Workshop, where the time dedicated to discussions and the interest in deeply\nunderstanding the topics were rather large.\n\n\n\\vspace{0.4cm}\n\n {\\it Question 1: may you comment about the ratio of the measured dark matter particles \n modulation amplitude to the total signal: the $S_m\/S_0$ ratio?}\n\n\\vspace{0.3cm}\n\nAnswer 1: the measured counting rate in the cumulative energy spectrum is about 1 cpd\/kg\/keV in the lowest energy bins; this \nis the sum of the background contribution and of the constant part of the signal S$_0$. As discussed e.g. in \nTAUP2011 \\cite{taupnoz}, the background in the 2-4 keV energy region is estimated to be \nnot lower than about 0.75 cpd\/kg\/keV; this gives an upper limit on S$_0$ of about 0.25 cpd\/kg\/keV. Thus, the S$_m$\/S$_0$ ratio is equal or larger than \nabout 0.01\/0.25 $\\simeq$ 4 \\%. \n\n\t\n\\vspace{0.4cm}\n\n {\\it Question 2: may you comment on the quenching factors, on their dependence on the type of the particles, and on \n some typical examples of extreme properties?}\n\n\\vspace{0.3cm}\n\n Answer 2: The quenching factor values play a role only when corollary model-dependent analyses for DM candidates inducing just nuclear recoils are carried out,\nin order to derive the energy scale in terms of nuclear recoil energy.\n\nAs is widely known, the quenching factor is a specific property of \nthe employed detector and not a general quantity universal \nfor a given material. For example, in liquid noble-gas detectors, it depends -- among others -- on the \nlevel of trace contaminants which can vary in time and from one liquefaction process to another, on the cryogenic microscopic conditions, etc..\nIn bolometers it depends for instance on specific properties, trace contaminants, cryogenic conditions, \netc. of each specific detector, while generally it is assumed exactly equal to unity (the maximum possible value). \nThe quenching factors in scintillators depend, for example, on the dopant concentration, on the growing method\/procedures, on residual trace contaminants, \netc., \nand are expected to be energy dependent. Thus, all these aspects are already by themselves relevant sources of uncertainties \nwhen interpreting whatever result in terms of DM candidates inducing just nuclear recoils. \nSimilar arguments have been addressed e.g. in Ref. \\cite{modlibra,bot11,RNC,chan,tretyak}. \n\n\\vspace{0.4cm}\n\n {\\it Question 3: May you comment under which extreme conditions your experiment is successful and\n comment what can at most the experiment which does not fulfil one of the\n conditions or more than one of them at most can ``see\"?}\n\n\\vspace{0.3cm}\n\n Answer 3: The full description and potentiality of the DAMA\/LIBRA set-up have been discussed in details in Refs. \\cite{perflibra,modlibra,pmts}\nand references therein. Obviously all the set-up specific features and adopted procedures contribute to the possibility to point out the signal\nthrough the model independent DM annual modulation signature. \nThe absence\/difference of one of them would limit whatever else result. \n\n\\vspace{0.4cm} \n\n {\\it Question 4: May you comment about muons?}\n\n \\vspace{0.3cm} \n\nAnswer 4: An extensive discussion on this topics can be found in the dedicated Ref. \\cite{mu,norole}, where its has been quantitatively demonstrated \n(see also Table \\ref{table:tab12} in this paper) that -- for many reasons (and just one would suffice) - muons cannot play (directly or indirectly) \nany role in the DAMA annual modulation effect. \n\n\n\\vspace{0.4cm}\n {\\it Question 5: May you comment about neutrinos?}\n\\vspace{0.3cm}\n\nAnswer 5: The contribution from solar, atmospheric, .. neutrinos is obviously negligible; a quantitative discussion can be found in Ref. \\cite{norole}\n(see also Table \\ref{table:tab12} in this paper). \n\n\\vspace{0.4cm}\n\n {\\it Question 6: May you comment about the operating temperature of your measuring apparatus?}\n\n \\vspace{0.3cm} \n\nAnswer 6: The DAMA set-ups operate at environmental temperature maintained stable by suitable and redundant air-conditioning system \n(2 independent devices for redundancy); moreover, the Cu housings of the detectors are in direct contact with the multi-ton metallic shield, thus \nthere is a huge heat capacity ($\\sim$ 10$^6$ cal\/$^0$C). In addition, the operating temperature of the detectors is continuously \nmonitored and analysed as the production data. A discussion on temperature in \noperating condition can be found e.g. in Ref. \\cite{modlibra,review}.\n\n\\vspace{0.4cm} \n \n {\\it Question 7: May you comment about the Snowmass plots and its meaning?}\n \\vspace{0.3cm}\n\nAnswer 7:\nThe recent plot from Snowmass and that in Ref. \\cite{RPP}\nabout the ``status of the Dark Matter search\"\ndo not point out at all the real status of Dark Matter searches since e.g.: i) Dark Matter has \nwider possibilities than WIMPs inducing just nuclear recoil with spin-independent interaction under single (largely \narbitrary) set of assumptions; \nii) neither the uncertainties for existing experimental and theoretical aspects nor alternative possible assumptions \nare accounted for; iii) they do not include possible systematic errors affecting the data \n(such as e.g. ``extrapolations\" of energy threshold, of energy resolution and of\nefficiencies, quenching factors values, \nconvolution with poor energy resolution, correction for non-uniformity of the detector, multiple subtractions\/selection of \ndetectors and\/or data, \nassumptions on quantities related to halo model, form factors, scaling laws, etc.); \niv) the DAMA implications -- even adopting the many arbitrary assumptions considered there -- appear incorrect, for example the \nS$_0$ prior is not accounted for, etc., etc.. The perspectives as well appear incorrect\/too optimistic.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nA key aspect in microfluidics is to understand the theoretical fundamentals of flow at \nlow-Reynolds numbers and to exploit them directly in applications and devices.\nAs the miniaturisation is an ongoing trend, especially lab-on-a-chip (LOC) or micro \ntotal analysis systems (\\textmu TAS) provide a fast reaction \ntime, low amount of reagents and low waste generation as well as high detection \naccuracy and verifiability \\cite{Gijs2004}.\nTherefore, the analysis and diagnostics of reagents within microfluidic channels require \nswitching, driving and mixing of the liquid fluids \\cite{Whitesides2006}.\nOptical forces, for instance, provide a non-contact trapping and \nmanipulation of microobjects or biological cells and are sufficient for usage \nin different microfluidic applications. The latter are \ne.g., pumping due to rotation of microstructures, \nsorting cells and intermixing different fluids \\cite{Mohanty2012,Wu2011}.\nPadgett \\textit{et al.} identify the use of optical forces in so-called \noptical tweezers (OT) as a technology with high potential for \nmicrofluidic applications in biology, medicine or chemistry \\cite{Padgett2011}.\n\nFurthermore, the need of measuring systems within microfluidic channels \nwith high precision arises. Besides actuation, systems based on optical \nforces can be used for sensing and characterizing of fluid properties.\nFor example, Di Leonardo \\textit{et al.} and Kn\\\"{o}ner \\textit{et al.} \nhave measured the flow field in a microfluidic \ndevice with OT's \\cite{Padgett2011,DiLeonardo2006,Knoener2005}. \nIn both studies a spherical microrotor has been activated by interaction of polarized laser light \nwith a birefringent vaterite crystal. The rotation has led to a fluid flow and \na displacement of an inserted probe particle. The azimuthal flow field around the microsphere\nhas been determined by measuring the shift of the probe particle. \n\nHowever, no radial component has been detected in these studies. \nObjects in microfluidic systems rotating about their \ncenter in axisymmetric manner can easily produce azimuthal flow, which is also \nsupported by theory \\cite{Landau1966,Jeffery1922}. Whether a rotating microstructure can \ninduce a radial flow component at low-Reynolds numbers, however, \nseems to be far less understood. Moreover, for axisymmetric rotations,\nsimple\nsymmetry arguments suggest that such a radial flow\ncan not occur.\nIn addition, theoretical works \\cite{camassa,feng,viana} on \nnon-axisymmmetric rotations appear to be restricted to ellipsoidal bodies and prove to be quite challenging from\na mathematical point of view. The work of Camassa \\emph{et al.} \\cite{camassa}, for instance,\ntreats the problem of an ellipsoid\nsweeping out a double cone in Stokes flow. Here, a rather complex spatio-temporal velocity field\nis obtained and explicitly analyzed for certain limit cases (small cone angles, far-field behavior etc.). Although\na radial flow component is predicted, the main focus of this works lies on the advection of tracers and\nthe establishment of certain invariants for the above mentioned limit cases.\nConsequently, the main purpose of this paper is to gain some insight on the persistence of such \nradial flow for the case of non-axisymmetric rotations\nat low-Reynolds numbers. Moreover, the azimuthal velocity field component is investigated and compared to\nthe case of a purely axisymmetric rotation. To this end, a comparatively \nsimple model that is based on the superposition\nof two rotlets \\cite{blake,pozrikidis} is used in order to approximate the rod-shaped\nrotor geometry and to make statements about the induced velocity fields.\n \nThe mere fact that the rotlet solution as a fundamental solution to the Stokes equation \nfor low-Reynolds number flows basically shares the same velocity profile as the \npoint-vortex solution of the two-dimensional Euler equation \\cite{aref,saffman,chaos} allows to draw an \ninteresting analogy to a recently proposed generalized vortex model \\cite{Friedrich2013}.\nHere, the model consists of pairs of like-signed point-vortices and it has been shown\nthat the straining of these small-scale vortical structures leads to non-vanishing radial \nvelocity components that prove to be of great importance for the energy transport in \nsuch a two-dimensional flow. Although the rotlet and the point-vortex solution\napply to two very different Reynolds number regimes,\nthis striking analogy allows to predict a possible radial flow also\nfor the Stokes equation.\n\nIn this paper, the predictions of this rotlet model\nare compared to experimental results that allow the detection of azimuthal and radial flow components\ninfluenced by microrotors. \nThe experiments were performed using a combination of optical and magnetic forces, which enables the \nmanufacture of rod-shaped microrotors and the investigation of the flow fields.\nRecently this method has been successfully applied for constructing a micro-\\\\ \npump for microfluidic channels \\cite{Koehler2014}.\n\n\n\\section{The rotlet model equations}\nThe proposed model captures the main effects of\nthe rotors investigated in the experiment. Here, we mainly follow the procedure\ndescribed by Chwang and Wu who approximated the velocity field of a rotating dumbbell-shaped\nbody via a superposition of two rotlets spinning about the main axis of symmetry \\cite{Chwang1974}; hence, \nthe resulting velocity field only possessed an azimuthal component.\nThe main difference in our approach is that we consider non-axisymmetric rotations,\nwhich turn out to induce an additional radial flow component. \nTo this end, we initially work in a frame of reference where the rotor is at rest and assume\nthat the entire fluid is undergoing a rotation about the $z$ axis with the angular velocity \n$\\omega$. We then consider the velocity field that emerges from two rotlets at \n${\\bf a}$ and ${-\\bf a}$\n \\begin{equation}\\label{rotlet-dipol}\n {\\bf u}({\\bf x})= \\Gamma \\omega \\left[ {\\bf e}_z \\times \\frac{{\\bf x}-{\\bf a}}{|{\\bf x}-{\\bf a}|^3} \n\t\t + {\\bf e}_z \\times \\frac{{\\bf x}+{\\bf a}}{|{\\bf x}+{\\bf a}|^3} \\right] \n\t\t - \\omega {\\bf e}_z \\times {\\bf x} \n \\end{equation}\n as an approximation of the velocity field that is induced by the rotor.\n Note that the last term has been added to ensure the correct fluid velocity in the far-field.\n Furthermore, the rotlet distance ${\\bf a}$ and the rotlet strength $\\Gamma$ have to be considered\n as free parameters yet to be determined by the rotor geometry.\n In order to see how these two paramters are fixed, we assume that the \n dumbbell is oriented in ${\\bf e}_x$-direction\n as it is depicted in Fig. \\ref{rot_image}. \n \\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{rotor.eps}\n \\caption{Approximation of a dumbbell-shaped body of length $b$ and neck length $c$\n by a pair of rotlets with strength $\\Gamma$ fixed at $\\pm {\\bf a}$. \n Initially,\n the body is supposed to be at rest as the surrounding fluid undergoes a rotation with the angular velocity\n $\\omega$ determining the parameters $\\Gamma$ and $a$ in terms of the rotor geometry $b$ and $c$.} \n \\label{rot_image}\n \\end{figure}\n As a consequence, the velocity field in Eq. (\\ref{rotlet-dipol}) has to satisfy no-slip\n boundary conditions at ${\\bf x}_b= \\pm b {\\bf e}_x$ and\n at ${\\bf x}_c= \\pm c {\\bf e}_y$, namely ${\\bf u}({\\bf x}_b) =0$ and\n ${\\bf u}({\\bf x}_c) = 0$. \n From these boundary conditions, we obtain \n \\begin{equation}\\label{boundary1}\n \\frac{b^2 + a^2}{(b^2-a^2)^2}= \\frac{b}{2 \\Gamma} \\qquad (a^2+c^2)^{-3\/2} = \\frac{1}{2\\Gamma}.\n \\end{equation}\nIn eliminating $\\Gamma$ from these equations, we obtain a relationship between \nthe geometrical parameters \n\\begin{equation}\\label{root}\n \\left(1 + \\frac{a^2}{b^2}\\right)\\left(\\frac{a^2}{b^2}+\\frac{c^2}{b^2}\\right)^{3\/2}=\n \\left(1-\\frac{a^2}{b^2}\\right)^2,\n\\end{equation}\nwhich can be solved for a variety of rotor geometries in order to obtain the non-dimensional parameter\n${a}\/{b}$ in dependence of ${c}\/{b}$. The corresponding solution for ${a}\/{b}$ determining\nthe rotlet distance from the center is depicted in Fig. \\ref{root_non_dimensional}.\n\nAt this point, it \nhas to be stressed that the approximation (\\ref{rotlet-dipol}) is not capable of reproducing the no-slip\nboundary conditions over the entire rotor body but is only strictly valid at the local points\n${\\bf x}_b= \\pm b {\\bf e}_x$ and\n at ${\\bf x}_c= \\pm c {\\bf e}_y$. \nHowever, in the following, we are not so concerned about the exact velocity field near the rotor\nbody but rather try to understand the immediate consequences of the velocity\nfield in Eq. (\\ref{rotlet-dipol}) on the flow pattern in the far-field. \nTo this end, we make use of the similarity of the rotlet velocity profile in Eq. (\\ref{rotlet-dipol}) \nto the point-vortex solution of two-dimensional turbulence, which only differ by the power law in \nthe denominator (quadratic instead of cubic for point-vortices). In this context, it has been \nshown recently that a generalized vortex model \\cite{Friedrich2013} consisting of pairs of \nlike signed point-vortices leads to a radial flow component in the far-field.\n\nApplying these results to the rotlet pair of Eq. (\\ref{rotlet-dipol}), a multi-pole\nexpansion for $|{\\bf a}|\/|{\\bf x}| \\ll 1 $ yields\n\\begin{equation}\\label{multi}\n {\\bf u}({\\bf x})= \\Gamma \\omega \\left[ 2 {\\bf e}_z \\times \\frac{{\\bf x}}{x^3} +\n {\\bf e}_z \\times( {\\bf a} \\cdot \\nabla_{\\bf x})^2 \\frac{{\\bf x}}{x^3} \\right]-\n \\omega {\\bf e}_z \\times {\\bf x}. \n\\end{equation}\n \\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{root_non_dimensional.ps}\n \\caption{Non-dimensional parameter solution for ${a}\/{b}$ of Eq. (\\ref{root}) in dependence\n of ${c}\/{b}$. \n In the limit of ${c}\/{b} \\rightarrow 1$, the solution of a rotating sphere $a=0$ is recovered.} \n \\label{root_non_dimensional}\n\\end{figure}\n\\begin{figure*}\n\\begin{minipage}[l]{0.325 \\textwidth}\n\\centering\n\\includegraphics[width=1 \\textwidth]{0.ps}\n\\end{minipage}\n\\begin{minipage}[l]{0.325 \\textwidth}\n\\centering\n\\includegraphics[width=1 \\textwidth]{0.15.ps}\n\\end{minipage}\n\\begin{minipage}[l]{0.325 \\textwidth}\n\\centering\n\\includegraphics[width=1 \\textwidth]{0.3.ps}\n\\end{minipage}\n\\begin{minipage}[l]{0.1625 \\textwidth}\n\\end{minipage}\\\\~\n\\\\\n~\n\\\\\n~\n\\\\\n\\begin{minipage}[l]{0.325 \\textwidth}\n\\centering\n\\includegraphics[width=1 \\textwidth]{0.45.ps}\n\\end{minipage}\n\\begin{minipage}[l]{0.325 \\textwidth}\n\\centering\n\\includegraphics[width=1 \\textwidth]{0.6.ps}\n\\end{minipage}\n\\begin{minipage}[l]{0.1625 \\textwidth}\n\\end{minipage}\n\\caption{Velocity field profiles in the $x$-$y$-plane determined by the far-field approximation (\\ref{u_r}) and\n(\\ref{u_phi}) for increasing $a\/b$ at $t=0$ (rotor is oriented in $x$-direction). The axisymmetric velocity field of the rotating sphere is recovered for $a=0$. As\n$a\/b$ approaches its maximum value determined by the solution of Eq. (\\ref{root}) in Fig. \\ref{root_non_dimensional} \nthe velocity field becomes more and more distorted in the $x$-direction.}\n\\label{profile}\n\\end{figure*}\nHere, the first term corresponds to the velocity field of a rotating sphere with radius \n$[2 \\Gamma \\omega]^{1\/3}$, which solely leads to an azimuthal component of the velocity \nfield.\nThe second term however, can lead to a component of the velocity in radial direction, which \ncan be seen in writing the multi-pole terms in Eq. (\\ref{multi}) explicitly as\n\\begin{equation}\n {\\bf u}({\\bf x})= \\Gamma \\omega {\\bf e}_z \\times \\left[ 2 \\frac{{\\bf x}}{x^3} \n -6 \\frac{{\\bf a}}{x^5} {\\bf a} \\cdot {\\bf x} +15 \\frac{{\\bf x}}{x^7} ({\\bf a} \\cdot {\\bf x})^2 -\n 3 \\frac{{\\bf x}}{x^5} a^2 \\right ]\n \\end{equation}\nThe term that allows the velocity field to escape from its solely azimuthal shape is the second one, \nwhich can be seen in projecting \nonto the radial direction. To this end, we introduce the unit vector \n${\\bf e}_a=\\frac{{\\bf\na}}{a}=\\left( \\begin{array}{l} \n \\cos \\varphi_a \\\\\n \\sin \\varphi_a \\\\ ~~~0\n \\end{array}\n \\right)$, as well as\n ${\\bf e}_r=\\frac{{\\bf x}}{x}=\\left( \\begin{array}{l} \n \\cos \\varphi \\sin \\theta \\\\\n \\sin \\varphi\\sin \\theta \\\\\n ~~~\\cos \\theta\n \\end{array} \\right)$, which yields\n\\begin{equation}\n{\\bf e}_r \\cdot \n[{\\bf e}_z \\times {\\bf e}_a]({\\bf e}_a\\cdot {\\bf e}_r)=\n\\frac{1}{2} \\sin(2 (\\varphi -\\varphi_a)) \\sin^2 \\theta.\n\\end{equation} \nAdditionally, we change to a frame of reference in which the fluid is at rest and \nthe rotor is undergoing rotation, i.e.\n$\\varphi - \\varphi_a \\rightarrow \\varphi - \\omega t$ \nand forget about additional centrifugal forces due to the neglection of inertia\nin the Stokes equation ( $\\rho | \\boldsymbol \\omega \\times {\\bf u}|$ small compared to\nthe viscous force $\\mu |\\nabla^2 {\\bf u}|$, \nwhere $\\rho$ is the density and $\\mu$ is the viscosity of the fluid). \nWe obtain\n\\begin{eqnarray} \\label{u_r}\n u_r ={\\bf e}_{r} \\cdot {\\bf u}({\\bf x})\n= -3 \\Gamma \\omega \\frac{a^2}{r^4} \\sin(2 ( \\varphi-\\omega t )) \\sin^2 \\theta \n\\end{eqnarray}\nfor the radial velocity component and \n\\begin{eqnarray} \\label{u_phi}\n &u_{\\varphi}& = {\\bf e}_{\\varphi} \\cdot {\\bf u}({\\bf x}) \\\\ \\nonumber\n &=& \\Gamma \\omega \\left( \\frac{2}{r^2}\\sin \\theta \n - 3 \\frac{a^2}{r^4}\\sin \\theta\n +15 \\frac{a^2}{r^4} \\sin^2 \\theta \\sin^2 ( \\varphi-\\omega t) \\right)\n\\end{eqnarray}\nfor the azimuthal velocity component. It can be seen that the radial component oscillates \nwith twice the rotor frequency, whereas the azimuthal component is a superposition of\na time-independent part and a time-dependent part that does not vanish in average, since\n$\\langle \\sin^2 ( \\varphi-\\omega t) \\rangle$ is a constant. Thereby the time-dependence\nof the velocity field not inherently contained in the Stokes equation is due to the inclusion\nof time-dependent boundary conditions similarly to the case of Taylor's swimming sheet \\cite{taylor}.\nThe corresponding velocity field determined by the radial (\\ref{u_r}) and azimuthal (\\ref{u_phi}) flow components \nfor increasing ${a}\/{b}$ are shown in Fig. \\ref{profile} for $t=0$.\nIt can be seen that the axisymmetric velocity field of the sphere ${a}\/{b}=0$ gets distorted in the $x$-direction\nfor increasing ${a}\/{b}$ as the two rotlets in Eq. (\\ref{rotlet-dipol}) become more and more remote. \n\n\n\\section{Assembling, actuation and measurements}\n\\begin{figure}[t!]\n\\centering\n \\includegraphics[width=0.45\\textwidth]{laser_setup.eps}\n \\caption{(color online) Schematic depiction of the\nholographic optical tweezer (HOT) setup. The laser beam is modulated\nand reflected by the spatial light modulator (SLM) and focused with a high-NA microscope objective into the sample chamber. \nThe high-speed camera is used to observe and detect the particles in the\ntrapping plane. BE: beam expander, L: lens, M: mirror, DM: dichroic mirror. \nInset: Microscope image of probe particle next to assembled five-particle rotor (scale bar is 5\\,\\textmu m).}\n \\label{laser_setup}\n\\end{figure}\nTo verify the established theoretical model a microrotor has to be assembled, actuated and the generated fluid flow has to be measured experimentally in a precise manner.\nThe OT is used to assemble chain-shaped magnetic microrotors by a simple ``pick-and-place'' technique \\cite{Ghadiri2012}.\nBased on a focused laser beam, OT provide non-contact trapping, moving and rotation of dielectric microparticles \\cite{Ashkin.1986}. The force acting on a particle is typically in the range of several piconewtons and can be described by\n\\begin{equation}\n {\\bf F}_{opt} = -k \\mathfrak{r}\n \\label{hook}\n\\end{equation}\nwith the trap stiffness $k$ and the displacement of the particle ${\\bf \\mathfrak{r}}$ from its equilibrium position.\nA Biotin (B) coated transparent microsphere (SiO2-Biotin-AR722, $\\varnothing$ = 2.73 $\\pm$ 0.13 \\textmu m, microParticles GmbH) is trapped by OT and fused together with Streptavidin (SA) coated magnetic microspheres (SiO2-MAG-SA-S2548, $\\O$ = 2.61 $\\pm$ 0.12 \\textmu m, microParticles GmbH) by bringing them into contact. The mutual high affinity of the biomolecules leads to a stable binding of the complementary particles and a magnetic three-particle rotor can be assembled \\cite{Ghadiri2012}.\nThe embedded transparent microsphere in the middle of the rotor provides stable trapping and friction-free rotation about the rotational axis of the rotor.\nFor bigger rotors magnetic interaction is used to attach magnetic particles at both ends of the microstructure (Fig. \\ref{laser_setup} inset). Although, the outer magnetic particles of the five-particle rotor are not connected with the B-SA binding, there is no deformation of the rotor up to the rotational frequency of 6.5\\,s$^{-1}$.\nThe following experimental investigation is based on chain-shaped rotors with three and five particles.\n\\begin{figure}[t!]\n\\centering\n \\includegraphics[width=0.45\\textwidth]{1ab_azi_displ_velo.ps}\n \\caption{(color online) a) The displacement of the probe particle increases with higher rotational frequency of the five-particle rotor. b) The calculated fluid velocities of five- and three-particle rotors show linear behavior.}\n \\label{a_flow1}\n\\end{figure}\n\nA cw laser (Verdi G5, Coherent) with a wavelength of $\\lambda = 532$\\,nm is used and with integration of a spatial light modulator (SLM) multiple optical trapping spots can be generated (Fig. \\ref{laser_setup}). Computer generated holograms implemented into the SLM allow simultaneous manipulation of multiple microobjects using a single laser source. The SLM based OT's are commonly referred to as holographic optical tweezers (HOT) \\cite{Hayasaki1999,Curtis2002,Grier2006}. \nThe laser beam is expanded to utilize the entire active area of the SLM (Pluto-VIS, Holoeye) and the reflected modulated beam is projected onto the back aperture of a 100\\,x 1.4\\,NA microscope objective using two lenses (L$_1$ and L$_2$) with 4-f-configuration. The beam is focused into the sample chamber and the microparticles can be trapped in an aqueous environment. The sample chamber has a circular aperture\nof 8\\,mm diameter, a height of 5\\,mm and is closed on the top and bottom with cover\nglasses. A high-speed CMOS camera (USB UI-1225LE, IDS) is used to observe the particle position and to analyze it with sub-pixel accuracy down to 5\\,nm \\cite{Neuman2004}.\nThe rotation of the assembled microstructures is introduced by a controllable external magnetic field, which is generated by four electromagnetic coils arranged around the sample chamber and addressed by a stepper motor driver (TMCM-1110, Trinamic). Due to the magnetic torque, the microrotor follows the external magnetic field and rotates with desired rotational speed.\n\\begin{figure}[t!]\n\\centering\n \\includegraphics[width=0.45\\textwidth]{2ab_azi_dist.ps}\n \\caption{(color online) The azimuthal fluid velocity decreases with increasing distance to the rotor. \n The results are in good agreement to established Eq. \\eqref{u_phi} which is illustrated by the fits (black lines).}\n \\label{a_distance1}\n\\end{figure}\n\\begin{figure}[h!]\n\\centering\n \\includegraphics[width=0.45\\textwidth]{radial_auslenkung.ps}\n \\caption{(color online) a) The oscillating radial displacement of the probe particle (green dots) is presented at a rotational frequency of 4\\,s$^{-1}$. The black curve represents a sine function fit which reaches its maximum (minimum) when the rotor is vertically (horizontally) aligned, respectively. b) The radial displacement of the probe particle decreases with increasing the distance to the rotor.\n The dashed lines connect the corresponding points for the sake of clarity.}\n \\label{r_displacement}\n\\end{figure}\nThe rotation leads to a flow field in the liquid environment, which is measured by introducing a non-magnetic probe particle as illustrated in the inset of Fig. \\ref{laser_setup}. The probe particle is trapped by a second optical spot at a variable distance to the rotor axis. \nThe generated fluid flow ${\\bf u}$ leads to a displacement of the probe particle which can be calculated using Stroke's law (drag force),\n\\begin{equation}\n {\\bf F}_{drag} = 6 \\pi \\eta R_p c_F {\\bf u}\n\\end{equation}\nwith the fluid viscosity $\\eta$ (the experiments are performed in buffer solution contains approx. 99\\% water), the probe particle radius $R_p$ and the correction factor $c_F$. Due to boundary effects of the surfaces the correction coefficient has to be implemented according to Faxen's law \\cite{Svoboda1994}, which is calculated to be $c_F$ = 1.62 for these experiments. As the probe particle will remain in the equilibrium position defined by the optical force and the drag force, the fluid velocity can than be expressed by\n\\begin{equation}\n {\\bf u} = \\frac{k\\, \\mathfrak{r}}{6\\, \\pi\\, \\mu\\, R_p \\, c_F}.\n \\label{flow_eq}\n\\end{equation}\nThe equipartition theorem \\cite{Neuman2004} is used in order to evaluate the trap stiffness $k$.\n\n\\section{Results}\nFig. \\ref{a_flow1} a) shows the displacement of the probe particle in azimuthal direction by varying the rotational frequency from 1\\,$^{-1}$ to 4\\,s$^{-1}$ of a five-particle rotor. The rotational magnetic field is applied for approx. 4\\,s for each measurement followed by a pause period to clearly identify the distinct regions. Here, the rotor radius is $R_r$ = 7.2\\,\\textmu m and the distance of the probe particle to the rotor center is $r$ = 9.8\\,\\textmu m. Note that the measured shift of the probe particle is constant for a steady rotation (gray shaded areas) and is independent on the instantaneous rotor orientation. That indicates that the last term of Eq. \\eqref{u_phi} is averaged during the measurements.\nAs depicted in Fig. \\ref{a_flow1} b) the azimuthal velocity linearly increases with the rotational frequency.\nThe averaged azimuthal velocities are shown for the five-particle rotor corresponding to Fig. \\ref{a_flow1} a) and a three-particle rotor with $R_r$ = 4.4\\,\\textmu m at the distance of $r$ = 7.1\\,\\textmu m at different rotational frequencies, exemplary.\nIn evidence, the five-particle rotor generates higher fluid flow as the three-particle rotor.\n\nIn Fig. \\ref{a_distance1} the azimuthal fluid velocity in dependence of the distance to the rotor is illustrated for the two different rotor types. The measured points represent the average values of five measurements at constant conditions.\nThe graphs reveal that there is a marked decrease of the fluid velocity with increasing the distance to the rotor. The plotted black lines correspond to fits using Eq. \\eqref{u_phi} namely $u_{\\varphi} = \\alpha r^{-2} + \\beta r^{-4}$, whereas in \\cite{DiLeonardo2006,Landau1966} for a rotating sphere only decrease as $r^{-2}$ has been reported.\n\n\nEq. \\eqref{u_r} indicates that the radial velocity component leads to an oscillating fluid flow with twice the rotor frequency. Fig. \\ref{r_displacement} a) highlights this characteristic by showing the radial displacement of the probe particle relative to the period $T = 2 \\pi \/ \\omega$.\nThe oscillation is measured by a rotor with $R_r = 7.2$\\,\\textmu m at the distance $r = 9.8$\\,\\textmu m and rotational frequency of 4\\,s$^{-1}$.\nThe black curve represents a sine function fit.\nThe period calculated with discrete Fourier transform (DFT) is the half of the period of the external magnetic field and thus confirms to the theoretical model.\nThe maximum positive displacement of the probe particle is reached when the rotor is vertical aligned. Vice versa the probe particle experiences its maximum negative displacement when the rotor is horizontal aligned.\n\nAt this point it is not possible to calculate the exact value of the radial fluid velocity. The constant shift of the probe particle in azimuthal direction (cf. Fig. \\ref{a_flow1} a)) leads to an unknown change of optical trapping force. Hence, Eq. \\eqref{flow_eq} can not be applied here. However, it can be noted that higher rotating frequencies lead to an increased amplitude of the displacement (Fig. \\ref{r_displacement} b)).\n\n\n\n\\section{Conclusion}\nIn conclusion, we have presented theoretical and experimental evidence that a non-axisymmetric rotation\nof a rod-like rotor at low-Reynolds number flow leads to a quite complex\nspatio-temporal flow pattern.\nThe validity of the established theoretical model is demonstrated by means of experimental investigations for azimuthal and radial flow fields.\nIn particular, the presence of an oscillating radial velocity component not generated in purely axisymmetric rotations \\cite{Landau1966} is a quite intriguing new feature.\nIt will be a task for future experiments to further classify the radial flow component especially with respect to its power law behavior with increasing distance from the rotor.\nThese findings might be important for mixing and driving \nin microfluidic devices as well as for the theory of \nswimming at low-Reynolds numbers.\nFurthermore, this experimental technique provides the measurement of fluid flow in microfluidic systems with high precision.\n\n\\section{Ackowledgments}\nJ.K. and A.O. are greatful for support within the Reinhardt Koeselleck project (OS 188\/28-1) of the German Research Foundation \n(DFG).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}