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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nMetal--line absorption in the intergalactic medium, or\nIGM,\\footnote{Throughout this paper, we use the terms ``{\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}\ncloud'', ``forest cloud'', and ``IGM'' somewhat interchangeably to\ndesignate {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} absorption with $\\tau_{912} < 1$.} is\nastrophysically interesting because the absorption properties can be\nexploited to reveal the star formation, chemical enrichment, and\nionization histories of the universe.\nThis provides a motivation for studying metal lines in {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} absorbers\nover as wide a redshift range as possible and for sampling transitions\ncovering as many ionization levels and chemical species as possible\n(cf.~{\\cite{uffe}; \\cite{rauch97}).\nObservations (\\cite{hulya}; \\cite{lulya}; \\cite{kim}) and numerical\nsimulations (\\cite{jordi}; \\cite{zhang}; \\cite{rad}; \\cite{norman})\nhave revealed that the forest is rapidly evolving with\nredshift from $z\\sim4$ to $z\\sim1$, and that the absorbing gas is\nhoused in a wide range of cosmic structures undergoing a wide range of\ndynamical processes. \nAt $z\\sim 2$, {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} clouds contain the majority of the baryon content\nof the universe.\nAt lower redshifts, {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} clouds are thought to be more directly\nassociated with low surface brightness and\/or dwarf galaxies\n(\\cite{salpeter}; \\cite{shull96}; \\cite{linder}), with the outer disks\nand halos of high surface brightness galaxies (\\cite{lanzetta};\n\\cite{lebrunlya}), or with the remnant material left over from the\nformation of galaxies and\/or small galaxy groups (\\cite{vangorkem};\n\\cite{bowen}; \\cite{lebrunlya}).\nStudies of the metal content and ionization conditions in these low\nredshift forest clouds, especially in the context of their association\n(or lack of association) with galaxies, could provide the\n``missing--link'' evidence necessary for inferring the evolving\ninterplay between the IGM and galaxies or the presence of low surface\nbrightness galaxies at higher redshifts.\n\nA limited number of strong metal--line species have now been\nseen in high ionization transitions at $z\\sim 2.5$ (\\cite{tytlereso};\n\\cite{cowie}; \\cite{songaila}).\nHowever, the chemical and ionization conditions of {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} clouds at\nlow redshifts ($z\\leq1$) remain unexplored because they require\ntime--intensive programs using HST.  \nRelative to $z\\sim2.5$, the meta--galactic UV background flux (UVB) at\n$z<1$ is reduced by a factor of $\\sim 5$ and its shape may be softened\nby stellar photons escaping bright field galaxies\n(\\cite{jeanmichel97}; \\cite{giallongo97}; \\cite{bergeronkp94}, and\nreferences therein).\nThus, the IGM ionization conditions may have evolved so that low\nionization species, especially the resonant {{\\rm Mg}\\kern 0.1em{\\sc ii}~$\\lambda\\lambda 2976, 2803$} doublet and\nseveral of the stronger {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} transitions, are detectable  in\n{\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} clouds.  \nAs discussed below, these particular species are well suited for\nunderstanding chemical enrichment histories.\n\nSongaila \\& Cowie (1996, hereafter SC96) detected {\\hbox{{\\rm C}\\kern 0.1em{\\sc iv}}} absorption in\n$\\simeq 75$\\% of all {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} clouds at $z\\sim2.5$ with $\\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\geq\n14.5$~{\\hbox{cm$^{-2}$}} and concluded that roughly 50\\% of $\\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq\n14.5$~{\\hbox{cm$^{-2}$}} clouds could have primordial abundances.\nThey also reported {\\hbox{{\\rm Si}\\kern 0.1em{\\sc iv}}} and {\\hbox{{\\rm C}\\kern 0.1em{\\sc ii}}} absorption in a fraction of the\n{\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} clouds (including ``partial'' Lyman limit systems).\nBased upon the photoionization models of Bergeron \\& Stasi\\'nska\n(1986)\\nocite{jbands}, SC96 find the metallicity of $z\\sim2.5$ {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}\nclouds to be $[Z\/Z_{\\odot}] \\sim -2$ and to be fairly uniform, with\nabout 1~dex of scatter.\\footnote{Throughout this paper, we use the\nnotation $[Z\/Z{\\odot}] = \\log Z - \\log Z_{\\odot}$, and $[X\/Y] = \\log\n(X\/Y) - \\log (X\/Y)_{\\odot}$, where $X$ and $Y$ are any two elements.}\nThey also report [Si\/C] ratios consistent with Galactic Halo stars\n(metal poor late--type stars), in that the $\\alpha$--group silicon is\nenhanced by a factor of three over the carbon.\nThis conclusion, however, is sensitive to the assumed UVB continuum\nshape, especially the question of how much star bursting galaxies\ncontribute to the UVB, and its non--uniformity, at higher redshifts\n(\\cite{giroux}).\n\nConsidering the mechanisms and range of environments that could\nplausibly give rise to metals in what are traditionally known as\n{\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} clouds, it is difficult to understand a high level of\nuniformity in their chemical enrichment histories.\nAs proposed by Tytler et~al.\\  (1995)\\nocite{tytlereso}, there are at\nleast three obvious mechanisms for the enrichment.\n\n(1) The larger $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$ clouds may be gravitationally bound with\ninternal gravitational instabilities in which they produce their\nown stars, which in turn distribute the metals throughout the cloud.\nThis type of object has little distinction from a galaxy.\nThis {\\it in situ\\\/} process would likely give rise to a strong\nmetallicity dependency with {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} column density, unless a well--tuned\nmechanism governing star formation yielded a uniform chemical\nenrichment history of the IGM, as suggested by Cowie et~al.\\  (1995).\nSuch a mechanism would likely represent non--standard star formation\nprocesses.\n\n(2) The metals may be produced in protogalaxies and then be\nwidely distributed via mechanical ejection from merging events\n(\\cite{gnedin96}) or from correlated supernovae (SNe) (\\cite{cen92}).  \nThis implies that {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} clouds formed after the metals were\ndistributed around the metal producing galaxies.  \nThe scenario also predicts that the metal enriched {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} clouds, as\nopposed to ``{\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}--only'' clouds, would cluster like galaxies.\n\n(3) Population III stars, formed at $z> 10$ and somewhat uniformly\nspread throughout the IGM, may have distributed metals into the IGM\nprior to the first protogalaxies.\nA population of {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest clouds that have been enriched by\nPopulation III stars may exhibit IGM chemical conditions that are\nrelatively unchanged from the epoch of the first stars.\nIf so, this population would be ideal for studying the intensity and\ncontinuum shape evolution of the UVB from $0<z<4$, since the\nchanging ionization conditions could be used to deconvolve the\nnon--evolving chemical conditions from the evolving UVB.\nAs such, the detection of extremely metal poor stars in the Galaxy\nhalo would also be very interesting, since their presence would\nsuggest the presence of Population III stars (cf.~\\cite{ostriker}).\n\nThe chemical abundance pattern can serve as a clue to the origin,\nphysical environment, and chemical enrichment history of any given\n{\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} absorber.\nThere are at least two major uncertainties involved in measuring \nrelative abundances using QSO absorption lines: ionization\ncorrections and dust grain depletion patterns.  \nThe ionization correction provides good reason for observing a wide\nrange of ionization levels.\nMoreover, the ionization corrections are sensitive to the intensity\nand continuum shape of the ionizing radiation, possibly providing even\nfurther leverage for understanding local environments and chemical\nenrichment history.\nDust depletion does not effect all $\\alpha$--group elements (for\nexample, sulfur is not readily incorporated onto dust grains) nor all\nFe--group elements (for example, zinc).  \nHowever, those elements provide neither the strong UV absorption lines\nneeded to accurately probe clouds with $\\tau_{912} < 1$, nor the\ntransitions observable from the ground for redshifts below $z\\sim 1$.\nThe strongest observable transitions are the {{\\rm Mg}\\kern 0.1em{\\sc ii}~$\\lambda\\lambda 2976, 2803$} doublet\n($\\alpha$--group), and {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} $\\lambda 2344$, 2382, and 2600.\nUnfortunately, both magnesium and iron can deplete onto dust grains\nand their depletion levels are environment dependent.\n\nFor each of the three scenarios suggested above, it is expected that\nthe relative elemental abundances of an enriched cloud should reflect\nthe $\\alpha$--group enhanced yield of Type II SNe (note that this is\nconsistent with the results of SC96).\nIn essence, the picture is simple: if the $\\alpha$--group elements are\nenhanced relative to the Fe--group then the chemical enrichment has\nbeen dominated by Type II SNe.  \nBased upon the $\\hbox{[Si\/Fe]}$, $\\hbox{[S\/Fe]}$, $\\hbox{[O\/Fe]}$\nratios measured in Galactic halo stars, this pattern is seen for\n$\\hbox{[Fe\/H]} \\leq -1$ (see \\cite{lauroesch}).\nIn the case of scenario (3) presented above, it is likely that only a\nsingle burst, or episode, of star formation would have occurred and\nthat the metal production arise exclusively from Type II SNe.  \nIn the case of scenario (2), the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} clouds would be far from the\ngalaxies; the only metal enriched gas that could infiltrate clouds\nforming from the primordial IGM would necessarily be ejected from\ncorrelated Type II SNe bursts.\n\nIf the abundance pattern is more in line with solar proportions\n(i.e,~$\\hbox{[Mg\/Fe]} \\sim \\hbox{[Si\/Fe]} \\sim \\hbox{[Fe\/H]} \\sim 0$),\nthen the picture is that Fe--group elements have been built up over a\nlonger time scale via Type Ia SNe.\nThis implies a star formation history local to the cloud that would\nhave been relatively quiescent for $\\sim \\hbox{Gyr}$ prior to the\nepoch of the observed absorption.\nThus, if a given {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} cloud is measured to have $\\hbox{[Fe\/H]} \\geq\n-1$ and $\\hbox{[$\\alpha$\/Fe]}$--group abundance ratios near solar\nproportions, then one might infer that Type Ia SNe have played a role\nin the cloud's chemical enrichment history.\nHowever, based upon uncertainties in Type II SNe yields, Gibson,\nLoewenstein, \\& Mushotzky (1997\\nocite{gibson}) have cautioned that \nthe relative importance of Type Ia and Type II SNe as inter--cluster\npolluters remains uncertain.\n\nA type of extended gas--rich object that is seen to have\n$\\hbox{[Fe\/H]} \\geq -1$, and even $\\hbox{[Fe\/H]} \\geq 0$, is the class\nof giant low surface brightness galaxies (\\cite{bothun};\n\\cite{pickering95}; \\cite{mcgaugh}).\nAt low redshifts, the general population of low surface brightness\ngalaxies are seen to outnumber high surface brightness galaxies by a\nfactor of at least two (\\cite{dalcanton}).\nAs such, these galaxies are important tracers of low density dark\nmatter halos and structure formation from small over--density\nfluctuations. \nThey also may represent environments where the pathways of star\nformation and chemical evolution reflect non--standard\nastrophysical processes (Bothun et~al.\\  1997\\nocite{bothun}).\nIf these objects are observable in absorption over a wide range of\nredshifts, they likely will provide us a unique astrophysical\nlaboratory for broadening our present perspective on star and galaxy\nformation.\n\nIn this paper, we report the search for and discovery of {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} and\n{\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublet absorption in the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest along {PKS~0454+039} sight\nline.\nIn \\S\\ref{sec:data}, we describe the data and analysis.\nIn \\S\\ref{sec:search}, we describe the sample of {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} lines and\nour search method.\nThe absorption properties of the detected systems are presented in\n\\S\\ref{sec:systems}.\nWe apply photoionization models to the detected metal--line systems in\n\\S\\ref{sec:models}, and briefly discuss the model results in\n\\S\\ref{sec:results}.\nThe implications of the results are addressed in\n\\S\\ref{sec:discuss}.\nA brief summary is provided in \\S\\ref{sec:conclude}.\n\n\n\n\n\\section{Observations and Data Analysis}\n\\label{sec:data}\n\nThe {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} transitions were searched for in an $R=45,000$\noptical HIRES (\\cite{vogtspie}) spectrum.\nThe {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} line list was obtained from the $R=1300$ G190H and G270H\nFOS\/HST spectra of Boiss\\'e et~al.\\  (1997, hereafter BBLD).\nThree images of the {PKS~0454+039} field have been incorporated into our\nstudy so that we may attempt to identify the luminous objects giving\nrise to the absorption.\nTwo are high--spatial resolution WFPC2\/HST images using the F450W and\nF702W filters (Le~Brun et~al.\\  1997, hereafter LBBD).\nThe third is the deep $R[6930\/1500]$ image (centered on $\\lambda 6930$\nwith a FWHM band pass of 1500~{\\AA}) taken from Steidel et~al.\\ \n(1995\\nocite{ccs0454}). \nWe also draw upon the published (\\cite{ccs0454}) and\nunpublished (\\cite{chuckprivcomm}) spectroscopic redshift measurements\nof the many objects along the line of sight to the QSO.\n\nThe HIRES spectrum was obtained and reduced as described in\nChurchill (1997a\\nocite{mythesis}) and in Churchill, Vogt, \\& Charlton\n(1998\\nocite{cvc98}).\nThe HIRES spectrum has wavelength coverage $3767-6198$~{\\AA}, though\nthere are some breaks in the coverage redward of 5100~{\\AA} because\nthe single setting of the 2048X2048 CCD did not capture the complete\nfree spectral range at these wavelengths.\nThe FOS spectra were obtained, reduced, and the list of\n{\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest lines used for this study were produced as described in\nBBLD.\nThe acquisition and analysis of the high--spatial resolution WFPC2\nimages are described in LBBD.\nThey also present a synopsis of candidate galaxies along the line of\nsight to the quasar available from their study and the literature.\nThe $R$--band ground--based image and the spectroscopic identifications\nof objects in the {PKS~0454+039} field are described in Steidel et~al.\\ \n(1995).\n\n\n\n\n\\section{Searching the Forest}\n\\label{sec:search}\n \nWe searched the HIRES\/Keck spectrum of {PKS~0454+039} for {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorption\nin the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} lines reported by BBLD.\nIn the FOS spectra, the redshift range over which {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} was detected\nis $0.4163 \\leq z(\\lambda 1216) \\leq 1.3431$.\nThe redshift range over which the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} $\\lambda 1215$ transition\ncould have been detected was $0.41 \\leq z \\leq 1.69$.\nThe line list is presented in Table~\\ref{tab:mgiilimits}. \nIn all cases, they are {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}--only systems (no other corroborating\ntransitions are detected in absorption).\nWe have included only those lines from BBLD that are not\nconfused by blending or are not coincident in wavelength with \nstrong metal--line transitions from the four known metal--line\nsystems ($z = 0.072, 0.859, 1.068, 1.153$) along the line of sight.\nThe first two columns of Table~\\ref{tab:mgiilimits} are the redshift of\n{\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} absorption and the rest--frame $\\lambda 1215.67$ equivalent\nwidth, respectively.\nIn columns 3--5, values of the neutral hydrogen column density,\n$\\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$, are tabulated for Doppler $b$ values of 80, 30, and\n15~{\\kms}, respectively.\nThese are shown only to illustrate the plausible $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$ range that\nmight be inferred from the equivalent widths.\nA $b=30$~{\\kms} is representative of the median $b$ value of 34~{\\kms}\nfound by Kim et~al.\\  (1997\\nocite{kim}) at redshifts $2<z<3$.  \nThere is evidence that the median $b$ value increases with decreasing\nredshift, which is believed to be due to kinematic substructure\nevolution for $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 14.0$~{\\hbox{cm$^{-2}$}} clouds. \nFor the higher column density clouds, the widths likely reflect\nnon--shock heated cloud temperatures (\\cite{haehnelt96}).\nA $b=80$~{\\kms} is plausible for kinematically broadened and\/or\nshock heated clouds (\\cite{kim}).\nDoppler parameters greater than $\\sim 80$~{\\kms} are likely due to\nblending (\\cite{lulya}), possibly of physically distinct systems.\nThe $b=15$~{\\kms} value is the ``lower cut off'' value found at high\nredshift (\\cite{lulya}, but also see \\cite{hulya}).\n\nThe sensitivity of the search, as a function of redshift, is\nquantified in terms of the rest--frame limiting equivalent width \nof the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} $\\lambda 2796$ transition.\nIn Figure~\\ref{fig:ewlimits}, we present the sensitivity curve, where\nwe have chosen to use a 5$\\sigma$ significance level.\nThe redshift range over which {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublets could be detected in\nthe HIRES spectrum is $0.3466 \\leq z(\\lambda 2796) \\leq 1.2134$.\nThere are small gaps in the coverage above $z(\\lambda 2796)=0.8340$\nthat increase with increasing redshift.\nThe 5$\\sigma$ observed equivalent width detection limit ranged from\n0.007 to 0.020~{\\AA}, except for $z(\\lambda 2796) \\leq 0.4662$, where\nit ranges from 0.020 to 0.035~{\\AA}.\nThe results, including the detection limits and the limiting column\ndensities of {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}, which are based upon linear curve of growth\nanalysis, are presented in columns 6 and 7 of\nTable~\\ref{tab:mgiilimits}.\nOnly those redshifts for which both transitions of the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublet\ncould be observed are tabulated.\n\nAbsorption features were defined using our own interactive software,\nwhich is based upon the detection algorithms of the QSO Absorption Line\nKey Project (\\cite{donspaper}).\nThe criteria that define a confirmed {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublet are presented by \nChurchill et~al.\\  (1997)\\nocite{crcv97}, who have searched 26 HIRES\/Keck\nQSO spectra for weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} systems.\nThese lines are fit with Gaussians to obtain their equivalent widths\nand observed central wavelengths.\nTo locate candidate {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublets, the candidate $\\lambda 2796$\nline centroid and detection aperture (full width at the continuum) is\nshifted to the expected location of the $\\lambda 2803$ line and the\nsignificance level is measured.\nAn acceptable candidate for the weaker member of the doublet occurs\nwhen the detection significance level is greater than or equal to the\nratio of the transition $f\\lambda$ times the significance level of the\nstronger member.\nThen, a ``chance probability'' is computed by scanning the spectrum\nwith the detection aperture over the spectrum for $\\sim 50$~{\\AA} to\nboth sides of the candidate and computing the fraction of detected\nfeatures (both emission and absorption) with a significance level\ngreater than or equal to the candidate $\\lambda 2803$ line.\nMost bonafide {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublets have chance probabilities of $\\leq\n10^{-6}$, though a very few have probabilities as large as $\\sim\n10^{-3}$.\n\nTo a 5$\\sigma$ limit of $\\log N({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) \\sim 11.3$~{\\hbox{cm$^{-2}$}}, the\n{\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublet was detected in two of the twenty--eight {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} lines\nin the list, which is a success rate of $\\sim 7$\\%.\nThe two weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} systems found in the HIRES spectra, have\n$z=0.6428$ and $z=0.9315$.\nThe data for these systems are presented in Figure~\\ref{fig:data} and their\nmeasured properties are listed in Tables~\\ref{tab:systems} and\n\\ref{tab:limits}.\nFor both, the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublet and at least one transition of {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}}\nwas detected.  \nBelow, we describe the measured properties of the two detected systems.\n\n\n\n\n\\section{Properties of the Absorbers}\n\\label{sec:systems}\n\nIn this work, we concentrate on the properties of the two absorbers\nfor which {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} has been detected.\nHere we note that the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} equivalent widths are among the smallest\nin the sample of 28 (there are seven as small as or smaller than the\n$z=0.6428$ absorber and two as small as or smaller than the $z=0.9315$\nabsorber).\nAlso, we note that the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} transitions have been detected\na factor of five to ten above the $5\\sigma$ detection limits of the\nHIRES spectra.  \nGiven the stringent limits on the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} column densities for the\nremaining {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} absorbers, and the fact that majority appear to have\nhigher $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$ than the two exhibiting {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}, it may be that there\nis a large dynamic range in the $N({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}})\/N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$ ratio.\nAs noted in \\S\\ref{sec:intro}, this is not the case for {\\hbox{{\\rm C}\\kern 0.1em{\\sc iv}}}\nabsorption in the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest.\n\nHowever, we note that upper limits on the metallicities are not very\nrestrictive if we assume a typical $b$ parameter of 30~{\\kms} and \nphotoionization by the UVB (using CLOUDY; see Figure~11 of\nChurchill et~al.\\  1997\\nocite{crcv97}).\nFor a {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} cloud with $W_{\\rm r}({\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}) \\sim 0.6$~{\\AA} and\n$N({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) < 11.4$~{\\hbox{cm$^{-2}$}}, the upper limit on $[Z\/Z_{\\odot}]$ ranges\nfrom $-0.5$ to $-2.8$, depending upon ionization level.\nFor a {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} cloud with $W_{\\rm r}({\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}) \\sim 0.4$ and\n$N({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) < 11.4$~{\\hbox{cm$^{-2}$}}, the upper limit ranges from $+0.4$ to $-2.2$\nas the cloud becomes highly ionized.\nThus, not a great deal can be said about the range of metallicities in\nthe {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest at $z\\leq 1$ based upon our {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} upper limits.\n\n\\subsection{The $z=0.6428$ System}\n\nIn the left hand panels of Figure~\\ref{fig:data}, the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and\n{\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} HIRES profiles are presented.  \nAlso shown (top panel) is the FOS spectrum, with the position of the\ncorresponding {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} line marked with a tick.\nAlong with the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublet, {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} $\\lambda2383$ and $\\lambda\n2600$ were clearly detected.\nThe weaker {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} transitions were covered by the spectrum, but were\nnot found to the 5$\\sigma$ significance level.  \nIn part, this is due to the decreasing signal to noise below\n4000~{\\AA} where the HIRES sensitivity drops rapidly (see\nFigure~\\ref{fig:ewlimits}).\nIn the upper panel of Table~\\ref{tab:systems}, the measured\nrest--frame equivalent widths, column densities and Doppler $b$\nparameters are presented.  \n\nThe {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublet ratio is $1.4\\pm0.1$, and both the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and\n{\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} lines are partially resolved.\nBased upon the apparent optical depth profiles (cf.~\\cite{savage91}),\nthere is evidence for unresolved saturation in the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublet.\nIt may be that there are two or more very narrow absorbing components\ngiving rise the profile, but the signal--to--noise ratio is not high\nenough to model the data to this level.\nThe column densities and $b$ parameters are obtained using Voigt Profile\n(VP) fits that incorporated both the atomic physics and the\ninstrumental spread function.\nWe used the program {\\sc minfit} (\\cite{mythesis}), which performs an\niterative $\\chi^{2}$ minimization between the data and the model\nspectra [see Churchill (1997a\\nocite{mythesis}) for a detailed\ndescription of the convergence criteria and the error computations].\nA VP model with two components was handed to {\\sc minfit}, but it\nreturned a single component model based upon the criterion that there\nwas no statistically significant difference between the best fit\nsingle component and two component models.\nThe measured VP column densities for the HIRES profiles are $\\log\nN({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) = 12.74\\pm0.02$~{\\hbox{cm$^{-2}$}} and $\\log N({\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}}) =\n12.46\\pm0.06$~{\\hbox{cm$^{-2}$}}.\nTheir respective $b$ parameters are $b_{\\rm tot}({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) =\n5.7\\pm0.3$~{\\kms} and $b_{\\rm tot}({\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}}) = 4.3\\pm1.3$~{\\kms}. \nThe reduced $\\chi^{2}_{\\nu}$ for the simultaneous VP fit to the\n{\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublet and the two {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} transitions is 0.96, where the\ndegrees of freedom is $\\nu = 107$.\nIn principle, the contribution of turbulent broadening to the\nprofiles, $b_{\\rm turb}\/b_{\\rm tot}$, could be determined from the\nratio of the atomic masses of iron and magnesium (see\neq.~[\\ref{eq:btot}] and eq.~[\\ref{eq:bXofbtot}]), but the\nuncertainties are too large to directly place useful limits on $b_{\\rm\nturb}$.\n\nAs we will discuss below, well determined uncertainties in the VP\nquantities are central to constraining the ionization, thermal, and\nchemical conditions in the absorbing gas ``cloud''.\nVP fits are particularly robust for profiles in this regime of column\ndensity and width.\nAs shown in Churchill (1997a\\nocite{mythesis}), the quoted\nuncertainties in the VP quantities are consistent with the spread in\nthese quantities measured from VP fits to 1000 simulated spectra with\nsimilar signal to noise ratios.\nThus, the measured column density and line broadening, and their\nuncertainties, are considered to be robust.\n\nIn the FOS spectrum, the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} absorption is the central line in a\ntriple--blend feature.\nIn Figure~\\ref{fig:midas}, we show the deblending fit. \nThe adjacent lines in the blended feature are {\\hbox{{\\rm Ly}\\kern 0.1em$7$}} at $z=1.1536$,\nand {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} at $z=0.6448$.  \nThe lone feature at 2004.5~{\\AA} is {\\hbox{{\\rm Ly}\\kern 0.1em$6$}} at $z=1.1536$, which is\nthe redshift of a strong {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorber studied elsewhere\n(Churchill et~al.\\  1998\\nocite{cvc98}).\nThough the fit is not unique, we have adopted the presented result,\nwhich yielded a rest--frame {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} equivalent width of $W_{\\rm r} =\n0.33\\pm0.03$~{\\AA}.\nBecause of the low resolution of the FOS spectrum, we could not obtain\nestimates of the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} column density and $b$ parameter directly from\nthe data.  \nThere is no coverage at the expected position of the Lyman limit, so\nwe cannot place an observed upper limit on $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$.\nThe {\\hbox{{\\rm Ly}\\kern 0.1em$\\beta$}} falls just blueward of the Lyman limit at 1695~{\\AA} due to\nthe $z=0.8596$ damped {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} absorber, and thus cannot be detected.\nIf the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} and the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} arise in the same physical locations in\nthe absorber, then the total {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} $b$ parameter is constrained to be\n$5.7 \\leq b({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 29.8$~{\\kms} (see Figure~\\ref{fig:z06428}a)\nincluding the spread introduced by the uncertainty in the measured\n$b_{\\rm tot}({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}})$.\nThe lower limit corresponds to the case in which turbulence or bulk\nmotions dominate the line broadening and the upper limit corresponds\nto a thermal motions scaling, $\\sqrt{24} b_{\\rm tot}({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}})$.\nFrom the curve of growth, we have estimated that the inferred \n$b({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$ range translates to a neutral hydrogen column density range\nof $14.2 \\leq \\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 17.6$~{\\hbox{cm$^{-2}$}}.\nFor this estimate, we have included the spread introduced by the\nuncertainty in the measured $W_{\\rm r}({\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}})$, which dominates over\nthe uncertainty in $b_{\\rm tot}({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$.\n\nThe FOS spectrum also covers several other transitions from a variety\nof species and over a wide range of ionization potentials.  \nIt is important to thoroughly check for the presence of absorption from\nthese species and to place limits on their column densities when no\nabsorption is detected.\nThese limits may be useful for further\nconstraining the chemical and ionization conditions of the absorber,\neven if the sensitivity level of the FOS spectrum is not very high.\nThus, we have systematically searched the FOS spectrum for other\ntransitions associated with the $z=0.6428$ absorber, using the\ndetection method described by Schneider et~al.\\  (1993).\nDetails of the search are presented in Appendix~\\ref{app:fossearch}\nand selected results are tabulated in Table~\\ref{tab:limits}.\n\n\n\\subsection{The $z=0.9315$ System}\n\nIn the right hand panels of Figure~\\ref{fig:data}, the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and\n{\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} HIRES profiles are presented.  \nAlso shown (top panel) is the FOS spectrum, with the position of\nthe corresponding {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} line marked with a tick.\nAlong with the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublet, the {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} $\\lambda2383$ transition\nwas detected. \nThe {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} $\\lambda2600$ transition may have been measurable as well,\nbut its predicted location coincided by chance with that of the pen\nmark (i.e.~``The Blob'') on the HIRES Tektronic's CCD.\nThe weaker {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} transitions were covered by the spectrum, but were\nnot found to the 5$\\sigma$ significance level.  \nTheir limits are consistent with the {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} $\\lambda 2383$ detection.\n\nIn the lower panel of Table~\\ref{tab:systems}, the measured\nrest--frame equivalent widths, column densities and Doppler $b$\nparameters are presented.  \nThe measured {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublet ratio is $2.0\\pm0.1$, though it may \nbe closer to 1.8, as we discuss below.\nThe {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and the {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} lines are unresolved.\nFor unresolved lines, it is difficult to accurately determine the\ncolumn densities and $b$ parameters from profile fitting. \n\nWe have performed extensive VP fitting simulations of the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}\ndoublet for this system.\nThe constraints for adopting the best model were the measured\n{\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} $\\lambda 2796$ equivalent width and the doublet ratio.\nUsing the curve of growth, we explored a grid of column densities and\n$b$ parameters that were consistent with the measured equivalent width\nof the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} $\\lambda 2796$ transition.\nThe grid range was $12.1 \\leq \\log N({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) \\leq 14.2$~{\\hbox{cm$^{-2}$}},\ncorresponding to $4.0 \\geq b({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) \\geq 0.4$~{\\kms}.\nThe increments in column density were 0.1 dex.\nFor each grid location, 500 spectra were simulated, convolved with the\nHIRES instrument spread function, sampled at the HIRES pixelization,\nand degraded to the signal--to--noise ratio of the observed data.\nThe simulation output consisted of the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} $\\lambda 2796$\nequivalent width, the doublet ratio, and the VP column densities and\n$b$ parameters from {\\sc minfit}.\n\nFor $b\\leq1$~{\\kms}, we found that we could not recover the measured\nequivalent width, nor the doublet ratio; they both decrease\ndramatically with decreasing $b$.\nThis is due to the finite pixelization of HIRES.\nAs $b$ is reduced, the line depth increases.\nAs the pre--instrumentally broadened line width drops below that of a\nsingle pixel, saturation losses dominate.\nIt could be argued that the measured equivalent width already reflects\nthis (that the absorption is actually stronger than the\nmeasured value) and that larger equivalent widths should be explored.\nFortunately, the VP fits recovered the input $b$ values to both high\naccuracy and precision over the full range explored.\nFor $b\\leq1$~{\\kms}, no matter the value of the equivalent width, the\ndoublet ratio could never be made consistent with the data (within\n3$\\sigma$).\nFor the lower limit on $b$, we adopted the criterion that the measured\ndoublet ratio would be a 3$\\sigma$ outlier in the distribution of\nsimulated VP fits.\nFor the upper limit on $b$, we adopted the criterion that the measured\n$b$ would be a $3\\sigma$ outlier from the mode of the fitted $b$\ndistribution.\n\nA caveat is worth noting.\nWe also explored simulations in which the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublets were fit\nindividually, rather than simultaneously.\nFrom this we concluded that the observed $\\lambda 2803$ transition is\nlikely compromised by a possible flat fielding artifact in its blue\nwing.\nThis is consistent with the visual appearance of the data in\ncomparison to the many simulated spectra and with the measured $\\chi\n^{2}_{\\nu}$ for the VP fits.\nThe value of $\\chi ^{2}_{\\nu}$ was 1.29 with $\\nu = 74$ when all three\ntransitions were fit simultaneously.\nIf just the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublet was fit, then $\\chi ^{2}_{\\nu} = 1.47$\nwith $\\nu = 51$.\nThis value is dominated by the ``poorer'' fit to the $\\lambda 2803$\ntransition, which by itself was  $\\chi ^{2}_{\\nu} = 2.13$ with\n$\\nu = 24$.\nIn contrast, the VP fit to just the $\\lambda 2796$ transition yielded\n$\\chi ^{2}_{\\nu} = 0.97$ with $\\nu = 24$.\nThe effect of this residual flux in the $\\lambda 2803$ transition was\nto push the measured $b$ value down to 1.5~{\\kms}, when the doublet\nwas fit simultaneously.\nWhen the observed $\\lambda 2803$ transition was omitted from the VP\nfit to the data, the resulting {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} $b$ parameter was consistent\nwith the mean of the simulations.\nBased upon these considerations, we have omitted the observed $\\lambda\n2803$ transition from the VP fit results; the measured $b$ parameter\nused for interpreting the simulations was obtained by a fit to the\n$\\lambda 2796$ transition only.\nSince we have adopted the assumption that the observed $\\lambda 2803$ \ntransition has been compromised, we have adopted the ``best'' doublet \nratio from the simulations, ${\\hbox{\\sc dr}}=1.8\\pm0.1$.\nThis implies that the equivalent width of the $\\lambda 2803$ line is \nslightly larger than that formally measured from the data.\nIf the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} {\\hbox{\\sc dr}} is 1.8, then rest frame $\\lambda 2803$ equivalent\nwidth is $\\sim 0.023$~{\\AA}.\n\nThe adopted VP column densities are $\\log N({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) =\n12.24\\pm0.09$~{\\hbox{cm$^{-2}$}} and $\\log N({\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}}) = 12.29\\pm0.08$~{\\hbox{cm$^{-2}$}}.\nTheir respective $b$ parameters $b_{\\rm tot}({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) =\n2.2\\pm0.5$~{\\kms} and $b_{\\rm tot}({\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}}) = 2.3\\pm1.6$~{\\kms}. \nAs with the $z=0.6428$, the value of $b_{\\rm turb}\/b_{\\rm tot}$ for\nthe system could be determined from the ratio of the atomic masses of\niron and magnesium (see eq.~[\\ref{eq:btot}] and\neq.~[\\ref{eq:bXofbtot}]), but the uncertainties are too large to\ndirectly place useful limits on $b_{\\rm turb}$.\n\nBecause of the low resolution of the FOS spectrum, we could not obtain\nestimates of the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} column density and $b$ parameter directly from\nthe data.  \nHowever, the wavelength range over which the Lyman limit break could\nbe observed is present in the spectrum at 1760.9~{\\AA}.\nThere is no apparent flux decrement at the expected position of the\nbreak. \nHowever, the signal--to--noise ratio is low, $\\sim 5$, and this places\na $3\\sigma$ limit of 1.6 on the flux ratio across the break.\nThis corresponds to an upper limit $\\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\sim 16.5$~{\\hbox{cm$^{-2}$}}.\nThe {\\hbox{{\\rm Ly}\\kern 0.1em$\\beta$}} transition is covered at 1981.2~{\\AA}, but the region is\ndominated by the $z=1.1537$ {\\hbox{{\\rm Ly}\\kern 0.1em$9$}} line at 1983.1~{\\AA}, so {\\hbox{{\\rm Ly}\\kern 0.1em$\\beta$}}\ndoes not provide a constraint on $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$.\nIf the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} and the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} arise in the same physical locations in\nthe absorber, then the total {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} $b$ parameter is constrained to be\n$1.8 \\leq b({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 13.0$~{\\kms} (see Figure~\\ref{fig:z09315}a),\nincluding the spread introduced by the uncertainty in the measured\n$b_{\\rm tot}({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}})$.\nThe lower limit corresponds to the case in which turbulence or bulk\nmotions dominate the line broadening and the upper limit corresponds\nto a thermal motions scaling, $\\sqrt{24} b_{\\rm tot}({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}})$.\nFrom the curve of growth, we have estimated that the inferred $b$\nrange translates to a neutral hydrogen column density range of \n$13.6 \\leq \\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 16.5$~{\\hbox{cm$^{-2}$}}, where we have adopted the\nupper limit from the Lyman limit break constraint.\nWe have included the spread introduced by the uncertainty in the\nmeasured $W_{\\rm r}({\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}})$ in this estimate,  which dominates over\nthe uncertainty in $b_{\\rm tot}({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}})$.\n\nAs with the $z=0.6428$ absorber, we have systematically searched the FOS\nspectrum for other transitions associated with the $z=0.9315$ absorber\nusing the detection technique described by Schneider et~al.\\  (1993).\nDetails of the search are presented in Appendix~\\ref{app:fossearch}\nand selected results are tabulated in Table~\\ref{tab:limits}.\n\n\n\n\n\\section{Modeling The Absorbers}\n\\label{sec:models}\n\nIn order to better understand the two absorbers, we have attempted\nto constrain their physical conditions, i.e.~ionization and chemical\nconditions, non--thermal motions, and sizes.\nIn particular, we are interested in the relationship between the\nionizing flux, whether it is UVB or stellar\/galaxy, and the inferred\nmetallicity\/abundance pattern.\nTaken together, constraints on these two quantities may reveal a great\ndeal about the origin, history, and local environment of the\nabsorbers.\n\nWe have modeled the absorbers as single--phase photoionized clouds\nusing CLOUDY (\\cite{ferland}).  \nThe clouds were assumed to have constant density and plane--parallel\ngeometry. \nA grid of models were produced; for each model cloud the specified\nphysical conditions were (1) the ionizing continuum shape and\nintensity, (2) the abundance pattern of the metals, and (3) the cloud\nneutral column density, $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$.\nThese input quantities constitute the biggest uncertainties in\nmodeling the absorbers.\nWe used CLOUDY in optimize mode, in which the residuals between the\nmodel and the measured {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} column densities were\nminimized.\nThe two quantities allowed to vary (optimized) were (1) the\nmetallicity of the assumed metal abundance patterns, and (2)\nthe total hydrogen density, $n_{\\rm H}$.\nFor those ionization species for which column density upper limits \nwere available, we applied the upper limits to the models.\n\n\\subsection{The Photoionizing Sources}\n\nThe two physical conditions within the absorbers that are the most\ntelling of its formation history are their abundance pattern\/metallicity\nand their photoionization source, either local stellar radiation or\nthe UV background (UVB).\nIn fact, the inferred chemical conditions are sensitive to the\nintensity and shape of the ionizing flux continuum.\nThere are several scenarios and we address a few of the more obvious\nones below.\n\n\\subsubsection{The UVB Scenario}\n\nThe two absorbers, whether associated with galaxies or not, may have\nphotoionization conditions dominated by the UVB. \nTo model this possibility, we have employed the UVB spectrum of Haardt\n\\& Madau (1996\\nocite{handm96}), where the intensity has been\nnormalized at $z=0.5$ and $z=1.0$ for the $z=0.6428$ and the\n$z=0.9315$ absorbers, respectively.\nThe Haardt \\& Madau UVB spectrum accounts not only for the UV flux\nemitted by QSOs and active galactic nuclei, but also for additional UV\nflux due to the reprocessing of soft X--rays (also from the QSOs and\nactive galactic nuclei) in intervening absorbers at all redshifts.\n\nThe UVB spectra are shown in Figure~\\ref{fig:uvb}, where only a\nselect range of energies is shown.\nAlso illustrated are the locations of the ionization potentials of a\nfew key ionization species.\nThe relevant ionization potentials that we are studying are all just \nabove 1~Ryd, the ionization potential of {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}.\n{\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} have ionization potentials of 1.11 and 1.19~Ryd,\nrespectively.\nThe {\\hbox{{\\rm C}\\kern 0.1em{\\sc ii}}} ionization potential is 1.79~Ryd, and for {\\hbox{{\\rm C}\\kern 0.1em{\\sc iii}}} is\n3.52~Ryd. \nWe mention {\\hbox{{\\rm C}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm C}\\kern 0.1em{\\sc iii}}} because they probe the {\\hbox{{\\rm He}\\kern 0.1em{\\sc i}}} edge, at\n1.81~Ryd, and because we have limits on the {\\hbox{{\\rm C}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm C}\\kern 0.1em{\\sc iii}}}\nabsorption strengths.\n\n\\subsubsection{The Stellar\/Galaxy Scenario}\n\\label{subsec:stars}\n\nThese particular clouds could be embedded within galaxies that are\naligned with the QSO on the plane of the sky (zero impact parameter),\nor they could be in the outskirts of the galaxies, i.e.~in the\nextended halo or outer disks.\nIn the latter scenario, the radiating stars can be treated as if they\nare all equidistant from the clouds.\nFor any stellar\/galaxy scenario, the number of stars and their\nspectral types, metallicities, and distances (quantities that\ndetermine the intensity and continuum shape of the ionizing flux) \nmust be consistent with known objects in the universe.\n\nThe stellar\/galactic UV flux could arise from a late--type\nsolar--metallicity stellar population, which would give rise to\na rapidly falling continuum with large {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} and {\\hbox{{\\rm He}\\kern 0.1em{\\sc i}}} breaks.\nA ``soft'' spectrum is required by the observed upper limits on the high\nionization species. \nA central question defining the scenario is: what level\ncould stellar\/galaxy flux be contributing to the UVB or\nbe completely dominating the UVB?\nWe have explored this question and have outlined the astrophysical\nprinciples in Appendix~\\ref{app:numstars}.\nWe constructed stellar\/galactic CLOUDY grids that included three\ngalactic spectral energy distribution models over a range of\nintensities and covered cases in which the stellar\/galaxy flux was\nprogressively stronger compared to the UVB and the case in which the\nUVB was locally ``extinct''.\n\nFor the ``dominant stellar--type'' scenarios, in which the cloud could\nbe near a dominant single star, we used Atlas stellar models\n(\\cite{kurucz}). \nWe produced a grid of optimized CLOUDY models using solar metallicity\nstars with $T_{\\rm eff} = 6,000$, 10,000, 15,000, 20,000, 30,000~K,\nand $\\log g = 4.4$ (the solar value).\nThe spectral shape is not sensitive to the surface gravity, but\nis quite sensitive to the metallicity and the effective surface\ntemperature.\nThe continuum falls more rapidly toward the UV for solar metallicity\nstars, so these stars have ``softer'' continua than what would be\nexpected in a low metallicity early--type galaxy.\n\nTo account for various metallicities and\/or stellar populations, we\nalso produced optimized CLOUDY grids using synthetic galaxy spectra.\nWe used a 12~Gyr single--burst Worthey (1994) model with metallicity\n$[Z\/Z_{\\odot}]=-0.7$, and a somewhat younger Worthey model with a\n8~Gyr single burst with metallicity $[Z\/Z_{\\odot}]=-2$.\nWe also used a later--type galaxy model from Bruzual \\& Charlot\n(1993\\nocite{bruzual}) with an exponentially decreasing star formation\nrate (SFR).\nThis model is a 16~Gyr stellar population with 1\\% of the total\nstar--forming mass in stars after a Gyr.\nThese models serve to bracket a reasonable spread in galaxy spectral\nproperties, given that an extreme scenario such as a star bursting\ngalaxy can be ruled out for two reasons. \nFirst, a star burst spectrum would highly ionize the gas, giving rise\nto {\\hbox{{\\rm Si}\\kern 0.1em{\\sc iv}}} and {\\hbox{{\\rm C}\\kern 0.1em{\\sc iv}}} absorption out to a galactocentric distance\nof $\\sim 100$~kpc (cf.~eq.~[2] of \\cite{giroux}).\nIn fact, we found that it was difficult to not produce too much\n{\\hbox{{\\rm Si}\\kern 0.1em{\\sc iv}}} and {\\hbox{{\\rm C}\\kern 0.1em{\\sc iv}}} even with the exponential SFR model.\nSecond, images of the {PKS~0454+039} field, in which point spread function\nremoval of the QSO has been performed to high accuracy, reveal no\nunidentified luminous objects with the characteristics of a star\nbursting galaxy to a limiting $K$ magnitude of $\\sim 20.5$.\n\n\\subsection{The Metallicity and Abundance Pattern}\n\nDifferent chemical enrichment histories and different environments can\ngive rise to a wide variety of chemical and ionization conditions.\nGiven the possible high iron to magnesium abundance ratio in these\nabsorbers, it is reasonable to assume that the clouds could arise in\nor near galaxies.  \nThus, to better understand the origin of the absorbers, we modeled\nthree abundance patterns that are taken from typical gaseous objects\nin galaxies.\nThe first is the solar abundance pattern, taken from Grevesse \\&\nAnders (1989\\nocite{grevanders89}) and Grevesse \\& Noels\n(1993\\nocite{grevnoels93}).\n\nIn the interstellar medium (ISM), both magnesium and iron deplete onto\ndust grains (cf.~{\\cite{lauroesch}; \\cite{savagearaa}).\nSince the main constraints on the CLOUDY optimization are the measured\n{\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} column densities, we also considered the effects of\nheating and cooling by grains and the dust depleted abundance patterns\nof two common interstellar environments.\nWe used the {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} abundance depletion pattern taken from Baldwin\net~al.\\  (1991\\nocite{baldwin91}), Rubin et~al.\\ \n(1991\\nocite{rubin91}), and Osterbrock, Tran, \\& Vielleux\n(1992\\nocite{osterbrock92}).  \nFor this abundance pattern we used the ``large--R'' grains\n(\\cite{baldwin91}), which are characterized by a more or less grey UV\nextinction.\nLike the solar abundances, the {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} pattern, which has\n$[\\hbox{Mg\/Fe}] = -0.07$, provides a good template for an iron--group\nenriched chemical evolution history.\n\nWe have also explored the possibility that these absorbers may\nactually be $\\alpha$--group enhanced.\nThus, we used the abundance patterns reported by Cowie \\& Songaila\n(1986\\nocite{cowie86}) for the cold and warm phases of the ISM.\nThis pattern is characterized by $[\\hbox{Mg\/Fe}] = 1.23$, and overall\nenhanced $\\alpha$--group elements.  \nThe dust grains used in these ISM models are a mixture of the graphite\nand silicates.\n\nThe presence of grains also effects the cooling and heating balance,\nand thus the ionization balance, of the clouds.\nAs mentioned, the metallicity, or the scaling factor of the input\nabundance pattern for elements heavier than helium, was allowed to\nvary.\nAs the metallicity in a cloud is increased by factors of a few, the\ncooling rates are dramatically increased and the cloud equilibrium\ntemperatures drop significantly.\nFor some clouds, the optimal metallicity was 10 to 100 times the\ninitial input and the cloud equilibrium temperature in the\n100~K range.\n\n\\subsection{Turbulent and\/or Bulk Motions}\n\nIt is not possible to obtain a direct determination of the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}\ncolumn densities for the two {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers.\nFor each, the extreme range of plausible {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} column densities can be\nestimated from the measured {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} $b$ parameter for the assumption\nof a thermal line broadening [lower $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$ limit] or turbulent\nbroadening [upper $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$ limit].\nThe range turns out to be large, $13.8 \\leq \\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) <\n17.0$~{\\hbox{cm$^{-2}$}}, when the uncertainties in $b({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$ and $W_{\\rm r}\n({\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}})$ are considered.\nWe have modeled clouds with $\\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) = 14.50$, 15.50, 15.75,\n16.00, 16.25, 16.50, 16.75, 17.00, and 17.50~{\\hbox{cm$^{-2}$}}.\nAs will be shown, there were no $\\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 15.75$~{\\hbox{cm$^{-2}$}} cloud\nmodels consistent with the data.\nSince, for a known equivalent width, the curve of growth provides a\ndirect relationship between the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} column density and $b$\nparameter, both quantities should be considered for constraining viable\ncloud models.\nThus, we have developed a technique to further constrain the\nacceptable model cloud properties using the relationship between the\nturbulent and thermal line broadening mechanisms.\n\nUsing the equilibrium temperature of the CLOUDY models and the\n$b_{\\rm tot}$ of {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}, one can narrow parameter space to those\nmodels that are self--consistent in $b_{\\rm turb}\/b_{\\rm tot}$,\ntemperature, and $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$.\nHere, $b_{\\rm turb}$ is the non--thermal contribution to the line\nbroadening.\nAssuming the contributions are Gaussian, $b_{\\rm tot}$ is written,\n\\begin{equation}\nb_{\\rm tot}^{2} = b_{\\rm therm}^{2} + b_{\\rm turb}^{2} \n                = \\frac{2kT}{m} + b_{\\rm turb}^{2} ,\n\\label{eq:btot}\n\\end{equation}\nwhere $T$ is the equilibrium kinetic temperature, and $m$ is the ion\nmass.\nIn reality, non--thermal motions are not expected to be Gaussian\n(see discussion in \\S\\ref{sec:caveats}), so eq.~[\\ref{eq:btot}] is\nsimply a parameterization of the relative contributions.\nDefining $f=b_{\\rm turb}\/b_{\\rm tot}$, the kinetic temperature of the\ncloud is written,\n\\begin{equation}\nT \\simeq 1450 \\left( 1 - f^{2}\\right) b^{2}_{\\rm tot}({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) \\quad\n\\hbox{K} .\n\\label{eq:tofbtot}\n\\end{equation}\n\nIn Figure~\\ref{fig:temp}, the predicted kinetic temperature for\nabsorbers are shown as a function of $f$.\nSchematically superimposed are the kinetic temperature ranges of\ntypical gaseous structures in the Galaxy (\\cite{savagearaa}; \n\\cite{fitzpatrickII}; \\cite{deoagn2};  \\cite{spitzer}).\nAlso shown is the inferred range of {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers, as observed\nusing QSO absorption lines (\\cite{mythesis}; Churchill et~al.\\ \n1998\\nocite{cvc98}).\nThis diagram is discussed further in \\S\\ref{sec:results}.\n\nThe $b$ parameter of each species is required in order to obtain\nits estimated column density from the curve of growth.\nUsage of the curve of growth method is consistent with our assumption\nof single--phase clouds, given that the velocity structure of the\nlow ionization absorption profiles are not complicated and are\nsuggestive of a single absorbing component.\nAssuming that all species are subject to the same thermal and\nnon--thermal conditions, the total Doppler $b$ parameter of any\nspecies, X, can be obtained from,\n\\begin{equation}\nb_{\\rm tot}({\\rm X}) = \\left\\{ \\frac{A_{\\rm Mg}}{A_{\\rm X}}\n  \\left( 1 - f^{2} \\right) + f^{2} \\right\\} ^{1\/2} b_{\\rm\n  tot}({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}})  \\quad \\hbox{\\kms} ,\n\\label{eq:bXofbtot}\n\\end{equation}\nwhere $A$ is the nucleon number ($A_{\\rm Mg} = 24$).\n\n\\subsection{Application of Constraints to Models}\n\nIn the absence of direct measurements of $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$, the most acceptable\ncloud models are those that are self--consistent in that their\n$N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$, temperatures, and $f$ are simultaneously consistent with\nthose allowed by the data.\nWith the above formalism in hand, the application for constraining a\ngiven cloud model is as follows: \nThe range of acceptable $f$ values parameterize the clouds, and\ngive the inferred non--thermal component to the line broadening.\nThe range is determined from the model cloud kinetic temperature, $T$,\nusing a modified version of eq.~[\\ref{eq:tofbtot}], \n\\begin{equation}\nf_{(\\mp)} = 1 - \\frac{1}{b_{(\\pm)}^{2}({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}})}\n\\left( \\frac{T}{1450} \\right) ,\n\\end{equation}\nwhere $b_{(\\pm)}({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) = b_{\\rm tot}({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) \\pm \\sigma _{b}$.\nThe cloud temperature was taken as a simple average, given that the\nmodel clouds are constant density by definition.\nFor clouds with $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 16.5$~{\\hbox{cm$^{-2}$}}, the temperature was\nconstant as a function of depth.\nFor the higher column density clouds, ionization and temperature\nstructure was present, but by no more than 10\\%.\nEq.~[\\ref{eq:bXofbtot}] was then used to obtain the inferred $b_{\\rm\ntot}({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$, as illustrated in Figure~\\ref{fig:z06428}a and\nFigure~\\ref{fig:z09315}a.\nUsing curve of growth analysis, the range of acceptable\nhydrogen column densities is obtained from the measured {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}\nequivalent width (and its uncertainty) and the range of acceptable\n$b_{\\rm tot}({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$.\n\n\\subsection{Caveats Regarding the Model Design}\n\\label{sec:caveats}\n\nThe quoted upper limits on the ionization species covered by the FOS\nspectrum were estimated assuming a single--phase isothermal cloud.\nThere are {\\it a priori\\\/} reasons that we have adopted this\nassumption.\nFirst, if the absorbers are {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} clouds that are pressure confined\nby a warmer, more tenuous medium (more than one thermal phase giving\nrise to absorption), they cannot be heated by the UVB\n(\\cite{guilbert}; \\cite{donahue}).\nOn the other hand, under the assumption of photoionization by the UVB,\na very low ionization parameter, $U = n_{\\gamma}\/n_{\\rm H} \\leq\n10^{-3.5}$, is required in order for the absorbing gas to give rise to\n$N({\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}}) \\sim N({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}})$ (\\cite{jbands}; \\cite{mythesis}).\nTherefore, for UVB photoionization, clouds giving rise to the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and\ncomparable {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} absorption must have low ionization conditions. \nIf strong {\\hbox{{\\rm C}\\kern 0.1em{\\sc iv}}} and {\\hbox{{\\rm Si}\\kern 0.1em{\\sc iv}}} were to arise in a UVB photoionized\nabsorber exhibiting $N({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) \\simeq N({\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}})$, then the abundance\npattern would need to be very different from solar, with a $1-2$ dex\nenhancement of iron or decrement of magnesium. \nCommon chemical evolution paths do not give rise to this type of\nabundance pattern.\n\nNonetheless, the possibility remains that the absorbers could be\nmultiphased in their ionization structures (perhaps they are not\npressure confined or are ionized by stellar\/galactic flux), so that\nthe measured {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} equivalent width may be due to a kinematic complex\nof clouds in which only one is giving rise to the observed {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and\n{\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} absorption.\nAs reported by SC96, {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} clouds at $z=2.5$ are seen to exhibit\nkinematic structure in {\\hbox{{\\rm C}\\kern 0.1em{\\sc iv}}} absorption, with the highest velocity\ncomponents more highly ionized.\nThey suggest that this arises due to layered stratification in \ncoalescing clouds.\nIf this were the case for the low redshift absorbers reported here,\nthen the upper limits provided by the FOS spectrum would not apply to\nthis higher ionization phase; the limits would be lower than those\nquoted above on account of the expected larger $b$ parameters.\n\nIf a high ionization phase exists in either absorber, it has\napparently not been detected in the FOS spectrum, suggesting that\neither its ionization conditions or abundance patterns are not giving\nrise to strong absorption lines.\nFor the ionization conditions to give rise to a multiphase absorber,\nthe neutral hydrogen column density would need to approach the Lyman\nlimit value of $10^{17.3}$~{\\hbox{cm$^{-2}$}} to provide shielding to the low\nionization phase. \nIf there is no surrounding hot phase associated with these absorbers,\nit might be difficult to understand them in terms of pressure confined\nclouds; they could either be gravitationally bound or not be in\ndynamical equilibrium.\nObservationally, some ambiguity still remains for the detection of\n{\\hbox{{\\rm C}\\kern 0.1em{\\sc iv}}} and {\\hbox{{\\rm S}\\kern 0.1em{\\sc iv}}} in the $z=0.6428$ absorber.\nThis uncertainty could be checked by higher resolution and higher\nsignal--to--noise ratio observations with STIS\/HST.\n\nA main point here is that the physical region of the absorbers\ngiving rise to the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} must have insignificant {\\hbox{{\\rm C}\\kern 0.1em{\\sc iv}}}\nand {\\hbox{{\\rm Si}\\kern 0.1em{\\sc iv}}} abundances.\nA second main point is that the estimated {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} column density in this\nregion would be an upper limit to this low ionization phase, since\nsome fraction of {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} would be arising in the high ionization phase\nas well.\nUnless the path length through a possible high ionization phase is far\ngreater than that of the low ionization phase, the estimated ranges of\nthe {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} column densities for the two absorbers cannot be\nsignificantly incorrect.\n\nThere is also the possibility that the spatial alignment of the\nneutral hydrogen in these absorbers may not directly coincide with\nthat of the ionized magnesium (ionization structure), since the\nionization potential of {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} are slightly different\n(\\cite{pbprivcomm}). \nAgain, this would imply that the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} column densities in the region\nwhere {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} are giving rise to absorption is smaller\nthan the estimated values.\n\nThe presence of dust can effect the modeling in two ways.\nFirst, for the model clouds using the {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}}\/ISM dust--depleted\nabundance pattern, there is a breakdown in the inferred metallicity\nwhen the temperatures are below $T\\sim 1000$~K.\nThe condensation temperature (the temperature at which 50\\% of an\nelement is removed from the gas phase due to depletion) of both\nmagnesium and iron is $\\sim 1300$~K.\nThis means that these cool clouds would have a significantly higher \ndepletion than the warmer clouds, and this has not been accounted for\nin the CLOUDY modeling.\nThe implication is that the optimized metallicity is in fact\nan underestimate of the gas--phase abundances, given that the\ndepletion would be larger than that input into the model.\\footnote{These\n$T\\leq 1000$~K model clouds are already characterized by quite large\nmetallicities, even without accounting for increased condensation.}\nSecond, if dust is present, it can seriously modify the intensity and\nshape of the incident UV spectrum, which would have implications for\nthe inferred source of the radiation from the photoionization models.\nWe discuss this point further in \\S\\ref{sec:stellarfail}.\n\nCentral to the model interpretation is the assumption that the line\nbroadening is governed by Gaussian distribution functions for the\nparticle velocities.\nN--body simulations with hydrodynamics (\\cite{norman}; \\cite{rad};\n\\cite{zhang}) have shown that the absorbers are filamentary structures\nand that the concept of an absorbing ``cloud'' is not altogether valid.\nOften, the simulations result in gas which is collapsing towards\n(or expanding along) a filament; this results in hydrodynamic features\nand shocks in which the distribution function of velocities is not\nGaussian.\nIf the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorption profiles give any indication of the\nvelocity distribution function, then the absorbers are not\ninconsistent with a Gaussian; in fact, very quiescent gas is suggested\nby the HIRES profiles.\n\nA final caveat is that we did not include turbulence physics in the\nCLOUDY models (only micro--turbulence and not bulk motions could have\nbeen modeled using CLOUDY).  \nThus, turbulence is not treated self consistently within the framework\nused to infer the absorber physical conditions.\nThe inclusion of turbulence would result in an additional pressure\nsource, which would effect the balance between the model cloud density\nand its depth.\nHowever, the model cloud densities were allowed to vary in an\noptimized fashion, and since the cloud depth is adjusted with each\nequilibrium calculation the pressure adjustment has little effect.\nThe second consequence of modeling turbulence is that the line--center\noptical depths of absorption transitions decrease, while they\nincrease in the line wings.\nHowever, since the majority of the model clouds in our grids were\noptically thin to neutral hydrogen, any change in the line profile\nshape has little effect on the model cloud equilibrium.\n\n\n\n\n\\section{Model Results}\n\\label{sec:results}\n\nIn the final analysis, only the scenario in which the model clouds are\nphotoionized by the UVB was both astrophysically plausible and\nconsistent with the data.\nHere, we focus on the results for the UVB scenario with the solar\nand {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}}\/ISM abundance patterns and then address to what level the \nstellar\/galactic scenarios can be ruled out as viable ionizing\nsources.\nWe defer discussion of the implications of the model results until\n\\S\\ref{sec:discuss}.\n\nThe optimized metallicities and densities are given for\nboth abundance patterns in Tables~\\ref{tab:hm06428} and\n\\ref{tab:hm09314}.\nCloud models with the $\\alpha$--group enhanced abundance pattern\n(\\cite{cowie86}) did not converge within the allowed uncertainties in\nthe {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} column densities.\nThus, we conclude that neither the $z=0.6428$ nor the $z=0.9315$\nabsorber have $\\alpha$--group enhanced abundance patterns.\nThis implies a gas--phase $[\\hbox{Fe\/H}] \\geq -1$ ({\\cite{lauroesch};\n\\cite{savagearaa}), and possibly iron--group enrichment by Type Ia\nSNe, though this remains somewhat controversial (Gibson et~al.\\ \n1997\\nocite{gibson}).\nThe $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$ versus $f$ parameter space is illustrated in\nFigures~\\ref{fig:z06428} and \\ref{fig:z09315} for the data and for the\nmodel clouds.\nThe adopted ranges for the absorber {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} column densities are defined\nby the overlap of the allowed ranges constrained by the data and by\nthe CLOUDY models.\nInterestingly, the results indicate that the clouds have a substantial\nnon--thermal line broadening (i.e.~they are turbulent or are\nundergoing differential bulk motions).\nHowever, it is striking that the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} profiles reveal a \nvelocity dispersion of only a few {\\kms}.\nThis provides a counter example to the expected large $b$ parameters\nif the gas was not dynamically settled (as found in hydrodynamic\nsimulations of the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest).\nThe very quiet nature of these absorbers indicate that our assumption\nof a well defined temperature for a settled gas ``cloud'' is well\nfounded.\nA possible, though unlikely, counter example would be if the absorbers\nwere streaming filaments seen perpendicular to their elongation and\nstreaming motion.\n\nShown in Figure~\\ref{fig:temp} are the predicted kinetic temperatures\nof the absorbers as a function of $b_{\\rm turb}\/b_{\\rm tot}$.\nTemperature and turbulence limits may provide clues to the nature of\nthe absorbing gas when the range of inferred properties are compared\nto gaseous objects typically found in galaxies.\nPlanetary nebulae have $T \\geq 40,000$~K (\\cite{deoagn2};\n\\cite{spitzer}).\nIf the $z=0.6428$ absorber has $f \\leq 0.4$, then its inferred\ntemperature is consistent with that of a planetary nebulae.  \nTypical expansion velocities of these objects have line widths of\n$\\sim 20-30$~{\\kms} (\\cite{deoagn2}).\nSince the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} Doppler parameters would also reflect\nthe expansion velocities, it is highly unlikely that the $z=0.6428$\nabsorber arises in a planetary nebula.\nFor $0.4 \\leq f \\leq 0.9$, the $z=0.6428$ absorber temperature is\nconsistent with that of individual clouds in complex {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}\nsystems (\\cite{mythesis}; Churchill et~al.\\  1998\\nocite{cvc98}). \nBoth absorbers are consistent with the temperature range of {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}}\nregions and warm {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} clouds, $7000 \\leq T({\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}}) \\leq 14,000$~K and\n$4000 \\leq T({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 7000$~K, respectively (\\cite{fitzpatrickII};\n\\cite{deoagn2}; \\cite{spitzer}).\nFor the $z=0.9315$ absorber, the inferred $b_{\\rm turb}\/b_{\\rm tot}$\nwould be $f \\leq 0.8$, and for the $z=0.6428$ absorber $f$ would be\nconfined to the narrow range $0.90 \\leq f \\leq 0.95$.\nFor these $f$ values, the non--thermal broadening would be roughly \n$b _{\\rm turb}({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) \\leq 5$~{\\kms} and $\\leq 2$~{\\kms} for the\n$z=0.6428$ and the $z=0.9315$ absorber, respectively.\nThe typical sound speeds in {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} clouds are $\\sim\n10~{\\kms}$ (\\cite{spitzer}).\nIf these absorbers are {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} or {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} clouds similar to those found\nin the Galaxy, the line of sight non--thermal broadening is well below\nthat expected for propagating disturbances in the clouds.\nIf the absorbers are dominated by turbulent motions ($f>0.98$), then\nthey must have $T \\leq 150$~K.\nDiffuse ISM clouds have typical temperatures in the range $30\n\\leq T \\leq 150$~K (\\cite{spitzer}).\nIn this regime, it would be more likely that the broadening was\ndominated by bulk flows rather than internal turbulence, given that\nthe turbulent motion would propagate at the $\\sim 0.1$~{\\kms} sound\nspeed typical of diffuse clouds (\\cite{spitzer}).\n\n\\subsection{The $z=0.6428$ Absorber Properties}\n\\label{sec:0.6428}\n\nAssuming the solar abundance pattern, the neutral hydrogen of the\n$z=0.6428$ model cloud is in the range $16.3 \\leq \\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})\n\\leq 16.8$~{\\hbox{cm$^{-2}$}}.   \nThe range of $b_{\\rm turb}\/b_{\\rm tot}$ is $0.85 \\leq f \\leq 0.93$,\nwhich correspond to the kinetic temperatures $13,000 \\geq T \\geq\n6500$~K.\nThe metallicity and model cloud density are $-0.2 \\geq\n[Z\/Z_{\\odot}] \\geq -0.7$, and $0.01 \\leq n_{\\rm H} \\leq\n0.02$~cm$^{-3}$, for the range of $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$.\nFor the {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} abundance pattern, the inferred neutral hydrogen column\ndensity is slightly higher, in the range $16.7 \\leq \\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})\n\\leq 17.2$~{\\hbox{cm$^{-2}$}}.   \nThe range of $b_{\\rm turb}\/b_{\\rm tot}$ is $0.90 \\leq f \\leq 0.95$,\nmaking this cloud kinetic temperature somewhat lower than the \nsolar abundance model. \nThe density and metallicity are $n_{\\rm H} \\simeq\n0.008$~cm$^{-3}$ and $+0.4 \\geq [Z\/Z_{\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}}] \\geq 0.0$, for the\nrange of $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$.\nIt could be that this cloud has very enhanced metallicity with an \n{\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} abundance pattern, but this is a far reaching suggestion given\nthe range allowed by the solar abundance pattern.\nStill, the cloud is inferred to have gas--phase $[\\hbox{Fe\/H}] \\geq\n-1$.\n\nThese model clouds are difficult to understand in terms of objects\ntypical of the Galactic disk or the Magellanic Clouds.\nFor one, the typical $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$ observed in Galactic objects is $\\log\nN \\geq 19.5$~{\\hbox{cm$^{-2}$}} (\\cite{savagearaa}), two or more orders of\nmagnitude greater than what is inferred for this absorber.\nSecond, the typical density of $T \\sim 10,000$~K clouds (warm low\ndensity medium) is $\\left< n_{\\rm H} \\right> \\sim 0.2$~cm~$^{-3}$\n(\\cite{spitzer}), which is higher than allowed by the optimal models.\nThird, the inferred gas--phase abundances in the the warm and cool disk\nare $[\\hbox{Fe\/H}] \\sim -1.1 $ and $ \\sim -2.1$, respectively\n(\\cite{savagearaa}).\nThese fall well below those predicted by the models.\nHowever, the metallicity range of the solar abundance pattern model is\nconsistent with $[\\hbox{Fe\/H}] \\sim -0.6$ found for the Galactic Halo.\n\n\\subsection{The $z=0.9315$ Absorber Properties}\n\\label{sec:0.9315}\n\nThe $z=0.9315$ model cloud appears to have a high gas--phase\nmetallicity, whether it be super solar or an enhanced {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} pattern.\nAssuming the solar abundance pattern, the neutral hydrogen is in the\nrange $15.8 \\leq \\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 16.3$~{\\hbox{cm$^{-2}$}}.   \nNote that this is consistent with the upper limit of 16.5~{\\hbox{cm$^{-2}$}}\ninferred from the lack of a Lyman limit break in the FOS spectrum.\nThe range of $b_{\\rm turb}\/b_{\\rm tot}$ is $0.62 \\leq f \\leq 0.94$,\nwhich correspond to the kinetic temperatures $4600 \\geq T \\geq 870$~K.\nThe metallicity and model cloud density are $+0.7 \\geq\n[Z\/Z_{\\odot}] \\geq +0.1$, and $0.2 \\leq n_{\\rm H} \\leq\n0.4$~cm$^{-3}$, for the range of $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$.\nThe optimal solar abundance model cloud is relatively dense with\nup to five times solar abundance.\nThe inferred $[\\hbox{Fe\/H}]$ is even greater for the {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} abundance\npattern model.\nThe neutral hydrogen column density is slightly higher, in the range\n$16.0 \\leq \\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 16.5$~{\\hbox{cm$^{-2}$}} (also consistent with the\nlack of a Lyman limit).\nThe range of $b_{\\rm turb}\/b_{\\rm tot}$ is $0.75 \\leq f \\leq 0.97$,\nwhich corresponds to the kinetic temperatures $3300 \\geq T \\geq 450$~K.\nThere is an inversion in the cooling curve at $T\\sim 2000$~K, which\nimplies that this absorber cannot be stable across the full range of\nallowed temperatures.  \nIt must either be a few thousand degrees or several hundred degrees.\nThe density and metallicity are $n_{\\rm H} \\simeq\n0.1$~cm$^{-3}$ and $+1.6 \\geq [Z\/Z_{\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}}] \\geq +0.7$, for the\nrespective $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$.\nThis is a metallicity enhancement of five to 40 times over the typical\nvalues seen in Galactic {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} regions (\\cite{baldwin91};\n\\cite{rubin91}; Osterbrock et~al.\\  1992\\nocite{osterbrock92}).   \nTo date, no other intervening QSO absorption system with such a\nhigh metallicity has been reported.\n\n\\subsection{Why the Stellar\/Galaxy Scenarios Fail}\n\\label{sec:stellarfail}\n\nIn what follows, we discuss the difficulties with the stellar\/galaxy\nscenarios.\nThe constraints were a trade off between the number of stars, or their\nnumber density, and the stellar population.\nThe former is constrained by astrophysics, assuming a non--extreme\nstellar environment, and by the imaging data.\nThe latter is constrained by the absorption line data, which have\nlimited the UV ionizing flux to late--type stars and\/or early--type\ngalaxies.\n\nFirst, consider the case in which the stellar\/galactic flux contributes\nto the UVB intensity.\nIn order for a stellar\/galactic contribution to modify the properties\nof a UVB model cloud, the stellar\/galactic flux must exceed the UVB at\n$\\geq 1$~Ryd, particularly in the regions $1 \\leq h\\nu \\leq 1.2$~Ryd\n(from the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} edge up to and including the {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} ionization\npotential).\nAs outlined in Appendix~\\ref{app:numstars}, if the stellar population\nis dominated by A0III and A0V stars, this requires $\\sim 10^{12}$\nstars confined to a region of space $\\sim 1$~kpc in radius. \nThis implies a stellar number density $n_{\\ast} \\geq 500$\nstars~pc$^{-3}$, which is five orders of magnitude greater than the\ndensity of A0V stars in the solar neighborhood (\\cite{allen}).\nThe required number density increases dramatically for later spectral\ntypes.\nIf, on the other hand, the stars are dominated by early--type B0V\n(B0I,III) stars, then $\\sim 10,000$ (1000) stars would be required in\na volume of radius 1~kpc.\nFor the main sequence stars, this corresponds to a number density\nabout 100 times greater than that of the solar neighborhood\n(\\cite{allen}).  \nOnly O stars can provide the UV flux necessary to match the UVB at\n1~Ryd and have a number density of stars consistent with that of a\ntypical galaxy environment.\nHowever, early--type stars give rise to high ionization absorption\nproperties, especially {\\hbox{{\\rm C}\\kern 0.1em{\\sc iii}}}, {\\hbox{{\\rm C}\\kern 0.1em{\\sc iv}}}, and {\\hbox{{\\rm Si}\\kern 0.1em{\\sc iv}}}, so that the model\ncloud conditions are not consistent with the data.\n\nThe 12 Gyr $[Z\/Z_{\\odot}] = -0.7$ Worthey (1994\\nocite{worthey}) model\nis characterized by a steep continuum slope with $\\Delta \\log \\nu\nf_{\\nu} \\sim -5$ from 1 to 1.2~Ryd.\nThis continuum shape is a smooth continuation of the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} edge, so\nthat this galaxy model had to have $\\nu f_{\\nu} \\geq 10^{8}$ times\nthat of the UVB at 5500~{\\AA} before affecting the model cloud\nproperties.\nBased upon the arguments presented in Appendix~\\ref{app:numstars}, for\nthe expected distribution of main sequence and giant stars in these\ngalaxies (\\cite{worthey}), the stellar number densities would be\nextreme, $N_{\\ast} \\geq 10^{2}$ stars~pc$^{-3}$, or $\\sim 10^{12}$\nstars within a kpc.\nEven under these extreme conditions, the 12 Gyr Worthey models yielded\n$[\\hbox{Fe\/H}] \\geq -1$.  \n\nThe 8~Gyr Worthey model with metallicity $[Z\/Z_{\\odot}]=-2$ is\ncharacterized by a continuum with a smaller {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} edge of $\\Delta \\log\n\\nu f_{\\nu} \\sim -2$ at 1~Ryd and a power law with $\\sim \\nu ^{-7}$\nout to $\\sim 1.5$~Ryd (this is steep but not nearly as steep as the 12\nGyr model).\nThe flux must be elevated to $\\nu f_{\\nu} \\geq 10^{5}$ times that of\nthe UVB at 5500~{\\AA} before affecting the model cloud properties.\nAs the flux, $\\nu f_{\\nu}(0.17)$, is increased above $\\sim\n10^{4}$~ergs~{\\hbox{cm$^{-2}$}}~s$^{-1}$, the ratio $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})\/N({\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}})$ very\nquickly changes from unity to $\\sim 10^{-5}$; the model cloud\nbecomes highly ionized and the limits on {\\hbox{{\\rm C}\\kern 0.1em{\\sc iii}}} and\/or {\\hbox{{\\rm C}\\kern 0.1em{\\sc iv}}} are\nexceeded (depending upon the specific absorber). \nTo adopt the idea that the 8 Gyr Worthey galaxy is contributing to the\nionization conditions, we would be forced to accept a very narrow\nrange of acceptable $\\nu f_{\\nu}(0.17)$ arising from the galaxy. \nThis range implies the absorbers would be embedded in the galaxy\nitself (zero impact parameter), with an extremely unrealistic number\nof stars.\nSuch a scenario is ruled out.\n\nThe exponential SFR spectrum of Bruzual \\& Charlot (1993\\nocite{bruzual})\nhas a continuum shape at $1-3$~Ryd that is similar to the Haardt \\&\nMadau UVB spectrum.  \nThus, as the galactic spectrum is slowly increased over the UVB, the\nmodel cloud properties adjust such that the ionization parameter is\nheld constant (the cloud density increases, but the metallicity and\ntemperature do not change).\nAt $\\nu f_{\\nu}(0.17) \\sim 0.1$~ergs~{\\hbox{cm$^{-2}$}}~s$^{-1}$, there is a\nslow decrease in the ratio $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})\/N({\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}})$; the model cloud\nbecomes progressively highly ionized. \nThe dominant ionization species of magnesium and iron are then\n{\\hbox{{\\rm Mg}\\kern 0.1em{\\sc iii}}} and {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc iii}}}, respectively, and the {\\hbox{{\\rm C}\\kern 0.1em{\\sc iv}}} and {\\hbox{{\\rm Si}\\kern 0.1em{\\sc iv}}} column\ndensities exceed the limits allowed by the data.\nThe exponential SFR galaxy of Bruzual and Charlot is not a viable\nphotoionizing source.\nThe galaxy flux makes little modification to the inferred cloud\nproperties (does not modify UVB only models) for a large range of\nintensities; once it does modify the UVB models, the cloud\nproperties are inconsistent with the data.\n \nNow, consider the case in which the stellar\/galactic flux is dominant.\nWe examined this by excluding the Haardt \\& Madau UVB from the incident\nflux.\nThis would require environments that are more extreme than those\npresented above, or that are shielded from the UVB but not from the\nstellar flux.\nIs it possible that dust extinction could be playing a role?\nFor a dust absorption\/scattering cross section of\n$\\sigma(\\hbox{1~Ryd}) \\sim 10^{-22}$~{\\hbox{cm$^{-2}$}} (\\cite{mathis77}), the\ntotal hydrogen column density [$N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})+N({\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}})+2N({\\rm H}_{2})$]\nrequired for $\\tau _{\\rm dust} \\sim 3$ at 1 Ryd is $\\log N \\sim\n22.5$~{\\hbox{cm$^{-2}$}}.\nGiven the upper limits on $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})$ for the absorbers, this implies\nhighly ionized gas with $N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}})\/N_{\\rm tot}({\\rm H}) \\geq 10^{-5}$.\nIn all our model clouds, the fraction of molecular hydrogen never\nincreased above $f({\\rm H}_{2}) \\leq 10^{-6}$.\n\nIn principle, one could imagine an environment in which the absorbing\ngas is embedded in a late--type dwarf galaxy that itself is\nenshrouded in dust.\nThis scenario provides the steep continuum above 1~Ryd while not\nrequiring the spectrum intensity to be elevated above the UVB.\nFor some models it was possible to achieve $\\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}}) \\simeq\n22.5$~{\\hbox{cm$^{-2}$}}, but even with no competition from the UVB, these\nshielded models required high $\\nu f_{\\nu}(0.17)$.\nAgain, as outlined in Appendix~\\ref{app:numstars}, this implies\nunrealistic stellar densities.\nIt is not simply a matter of the continuum slope, but also the\nintensity that dictates the cloud ionization conditions.\nInterestingly, these models produced very low metallicities, with\n$[Z\/Z_{\\odot}] \\sim -3$.\nAll of the above stellar\/galaxy models represent our failed attempt to\nlocate a place in parameter space consistent with low metallicity\nclouds.\n\n\n\n\n\\section{Discussion}\n\\label{sec:discuss}\n\nBased upon the success rate of finding two {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers out of 28\n{\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} absorbers along the {PKS~0454+039} line of sight, we must conclude that\nthe two absorbers are unique in some manner with respect to the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}\nforest at large.\nWe note, as mentioned in \\S\\ref{sec:systems}, that not a great deal\ncan be quantified about the range of metallicities in the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}\nforest at $z\\le 1$ based upon our upper limits on the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} column\ndensities.\nHowever, if our assumption of $b\\sim30$~{\\kms} is not applicable, as\nfound for the two {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers, then our quoted estimates in the\nrange of metallicities would be quite wrong.\nIn fact, given that the upper limits on the metallicity are\nsuper--solar for the $W_{\\rm r}({\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}) \\leq 0.3$~{\\AA} absorbers, and\nthat it is not expected that many of the absorbers are super--solar,\nwe are lead to suggest that either the $b$ parameters of the $W_{\\rm\nr}({\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}) \\leq 0.3$~{\\AA} absorbers are significantly smaller than\n30~{\\kms} or the objects do not have a Gaussian velocity\ndistribution (simple curve of growth techniques are not applicable).\nIt could be that the uniqueness of the two {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers is that\nthey {\\it are\\\/} dynamically settled and have a small velocity\ndispersion.\n\nOther possibilities for the uniqueness of the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers, are:\n\n(1) {\\it Their ionization conditions are different}:\nIf the chemical enrichment histories of these absorbers are typical of\n{\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest clouds, then it might be inferred that their ionization\nconditions are governed by a local UV flux rather than the UVB.\nHowever, our models lead us to conclude that there is nothing\n``special'' about the photoionization conditions of these clouds;\nthey are best described as being photoionized by the UVB.\nThere is no evidence to suggest that these clouds are collisionally\nionized.\n\n(2) {\\it Their chemical conditions are different}:\nIf the ionization source is no different than that ionizing the other\n26 {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} systems we searched, then we are led to infer somewhat\nunique chemical enrichment histories for these two absorbers.\nIf these absorbers are IGM\/{\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} clouds, in that they are not\nassociated with galaxies, then the low metallicity results of SC96 at\nhigher redshift do not support the notion that these two absorbers\nhave undergone typical IGM chemical enrichment.\nIn other words, these two absorbers are not consistent with the\npicture in which the IGM was enriched at high redshift by a single\nburst of Population III stars that left the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest\n$\\alpha$--group enhanced\\footnote{Depending upon the Population III\nIMF, some pockets of the IGM may have experienced slow metallicity\nbuild up due to late type stars.  However, we find this scenario to be\nno different in name than if the process took place in ``galaxies''.}\nwith $[\\hbox{Fe\/H}] \\leq -2$.\nThus, these absorbers could constitute a metal--rich minority of the\nIGM\/{\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} population (less than $\\sim 10$\\%).\nIt then becomes a question of understanding what environments and\nevolutionary histories give rise to high metal content in some {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}\nclouds.\nIt is already known that at least some fraction of the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest\nat $z<1$ are associated with galaxies (\\cite{lebrunlya};\n\\cite{lanzetta}; Bowen et~al.\\  1996\\nocite{bowen}).\n\n(3) {\\it Both their ionization and chemical conditions are different}:\nInferring the chemical content of ionized absorbers requires\nionization corrections that are uncertain.\nIn the case of photoionization, the inferred conditions of the clouds\nare very sensitive to both the intensity and shape of the ionizing\ncontinuum.\nWe have explored this interplay between the chemical content of the\nclouds and the properties of the UV ionizing flux, and have concluded\nwith some certainty that the ionizing field for the two absorbers is\nconstrained to have slope and intensity consistent with the UVB.\nThis is tantamount to saying that it is only the chemical conditions\nthat are inferred to be unique in the two absorbers.\n \nOverall, it is difficult to understand these absorbers in terms of the\nclassic picture of {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest clouds.\nOne problem is that their inferred {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} column densities are higher\nthan ``typical'' forest clouds.\nA second is their high $[\\hbox{Fe\/H}]$ and iron--group enhance\nabundance pattern.\nSince the metal abundance patterns of these absorbers are not\n$\\alpha$--group enhanced, it is implied that their environments \nhave been influenced by Type~Ia SNe yields (\\cite{lauroesch};\n\\cite{fxt}; however, see {Gibson et~al.\\  1997\\nocite{gibson}).\nIt could be that these absorbers are associated with galaxies.\nHowever, based upon imaging and spectroscopic studies, there are no\ncandidate high surface brightness (HSB) objects in the {PKS~0454+039} field.\n\nThe extended luminous objects identified in the WFPC2 images (Figure~4\nof LBBD) and ground--based image (Figure~3 of Steidel et~al.\\  1995),\nhave now had their redshifts spectroscopically measured using LRIS on\nthe Keck~I telescope.\nNone of them have redshifts that match the two {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers\n(\\cite{chuckprivcomm}).\nObject \\#5 in LBBD has been confirmed by Steidel and\ncollaborators to be the strong {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbing galaxy at\n$z=1.1536$. \nAt impacts greater than $120h_{75}^{-1}$ there are three galaxies with\n$0.605 \\leq z \\leq 0.610$ (not presented in published images).\nThus, to a limiting of magnitude of $K\\sim 20.5$, the limit of the\nSteidel et~al.\\  image, there are no luminous candidates within $\\sim\n30${\\arcsec} of the QSO.\n\nBased upon the residuals following the point--spread function\nsubtraction of the QSO in both the WFPC and the ground--based images,\nthere is no evidence for luminous objects directly in front of the QSO\n(``zero--impact'' absorbers).\nHowever, dwarf galaxies of roughly $\\leq 0.01~L^{\\ast}$ at zero impact\ncannot be ruled out.\nIn general, it seems unlikely that two absorbing galaxies, separated\nby such a large redshift interval, would be aligned with the QSO on\nthe sky.\nAccording to the work of Bowen et~al.\\  (1997\\nocite{bowenleoI}), dwarf\nspheroid galaxies similar to Leo I are not massive enough to have\nhalos that can contribute significantly to the metal line absorption\ncross section of QSO absorbers seen at high redshift.\nBut the situation is not so clear overall, given the recent discovery\nby BBLD of the saturated {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} doublet associated with the\n$z=0.072$ dwarf galaxy at impact parameter $2.7h_{75}^{-1}$~kpc.\nThe emission line properties of this dwarf (\\cite{ccsdwarf}) suggest\nthat star formation in these objects may directly govern their gas cross\nsection.\nIf so, perhaps active star forming dwarf galaxies could contribute to\nthe overall metal line absorption cross section (cf.~\\cite{york};\n\\cite{yanny}).\nNaively, one would then expect that the abundance pattern arising from\na bursting dwarf would be $\\alpha$--group enhanced.\nAlso, it is likely that UV ionizing flux from the newly formed O and B\nstars would contribute to the ionization conditions in the absorbers,\nwhich is not what we find.\n\nIf we are to assume that these two absorbers are associated with\ngalaxies of some type, and if we accept the lack of evidence for \nHSB candidates in the {PKS~0454+039} field, then we\nmust explore the idea that low surface brightness (LSB) galaxies\n(cf.~Bothun et~al.\\  1997\\nocite{bothun}) could be giving rise to the\nabsorbing gas.\nParticularly, we are lead to consider the class of galaxies called \n``giant LSB galaxies'' (\\cite{sprayberry93}; \\cite{sprayberry95}).\n\nGreat progress in our knowledge of LSB galaxies, their number density,\nsizes, metallicities, and luminosity function in the local universe has\nbeen made over the past few years (\\cite{deblok}; \\cite{quillen};\n\\cite{dalcanton}; \\cite{sprayberry97}; \\cite{dejong};\n\\cite{sprayberry95}; \\cite{mcgaugh}).\nDalcanton et~al.\\  (1997) find that LSB galaxies have a space density of\nat least $n=0.03$~galaxies~$h_{75}^3$~Mpc$^{-3}$ and outnumber\ncomparable HSB galaxies by factors of $\\sim 2$ or more.\nLSB galaxies are a non--negligible component of the local universe\nbaryonic mass.\nSprayberry et~al.\\  (1995) find that LSB giants have larger disk scale\nlengths than HSB galaxies of comparable total luminosity.\nIn the cases of F568--6 and UGC~6614, the luminous spiral arms \nextend to $80h^{-1}_{75}$ and $50h^{-1}_{75}$~kpc, respectively\n(\\cite{quillen}). \nThe extent of the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} disks for the general population of LSB\ngalaxies is seen to be roughly 2.5 times that of their $D_{25}$, the\ndiameters of their $\\mu _{B} = 25$~mag~arcsec$^{-2}$ isophotes\n(\\cite{vanderhulst}).\nIf this scaling holds for F568--6, then it may have an {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} disk of\n$\\sim 200h^{-1}$~kpc.\nIn the case of the LSB galaxy $1226+0105$, the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} disk may extend  \nmore than four times its $D_{25}$ (\\cite{sprayberry95}).\n\nWhatever structures give rise to these two absorbers, the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} gas\nmust have a velocity dispersion of no more than $\\sim 30$~{\\kms}, as\ndictated by the inferred upper limit on the $b$ parameter of the\n$z=0.6428$ cloud.\nThe constraint is even as low as $\\sim 15$~{\\kms} based upon the\n$z=0.9315$ cloud.\nThe best values of the cloud $b$ parameters are $\\sim 12$~{\\kms} and \n$\\sim 9$~{\\kms}, respectively.\nThese $b$ parameters fall {\\it below\\\/} the lower cut offs in the\noverall {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest $b$ distribution (\\cite{kim}; \\cite{lulya};\n\\cite{hulya}).\nAgain, this suggest that these absorber possibly arise in a minority\nsub--class of the overall {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} cloud population.\nIn their study of the giant {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} disks of F568--6 and UGC~6614,\nQuillen and Pickering (1997a) found that the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} showed small\nvelocity dispersions of 10--30~{\\kms} and 10--20~{\\kms}, respectively,\nas compared to 60--90~{\\kms} measured for local HSB spirals\n(\\cite{vogel}; \\cite{canzian93}).\nThese dispersions were measured among the spiral arms; it could be\nthat the extended outer disks are even more quiescent.\nEven so, these values are consistent with the allowed ranges of the\ntwo absorbers.\nIf a sub--population of the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest is arising in LSB galaxies,\nthey may be characterized by having $b$ parameters scattered about the\nlow end of the distribution.\n\nThe metallicities of several LSB galaxies have been measured using\n{\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} regions.\nLSB galaxies with relatively smaller disk scale lengths are found to\nhave $[Z\/Z_{\\odot}] \\leq -0.3$, and are therefore metal poor\n(\\cite{mcgaugh}).\nHowever, McGaugh found near--solar and super--solar metallicities\nfor UGC~5709 and F568--6, respectively.\nThese two galaxies have large disk scale lengths, and classify as\ngiant LSB galaxies.\\footnote{The giant, or large scale length, LSB\ngalaxies are defined by Sprayberry et~al.\\  (1995) to have $\\mu _{B}(0) +\n5 \\log \\alpha ^{-1} > 27$, where $\\mu _{B}(0)$ is the central surface\nbrightness in the $B$ band and $\\alpha^{-1}$ is the scale length in\n$h^{-1}$kpc. The high metallicity LSB galaxies are seen to have\n$\\alpha^{-1} \\geq 13h^{-1}_{75}$~kpc, which constitute $\\sim 1\/3$ of\nthe known giant LSB galaxies defined in Sprayberry et~al.}\nPickering and Impey (1995; \\cite{impeyprivcomm}) have also found other\ngiant LSB galaxies have metallicities that scatter around solar.\nThese giant galaxies are found to have stellar surface densities at\nleast on the same order as their gas densities, which leads Pickering\nand Impey to suggest that these galaxies have been forming stars\nslowly.\n\nWe find these facts to be quite interesting in light of the two high\nmetallicity {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} systems we have found.\nIt is well established that {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorption with $W_{\\rm r} \\geq\n0.3$~{\\AA} selects the population of HSB galaxies\n(\\cite{csv96}; \\cite{steideleso}; \\cite{sdp94}; \\cite{bb91}).\nThese galaxies appear to be ``normal'' in their morphologies and to\nhave luminosities greater than $\\sim 0.05~L_{K}^{\\ast}$.\nBy comparison, the total luminosities of giant LSB galaxies scatter about\n$L^{\\ast}$ (cf.~\\cite{sprayberry95}).\nThough LSB galaxies have low luminosity densities, their disks are\nproportionally larger, giving them total luminosities on par with HSB\ngalaxies.\nThus, it seems reasonable that LSB galaxies could be part of a more \ngeneral {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorption selected galaxy population, where the\nLSB galaxies are selected by the smaller {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} equivalent widths.\n\nIn a recent survey to a limiting rest--frame equivalent width of\n0.02~{\\AA}, Churchill et~al.\\  (1997\\nocite{crcv97}) found that {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}\nabsorbers with $W_{\\rm r}(\\lambda 2796) \\leq 0.3$~{\\AA}  (hereafter\ncalled ``weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers'') account for $\\sim 65$\\% of all\n{\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers (also see \\cite{iapconf})\n{\\it Nothing is yet known about the type of luminous object they select};\nnone have luminous candidates to $\\leq 0.06~L^{*}_{K}$ [assuming the\nFreeman (1970\\nocite{freeman}) surface brightness] in the survey of\n{\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers by Steidel et~al.\\  (\\cite{chuckprivcomm}). \nThese systems also exhibit {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} absorption; for the sample, $\\left<\n\\log N({\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}})\/N({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) \\right> = -0.3\\pm 0.4$ (i.e.~they may\nhave $[\\hbox{Fe\/H}] \\geq -1$).\nWithout information on their {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} absorption, we can only speculate\nthat some fraction of the weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} systems have ionization and\nchemical conditions similar to the two absorbers studied in this work.\n\nIf weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers are selecting out a ``missing'' part of the\n{\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorption selected galaxy population, LSB galaxies are a\nlogical candidate for this missing portion, particularly the class of\ngiant LSB galaxies, or ``Malin--cousins'', as designated by Sprayberry\net~al.\\  (1993, 1995).\nThese galaxies are disk galaxies, and these disks are observed to have\na lower neutral {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} surface density than HSB galaxies\n(Bothun et~al.\\  1997\\nocite{bothun}).\nAs a result, the giant LSB galaxy disks have relatively quiescent\nstellar evolution.\nIn fact, the general population of LSB galaxies show a trend of\nincreasing red color with increasing disk scale length\n(\\cite{sprayberry95}).\nQuillen and Pickering (1997b) reported extremely red colors ($R-H = 2.2$\nand $B-H=3.5-4.2$) for the two giant LSB galaxies UGC~6614 and \nF568--6, which suggest that they have a dominating old component in their\nstellar populations.\nThese galaxies provide the precise type of environment in which there\nhas been ample time for iron--group enhancement and metallicity build\nup in the gas phase of their disks.\nFurther, the quiescent nature of their disks leads us to suggest that\nwe should {\\it not\\\/} expect to see the complex velocity structures seen\nin the majority of the stronger {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorption profiles\n(\\cite{mythesis}; Churchill et~al.\\  1998\\nocite{cvc98}).\nIndeed, the low {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} surface density of these galaxy disks should\nresult in weaker {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorption because there is less of the \nneutral hydrogen shielding required for {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} to survive.\nAlso, the quiescent nature of the interaction between the gas and\nstars in these galaxies suggests that the gas is not being stirred up,\nwhich would generate erratic gas kinematics.\nIt may be that such processes facilitate the generation of a high\nionization layer around the disk, as seen in the Galaxy\n(cf.~\\cite{savagearaa}).\nThe small $b$ parameters inferred for the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} and the apparent lack\nof {\\hbox{{\\rm C}\\kern 0.1em{\\sc iv}}} absorption in the two weak absorbers are consistent with a\nquiescent disk.\n\nIf weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers are selecting giant LSB galaxies, then\nthese absorbers provide a potential probe of the number density of\nthese massive galaxies.\nLSB galaxy disks may grow from isolated 1--2$\\sigma$ peaks in the\ninitial density fluctuation spectrum and may trace low density\nextended dark matter halos in a relatively unbiased way\n(Bothun et~al.\\  1997\\nocite{bothun}).\nThey also would provide a powerful probe of the chemical enrichment\nhistory of LSB galaxies, which appear to evolve at a significantly\nslower rate and may produce stars via conventional pathways [such as \nnot within molecular clouds (cf.~Bothun et~al.\\  1997\\nocite{bothun})].\n\nTo date, there is not enough known about the number density and\ngaseous cross sections of the class of giant LSB galaxies to compare\ntheir $dN\/dz$ directly to that of the weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} systems, or to\nplace meaningful limits on their number evolution if we assume they\nare selected by weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorption (Bothun et~al.\\ \n1997\\nocite{bothun}; \\cite{impeyprivcomm}).\nRoughly, the overall population of LSB galaxies appear to follow a\ntrend such that those with larger disk scale lengths are observed to\nhave smaller central surface brightness (\\cite{sprayberry95}).\nFollowing this relation, there is a significant gap between Malin~1\nand the remaining sub--population of giant LSB galaxies.\nMalin~1 has a disk scale length a factor of 30 times greater than that\nof  F568--6 and a central surface brightness a factor of 100 less than\nF568--6.\nDoes the gap in this parameter space reflect a true break, suggesting\nthat Malin~1 is a rare galaxy type?\nOr, is the gap an artifact of selection effects?\nAs Sprayberry et~al.\\  point out, it is important to explore this\nparameter space in order to determine the size and number density\ndistributions of giant LSB galaxies.\nIf LSB galaxies are considered to be a natural extension of the HSB\ngalaxy luminosity function, and their disk scale lengths and central\nsurface brightnesses exhibit similar behaviors to those found for HSB\ngalaxies, then the region of central surface brightness -- scale\nlength parameter space giving rise to Malin--type LSB galaxies is \ncontinuously populated (\\cite{slinderprivcomm}).\n\nWe can only tentatively suppose that weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} systems with\naccompanying {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} absorption may be selecting the class of giant\nLSB galaxies (\\cite{sprayberry93}) out to $z \\sim 1$.\nImpey \\& Bothun (1989), upon reexamining the selection effects and\nassumptions that go into the calculations of galaxy cross sections\nfrom QSO absorbers, found that LSB galaxies are expected to dominate\nthe absorption cross section.\nIf we assume a non--evolving absorber cross section, $n_0 \\sigma _0$,\nwe can write the projected radial extent of QSO absorbers as $R \\simeq\n7.5 (N\/n_0)^{1\/2}h_{75}^{-1}$~kpc, where we have integrated over the\nredshift interval $0.42 \\leq z \\leq 1.18$, and where $N$ is the number\nof observed absorbers in the redshift interval.\nIf we are to claim that the two {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} systems arise from the general\npopulation of LSB galaxies, we obtain $R\\sim 60$~kpc, where we have\nused the space density, $n=0.03$~galaxies~$h_{75}^3$~Mpc$^{-3}$, found\nby Dalcanton et~al.\\  (1997).\nSupposing that giant LSB galaxies comprise 1\\% of the LSB\npopulation, the size of the absorbers is inferred to be $R\\sim\n200$~kpc, which compares to the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} sizes of F568--6, UGC~6614, and\nUGC~5709 (the ``Malin cousins'').\n\nThe success rate of two weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers out of 28 {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} clouds\nleads us to tentatively suggest that 5--10\\% of the so--called {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}\nclouds in the forest at $\\left< z \\right> \\sim 0.7$ will have\ndetectable {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorption to our level of sensitivity.\nUsing the $dN\/dz$ for the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest from the Quasar Absorption\nLine Key Project (\\cite{buell}), this corresponds to non--evolving\npopulation with $dN\/dz \\sim 1.5-3$.\nInterestingly, this is not inconsistent with the number density of\n$dN\/dz \\simeq 1.74$ for {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers with $W_{\\rm r} \\leq\n0.3$~{\\AA}, as reported by Churchill et~al.\\  (1997\\nocite{crcv97}).\nAs with the {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} absorbers at $z\\leq1$, the weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} systems are\nconsistent with a non--evolving population.\nIt could be that our search through the {PKS~0454+039} forest has picked up\nthe same population of absorbers selected by a fair\nfraction of the weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} survey, whatever that population may be.\n{\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} absorption is present in many of these systems, suggesting\nthat many of these absorbers may have chemical and photoionization\nconditions similar to the two absorbers along the {PKS~0454+039} sight line\n(i.e.~they may arise in similar environments with similar evolutionary\nhistories).\n\nIn contrast to post--starbursting dwarfs, which have not formally been\nruled out if they both happen to be tightly aligned on the sky with\nthe QSO, the few giant LSB galaxies known to date have colors\nconsistent with a population of late--type stars (\\cite{quillen2}).\nThis is consistent with the inferred ionization conditions of the two\nabsorbers, since early--type stars are ruled out as sources of UV flux.\nMost giant LSB galaxies have active galactic nuclei [narrow emission\nlines (\\cite{sprayberry95}; Bothun et~al.\\  1997\\nocite{bothun})], and\nthis could very well be the case here.\nThere is a point--like\/stellar object (\\#7 in LBBD) with\nimpact parameter $\\sim 45$ or $60h_{75}^{-1}$ (assuming  $z=0.6$ or\n$z=0.9$.) that has a one--sided spiral arm like structure.\nThough many damped {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} absorbers are seen to be LSB galaxies and low\nluminosity dwarfs (\\cite{cohen}; \\cite{lebrundla}; \\cite{ccs3c336};\n\\cite{meyer}), this is not a necessary condition for the absorbers to be\nLSB galaxies; the high redshift LSB absorbing galaxies studied so far\nhave been selected by their {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} absorption and not by their {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}\nabsorption.\nWeak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorption that is accompanied by {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} absorption of\ncomparable strength may be selecting a well--defined population of\nluminous objects.\nThese objects probably do not include the damped {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} systems, but\nprobably do include sub--Lyman limit systems with high metallicity.\nThis is in contrast to the findings of BBLD, who find that\nstrong {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorption accompanied by strong {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} absorption\nselects damped {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} systems of low metallicity.\n\nThe weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers may in fact be selecting LSB galaxies,\ngiven that the weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers found by Churchill et~al.\\ \n(1997\\nocite{crcv97}) do not have HSB candidates to $\\leq\n0.06~L^{*}_{K}$ (\\cite{steideleso}; \\cite{chuckprivcomm}). \nVery deep imaging and faint object spectroscopy will be required if we\nare to identify the luminous objects selected by weak {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}\nabsorption.\nThese objects may represent a significant fraction of the galaxy\npopulation of the universe (and therefore dark matter content).\nThus, understanding their statistical properties is important for\ntheories of structure formation and galactic evolution.\n\nIn the future, a detailed comparison of the {\\it relative\\\/}\nabundances of $\\alpha$--group elements (O, Ne, Mg, Si, S, Ca) to\nFe--peak elements (Cr, Ni, Fe, Zn) in ``{\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest clouds'' holds\nthe promise of revealing their various origins and formation epochs.\nThe rate of Type~Ia SNe appears to be very low for the first Gyr\nin the history of a Milky--Way like galaxy (\\cite{truran};\n\\cite{smecker}).\nIf the delay is similar for the onset of Type Ia SNe in other galaxies\nas well, then it could be that [Mg\/Fe] abundance ratio tests are\nconfined to low redshifts.\nAs such, it is expected that $\\alpha$--group enhanced abundance ratios\nshould be almost exclusively seen at $z > 1.5$ (Timmes et~al.\\ \n1995\\nocite{fxt}).\nFor $q_0 = 0.5$ and $\\Lambda = 0$, if a galaxy formed at $z \\geq\n4$, then Type Ia SNe may  not {\\it begin\\\/} to contribute to its\nchemical enrichment until $z=1.5$ and may not influence the abundance\npatterns to a detectable level until $z\\sim1$ (1 Gyr later).\nFor $q_0 = 0.1$ and $\\Lambda = 0$, the Fe--group enrichment would\nlikely be seen no earlier than $z\\sim 1.5$.\n\n\n\n\n\\section{Summary}\n\\label{sec:conclude}\n\nWe searched for {{\\rm Mg}\\kern 0.1em{\\sc ii}~$\\lambda\\lambda 2976, 2803$} absorption in 28 {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest absorbers\nalong the {PKS~0454+039} line of sight.  \nThe spectrum studied for the metal line absorption was an optical\nHIRES spectrum (\\cite{mythesis}; Churchill et~al.\\ \n1998\\nocite{cvc98}).\nThe {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} line list were taken from the UV G190H and the G270H FOS\/HST\nspectra of BBLD.\nThe redshift range was $0.4163 \\leq z(\\lambda 2796) \\leq 1.1871$.\nThe doublets were identified and confirmed using the techniques\ndescribed in Schneider et~al.\\  (1993) and in Churchill et~al.\\ \n(1997\\nocite{crcv97}).\nWe found two {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorbers, one at $z=0.6428$, and one at\n$z=0.9315$.\nBoth these systems exhibit {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} absorption.\nWe carefully searched the FOS\/HST spectrum for the expected\nmetal--line transitions (cf.~\\cite{uffe}) that were covered in the\nforest.\nThere are currently no other high spectral resolution studies\nof metals in the $z<1$ {\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}} forest, and it also appears that there\nare no counter parts to these two systems at higher redshifts.\n\nIn Figure~\\ref{fig:ewlimits}, we present the 5$\\sigma$ rest--frame\nobserved equivalent width detection limit of the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} $\\lambda\n2796$ transition as a function of redshift.\nThe detection limit ranged from\n0.007 to 0.020~{\\AA}, except for $z(\\lambda 2796) \\leq 0.4662$, where\nit ranges from 0.020 to 0.035~{\\AA}.\nThe 5$\\sigma$ mean upper limit is $\\log N({\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}}) \\sim 11.3$~{\\hbox{cm$^{-2}$}}\nfor clouds with $0.1 \\leq W_{\\rm r}({\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}) \\leq 1.6$~{\\AA}, which\ncorresponds to neutral hydrogen column densities over the range $13.5\n\\leq \\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 18.5$~{\\hbox{cm$^{-2}$}}.\n\nWe studied the two discovered absorbers in some detail.\nVoigt profile fitting was performed to obtain the column densities and\nDoppler $b$ parameters of the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}}.\nSince the {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} are unresolved, we performed Monte Carlo\nmodeling of the Voigt profile fitting in order to best constrain the\nuncertainties in the Voigt profile parameters (\\cite{mythesis}; this\nwork).\nWe then used CLOUDY (\\cite{ferland}) to model the ionization and\nchemical conditions of the two absorbers. \nCLOUDY was used in its optimize mode, in which the residuals between\nthe model and the measured {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} and {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}} column densities were\nminimized.\nThe fixed quantities for each cloud, which constitute the grid\nparameters, were the UV flux intensity and continuum shape, the metal\nabundance pattern, and the ``observed'' neutral hydrogen column\ndensity.\nThe optimized output quantities were the total hydrogen density,\n$n_H$, and a scaling factor for the metallicity, $Z_{\\rm scale}$.\nWe constrained the model clouds using the {\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}} column density and the\nparameter $f=b_{\\rm turb}\/b_{\\rm tot}$, where $b_{\\rm turb}$ is the\nnon--thermal contribution to $b_{\\rm tot}$.\n\nThe UV continua considered were a Haardt \\& Madau (1996) UV background\n(UVB) spectrum, several Kurucz (1991) Atlas stellar models, late--type\ngalaxy models (\\cite{worthey}), and a star forming galaxy model\n(\\cite{bruzual}).\nWe investigated three abundance patterns: solar, {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} depletion, and\nISM depletion (the latter being $\\alpha$--group enhanced).  \nThe {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} and ISM CLOUDY models included grain physics.\nThe only UV ionizing scenario that yielded model clouds that were\nconsistent both with astrophysical constraints (numbers of stars, etc.)\nand with constraints imposed by the data was the Haardt \\& Madau UVB.\nWe conclude that the absorbers are photoionized by the UVB and not\nby stellar radiation.\nNeither absorber is consistent with having an $\\alpha$--group enhanced\nabundance pattern.\n\nAs described in \\S\\ref{sec:0.6428}, the $z=0.6428$ absorber may have a\nnear--solar or super--solar [Fe\/H].\nFor the solar abundance pattern, the model cloud has\n$16.3 \\leq \\log N({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 16.8$~{\\hbox{cm$^{-2}$}}, a $b_{\\rm turb}\/b_{\\rm tot}$\nof $0.85 \\leq f \\leq 0.93$, metallicity $-0.2 \\geq\n[Z\/Z_{\\odot}] \\geq -0.7$, and density $0.01 \\leq n_{\\rm H} \\leq\n0.02$~cm$^{-3}$.\nFor the {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} abundance pattern, the absorber has $16.7 \\leq \\log\nN({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 17.2$~{\\hbox{cm$^{-2}$}}, $b_{\\rm turb}\/b_{\\rm tot}$ in the range\n$0.90 \\leq f \\leq 0.95$, metallicity $+0.4 \\geq [Z\/Z_{\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}}]\n\\geq 0.0$, and density $n_{\\rm H} \\simeq 0.008$~cm$^{-3}$.\nIf this cloud is relatively free of dust depletion, so that the\nabundance pattern is close to solar, then the cloud has $[\\hbox{Fe\/H}]\n> -1$.\nIf the gas--phase abundance follows that of depleted clouds in our\nGalaxy, then the cloud could have $[\\hbox{Fe\/H}] > 0$.\n\nAs described in \\S\\ref{sec:0.9315}, the $z=0.9315$ model cloud appears\nto have a super--solar gas--phase [Fe\/H].\nFor the solar abundance pattern, the absorber has $15.8 \\leq \\log\nN({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 16.3$~{\\hbox{cm$^{-2}$}}, a $b_{\\rm turb}\/b_{\\rm tot}$ of $0.62 \\leq\nf \\leq 0.94$, metallicity $+0.7 \\geq [Z\/Z_{\\odot}] \\geq +0.1$,\nand density $0.2 \\leq n_{\\rm H} \\leq 0.4$~cm$^{-3}$.\nThe inferred $[\\hbox{Fe\/H}]$ is even greater for the {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} abundance\npattern model.\nFor the {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} abundance pattern, the absorber has $16.0 \\leq \\log\nN({\\hbox{{\\rm H}\\kern 0.1em{\\sc i}}}) \\leq 16.5$~{\\hbox{cm$^{-2}$}}, $b_{\\rm turb}\/b_{\\rm tot}$ in the range\n$0.75 \\leq f \\leq 0.97$, metallicity $+1.6 \\geq [Z\/Z_{\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}}]\n\\geq +0.7$, and density $n_{\\rm H} \\simeq 0.1$~cm$^{-3}$.\nThis is a metallicity enhancement of five to 40 times over the typical\nvalues seen in Galactic {\\hbox{{\\rm H}\\kern 0.1em{\\sc ii}}} regions (\\cite{baldwin91};\n\\cite{rubin91}; Osterbrock et~al.\\  1992\\nocite{osterbrock92}).   \nNo matter the abundance pattern, this cloud has $[\\hbox{Fe\/H}] > 0$.\n\nIn \\S\\ref{sec:discuss}, we discussed the possibility that these two\nabsorbers could arise in giant LSB galaxies.\nThese galaxies are seen to have metallicities that scatter about solar\n(\\cite{pickering95}) and large extended disks (\\cite{quillen}).\nWe tentatively suggest that 5\\% (at most 10\\%) of $z \\leq 1$ ``{\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}\nforest clouds'' with  $0.1 \\leq W_{\\rm r}({\\hbox{{\\rm Ly}\\kern 0.1em$\\alpha$}}) \\leq 1.6$~{\\AA} will\nexhibit {\\hbox{{\\rm Mg}\\kern 0.1em{\\sc ii}}} absorption to a 5$\\sigma$ $W_{\\rm r}$ detection limit\nof 0.02~{\\AA}.\nThe sub--sample of these systems that also exhibit comparable {\\hbox{{\\rm Fe}\\kern 0.1em{\\sc ii}}}\nabsorption may have iron--group enhanced metallicities with\n$[\\hbox{Fe\/H}] \\geq -1$, and may be selecting giant LSB galaxies at\nhigh redshifts.\n\n\n\\acknowledgments\nThis work has been supported in part by the National Science\nFoundation grant AST--9617185 at Penn State.\nCWC acknowledges support through the Eberly School of Science\nDistinguished Postdoctoral Fellowship at The Pennsylvania State\nUniversity.\nThanks to:\nP.~Boiss\\'e for providing the FOS spectrum prior to publication;\nJ.~Charlton for assistance with the Voigt profile fitting simulations\nand CLOUDY modeling; G.~Ferland for making CLOUDY a public tool,\nU.~Hellsten for providing the Haardt \\& Madau input spectra for CLOUDY;\nS.~Linder for generating plots of $\\mu_{B}(0)$ versus $\\log\n\\alpha$ of LSB galaxies for our inspection and for discussions about\nthe observed properties of LSB galaxies; and \nC.~Steidel for sharing unpublished data on the {PKS~0454+039} field.\nIt is a pleasure to acknowledge J.~Bergeron, M.~Bershady, P.~Boiss\\'e,\nJ.~Charlton, R.~Dav\\'e, U.~Hellsten, C.~Impey, J.~Lauroesch,\nP.~Petitjean, D.~Schneider, and M.~Shetrone for stimulating\ndiscussions and\/or comments.\nSpecial thanks to S.~Vogt for HIRES and for providing the opportunity\nto use it.\nWe thank C. Impey, the referee, for valuable comments that improved\nthe quality of this manuscript.\nFor J.~L.~N.\n\n\n\\section*{APPENDIX}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{introsec}\nOne of the cornerstones of modern cosmology is the framework of cosmological perturbation theory, which allows detailed predictions to be derived from fundamental models of the early universe. The theory of gauge-invariant cosmological perturbations was pioneered in \\cite{Bardeen:1980kt}, and the resulting power spectrum of curvature perturbations in single field inflation was first computed in \\cite{Sasaki:1986hm,Mukhanov:1988jd}; see \\cite{Mukhanov:1990me,Baumann:2009ds} for reviews. The three-point function (bispectrum) in single field inflation was computed in  \\cite{Maldacena:2002vr}, a calculation which has been extended to many other models of inflation. More recently there has been interest in the consequences of Ward identities for cosmological correlation functions, arising from spatial diffeomorphisms and conformal symmetry \\cite{Creminelli:2012qr,Hinterbichler:2013dpa,Berezhiani:2013ewa,Pimentel:2013gza}.\n\nTheoretical interest of a different kind is attached to gravity in two spacetime dimensions, where Newton's constant is dimensionless and the gravitational interaction is renormalizable. The canonical theory of gravity in 2d is Liouville theory \\cite{Polyakov:1981rd}, which describes the conformal factor of the 2d metric in conformal gauge. Liouville theory has attracted sustained interest as a toy model of quantum gravity, and as an interacting conformal field theory whose classical solutions are known in full \\cite{Braaten:1982fr,Braaten:1982yn,Teschner:2001rv,Polchinski:1989fn,Ginsparg:1993is} (although the status of timelike Liouville as a conformal field theory remains unclear \\cite{McElgin:2007ak,Harlow:2011ny}). It also has relevance for cosmology, providing a description of two-dimensional (A)dS spacetimes, and giving rise to 2d Friedmann equations when coupled to conformally invariant matter \\cite{DaCunha:2003fm}.\n\nGeneral covariance is a powerful organizing principle in the theory of cosmological perturbations \\cite{Bardeen:1980kt}. With this connection in mind, a generally covariant extension of Liouville theory was recently introduced \\cite{Martinec:2014uva}. At the classical level, this extended Liouville theory is a local version of Polyakov's covariant 2d gravity \\cite{Polyakov:1987zb}. It features an auxiliary scalar field in addition to the 2d metric, making for a total of four scalar degrees of freedom --- including two Lagrange multipliers --- subject to a gauge symmetry consisting of two-component diffeomorphisms. This counting is identical to the scalar sector of perturbations in 4d Einstein gravity \\cite{Bardeen:1980kt}, and indeed there is a precise map between perturbations in the two models.\n\nThe purpose of this paper is to continue the analysis of cosmological perturbations in extended Liouville theory coupled to a scalar inflaton, begun in \\cite{Martinec:2014uva}. We begin in sections \\ref{Liouvillesec} and \\ref{quadsec} with a review of the basic features of extended Liouville theory, its perturbative degrees of freedom and their correspondence with the scalar sector in 4d. We construct the systematic slow roll expansion and give examples of spatially homogeneous slow roll backgrounds. The power spectra of scalar curvature perturbations in 2d and 4d are derived in a unified manner, making transparent the characteristic scale invariance. We also confirm the freezing out of scalar curvature perturbations on superhorizon scales, to all orders in perturbation theory.\n\nIn section \\ref{cubicsec} the cubic fluctuation action is derived in spatially flat gauge and constant inflaton gauge. Following the method of \\cite{Maldacena:2002vr}, we confirm the suppression of the cubic action by two powers of the slow roll parameter $\\epsilon$. This allows us to evaluate, in section \\ref{3ptsec}, the three-point function of scalar curvature perturbations in the slow roll approximation, by working at tree level and using the cubic vertex only. The resulting non-Gaussianity has a local shape function analogous to that generated by single field inflation in 4d. Kinematic simplifications in 1+1d allow us to condense the shape function into a very compact form.\n\nFinally, in section \\ref{consistencysec} we discuss consistency relations for the three-point function. It turns out that there is no room in 1+1d for residual diffeomorphisms preserving the gauge choice and the physicality of metric fluctuations. However, we find that spatial dilations are revived by a global shift symmetry of extended Liouville theory, leading to the familiar consistency relation of Maldacena. We show how this shift symmetry can be interpreted as a global Weyl symmetry of the theory in cosmological gauges, which is related to the local Weyl symmetry of regular Liouville theory. There are no higher order consistency relations for the three-point function in 2d.\n\nEmphasis throughout is on the similarity of our methods and results to those familiar in the scalar sector in 4d, lending support to the use of extended Liouville theory as a theoretical laboratory. This approach has already yielded new insights into non-perturbative effects of quantum gravity in inflation \\cite{Martinec:2014uva}. It also provides a tool to study renormalized scalar perturbations in cosmological gauges, a topic which is the focus of ongoing work.\n\n\n\n\n\n\n\\section{Extended Liouville theory}\\label{Liouvillesec}\nHere we review the generally covariant extension of Liouville theory which was introduced in \\cite{Martinec:2014uva}, discussing its basic features and interpretation in worldsheet string theory.\n\n\\subsection{Action, equations of motion, and constraints}\nThe action for the extended Liouville theory introduced in \\cite{Martinec:2014uva} is\\footnote{The action \\eqref{Lcovpre} has been scaled by $4\\pi$ relative to that in \\cite{Martinec:2014uva}, to give the conventional normalization for the scalar curvature perturbation.}\n\\begin{equation}\n\\label{Lcovpre}\n{\\mathcal{S}}_{\\rm grav} = \\frac{2}{\\gamma^2} \\int \\sqrt{-g}\\left[  -(\\nabla\\Laux)^2 -  R \\Laux - {\\Lambda} \\right] \n\\end{equation}\nHere $\\Laux$ is an auxiliary scalar field, and $\\gamma^{2}$ is a dimensionless parameter which plays the role of Newton's constant $G_{N}$. Precisely speaking, $2\/\\gamma^{2}$ plays the role of $m_{p}^{2}\/2=1\/16\\pi G_{N}$, where $m_{p}$ is the Planck mass. All action integrals are over 1+1d spacetime, unless otherwise indicated. Let us see what sort of spacetimes are described by the theory \\eqref{Lcovpre}. The equation of motion for $\\Laux$ is\n\\begin{equation}\\label{chieom}\n0 = 2\\nabla^{2}\\Laux - R \n\\end{equation}\nThe scalar curvature acts as a source for $\\Laux$. The gravitational stress tensor\\footnote{Strictly speaking gravity does not have a stress tensor, but by an abuse of language we use this terminology to denote the variation of the gravitational action with respect to the metric --- that is, the metric equations of motion.}\n\\begin{equation}\nT_{ab}^{\\rm grav} \\equiv  \\frac{2}{\\sqrt{-g}} \\fd{{\\mathcal{S}}_{\\rm grav}}{g^{ab}}\n\\end{equation}\nis equal to\n\\begin{equation}\\label{Tgrav}\n\\gamma^{2} T_{ab}^{\\rm grav} =  {1\\over2} g_{ab}\\[ (\\nabla\\Laux)^{2}+\\Lambda \\]\n- \\nabla_{a}\\Laux\\nabla_{b}\\Laux\n+ \\nabla_{a}\\nabla_{b}\\Laux - g_{ab}\\nabla^{2}\\Laux\n\\end{equation}\nThe final two terms are ``improvement'' terms arising from the coupling to the Ricci scalar in \\eqref{Lcovpre}. In 2d there is only one component of curvature, which completely determines the tensor structures\n\\begin{equation}\\label{2dcurv}\nR_{abcd} = {1\\over2} R\\( g_{ac}g_{bd}-g_{ad}g_{bc} \\)~,\\qquad \nR_{ab} = {1\\over2} R\\, g_{ab}\n\\end{equation}\nThese relations cause the Einstein tensor to vanish identically in 2d. In other words, the Einstein-Hilbert action is a topological invariant of the 2d manifold $M$, called the Euler characteristic $\\chi_{\\rm Euler}$ (not to be confused with the auxiliary field $\\Laux$):\n\\begin{equation}\\label{Euler}\n\\chi_{\\rm Euler} (M) = \\int_{M}\\sqrt{-g}\\,R\n\\end{equation}\nIn light of the vanishing Einstein tensor, the variation of the Einstein-Hilbert Lagrangian density with respect to the metric reduces to\n\\begin{equation}\n\\delta(\\sqrt{-g}\\, R) = \\sqrt{-g}\\,g^{ab}\\delta R_{ab}\n\\end{equation}\nwhich is a total derivative. In the Einstein-Hilbert action this would be discarded, but in the action \\eqref{Lcovpre} it gives rise to the final two terms in the stress tensor \\eqref{Tgrav}. The metric equations of motion state that all components of \\eqref{Tgrav} must vanish. The trace component is\n\\begin{equation}\\label{phieom}\n0 = \\nabla^{2}\\Laux - \\Lambda\n\\end{equation}\nClearly the solutions to \\eqref{chieom} and \\eqref{phieom} are metrics of constant curvature:\n\\begin{equation}\\label{RLambda}\nR[g]=2\\Lambda\n\\end{equation}\nThis tells us everything about the curvature of the 1+1d manifold. The solutions are two-dimensional (anti-) de~Sitter  spacetimes. \n\nThe remaining components of the metric equations of motion \\eqref{Tgrav} are the Hamiltonian and momentum constraints, which enforce diffeomorphism invariance. The constraints are expressed most easily in terms of the conjugate momenta. Let us parametrize the metric in 1+1d as\n\\begin{equation}\n\\label{metparam}\ng_{ab}  = e^{2\\phican}\\left(  \\begin{matrix} -N_\\ct^2+N_x^2\\quad & N_x \\\\ N_x & 1 \\end{matrix}\\right) \n\\end{equation}\nwhere the lapse $N^{\\ct}$ and shift $N^{x}$ play the role of Lagrange multipliers in the Hamiltonian formalism. Here $\\ct$ is conformal time, and the scale factor is $a^{2}=g_{xx}=e^{2\\phican}$. For the study of cosmological perturbations, the theory \\eqref{Lcovpre} is coupled to a scalar inflaton field $\\inflaton$ with action\n\\begin{equation}\\label{SX}\n{\\mathcal{S}}_{\\infsub}=\\frac{1}{2}\\int \\sqrt{-g}\\,\\left[  -(\\nabla\\inflaton)^2 - {\\mathcal{V}}(\\inflaton) \\right]\n\\end{equation}\nThe potential ${\\mathcal{V}}(\\inflaton)$ contributes to the gravitational equations of motion by combining with the cosmological constant as $\\Lambda+(\\gamma^{2}\/4){\\mathcal{V}}(\\inflaton)$. The non-zero conjugate momenta of the combined gravity and matter system (with total action \\eqref{Lcovpre} plus \\eqref{SX}) are\n\\begin{equation}\\label{momenta}\n\\begin{aligned}\n\\pi_{\\Laux}&=\\frac{4}{\\gamma^{2}N^{\\ct}}\\Bigl( (\\Laux+\\phican)'-N^{x}\\partial_{x}(\\Laux+\\phican) - \\partial_{x} N^{x} \\Bigr)\\\\\n\\pi_{\\phican}&=\\frac{4}{\\gamma^{2}N^{\\ct}} \\Bigl( \\Laux'-N^{x}\\partial_{x}\\Laux \\Bigr)\\\\\n\\pi_{\\inflaton}&=\\frac{1}{N^{\\ct}} \\Bigl( \\inflaton'-N^{x}\\partial_{x}\\inflaton \\Bigr)\n\\end{aligned}\n\\end{equation}\nwhere a prime denotes the conformal time derivative $\\partial_{\\ct}$. The Hamiltonian and momentum constraints are then written as\n\\begin{equation}\\label{constraints}\n\\begin{aligned}\n{\\mathcal{H}} &= \\frac{\\gamma^2}{4}\\Bigl( -{1\\over2}\\pi_\\phican^2 + \\pi_\\phican\\pi_\\Laux \\Bigr) + \\frac{2}{\\gamma^2} \\Bigl( (\\partial_x\\Laux)^2 + 2 (\\partial_x\\phican)(\\partial_x\\Laux) - 2\\partial_x^2\\Laux + \\Lambda e^{2\\phican} \\Bigr)\\\\\n&\\qquad{\\;}\\qquad{\\;}\\qquad{\\;}\\qquad{\\;}\\qquad{\\;}\\qquad{\\;}\\qquad +{1\\over2}\\pi_{\\inflaton}^{2}+{1\\over2}(\\partial_{x}\\inflaton)^{2}+{1\\over2} e^{2\\phican}{\\mathcal{V}}(\\inflaton)\\\\\n\\mathcal{P}&=\\pi_{\\Laux}\\partial_{x}\\Laux+\\pi_{\\phican}\\partial_{x}\\phican-\\partial_{x}\\pi_{\\phican}+\\pi_{\\inflaton}\\partial_{x}\\inflaton\n\\end{aligned}\n\\end{equation}\nThe vanishing of these constraints will be imposed on cosmological perturbations to give the constrained effective action at each order in perturbation theory.\n\n\\subsubsection*{Global shift symmetry}\nThe existence of the topological Euler characteristic \\eqref{Euler} gives rise to an important symmetry of the extended Liouville action \\eqref{Lcovpre}. The coupling of the scalar curvature to the auxiliary field $\\Laux$ is non-trivial only when $\\Laux$ is not a constant. In other words, shifting $\\Laux$ by a constant changes the action \\eqref{Lcovpre} only by a multiple of the Euler characteristic, which does not affect the equations of motion. We will see in section \\ref{consistencysec} that in cosmological gauges such as spatially flat gauge or constant inflaton gauge, the shift symmetry of $\\Laux$ becomes an invariance under global rescalings (Weyl transformations) of the metric. This symmetry will turn out to be crucial in deriving Maldacena's consistency relation for the three-point function of curvature perturbations in 2d.\n\n\n\\subsection{Interpretation in worldsheet string theory}\nBefore moving on, let us briefly mention the connection between extended Liouville theory and more familiar theories of gravity in 2d. Firstly, the nonlocal Polyakov action for 2d gravity induced by minimally coupled matter \\cite{Polyakov:1987zb} is recovered from \\eqref{Lcovpre} upon eliminating $\\Laux$ by its equation of motion \\eqref{chieom}. Secondly, the extended Liouville theory \\eqref{Lcovpre} reduces in conformal gauge to regular timelike Liouville theory. To fix conformal gauge, the metric is written as $g_{ab}=e^{2\\phican}\\hat g_{ab}$, but without assuming $\\hat g_{xx}=1$ as in \\eqref{metparam}. The Liouville field $\\phican$ is allowed to fluctuate, while $\\hat g_{ab}$ is fixed by gauging diffeomorphisms. The scalar curvature decomposes into\n\\begin{equation}\\label{R2Rhat}\n\\sqrt{-g}\\,R = \\sqrt{-\\hat g}\\, \\bigl( \\hat R - 2\\hat\\nabla^{2}\\phican \\bigr)\n\\end{equation}\nwhere all hatted quantities are constructed from $\\hat g$. The action \\eqref{Lcovpre} becomes\n\\begin{equation}\\label{Sgravhat}\n{\\mathcal{S}}_{\\rm grav}  = \\frac{2}{\\gamma^2} \\int \\sqrt{-\\hat g}\\left[ (\\hat\\nabla\\phican)^2 -(\\hat\\nabla\\Theta)^2 + Q\\hatR (\\phican -\\Theta)   - {\\Lambda}e^{2\\phican} \\right]  \n\\end{equation}\nwhere $\\Theta\\equiv\\Laux+\\phican$, and $Q=1$ at the classical level. The vanishing trace component of the stress tensor, equation \\eqref{phieom}, is now the $\\phican$ equation of motion; hence \\eqref{Sgravhat} with fixed reference metric $\\hat g$ is classically conformally invariant. Imposing the $\\Laux$ equation of motion \\eqref{chieom}, and using \\eqref{R2Rhat}, fixes $\\Theta=0$ up to nonlocal terms in $\\hat g$. The remaining terms in \\eqref{Sgravhat} are precisely the action for timelike Liouville theory \\cite{Martinec:2014uva} --- hence they are classically equivalent in conformal gauge.\n\nThe action \\eqref{Sgravhat} may also be coupled to additional matter fields to give a nonlinear sigma model with vanishing conformal anomaly \\cite{Martinec:2014uva}. The Liouville field $\\phican$ becomes the time coordinate in target space, and the coupling to $\\hat R$ represents a null linear dilaton. This causes a cancellation between the conformal improvements of $\\phican$ and $\\Theta$, so the gravity sector has central charge $c_{\\phican}+c_{\\Theta}=2$. Twenty-four additional scalar fields are required to cancel the conformal anomaly of the Faddeev-Popov ghosts. The resulting nonlinear sigma model describes bosonic string propagation in the critical dimension $D=26$, in the presence of a null linear dilaton, and a tachyon condensate generating the worldsheet cosmological constant $\\Lambda$ (and any potential for the matter fields, such as a slow roll inflaton potential).\n\nThe above discussion of the conformal anomaly takes place in conformal gauge. In this paper, our interest will instead be the classical theory in cosmological gauges, where $\\hat g_{ab}$ fluctuates. Matter fields other than the inflaton \\eqref{SX} are surplus to our requirements, as the conformal anomaly does not arise in the context of the classical perturbation theory developed here. In any case, additional minimally coupled fields would be spectators in the analysis of cosmological perturbations.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Slow roll expansion and quadratic fluctuation action}\\label{quadsec}\nWe now explore the basic properties of cosmological perturbations in extended Liouville theory, emphasizing the strong parallel with the scalar sector in 4d Einstein gravity. This is partly a review of material presented in \\cite{Martinec:2014uva}, and sets the stage for the analysis of non-Gaussianity in the following sections.\n\n\\subsection{Slow roll background}\nTo get started, we must describe the background spacetime on which cosmological perturbations propagate. We are thus interested in spatially homogeneous solutions of the equations of motion of the combined gravity and matter system. The background metric is taken to be the de~Sitter  metric\n\\begin{equation}\\label{dSmetric}\nds^{2} = e^{2\\phib}\\( -d\\ct^{2}+dx^{2} \\)\n\\end{equation}\nwhere $-\\infty<\\ct<0$ is conformal time. A tilde denotes a spatially homogeneous background field, such as $\\phib=\\phib(\\ct)$. The scale factor is identified as $a(\\ct)=e^{\\phib(\\ct)}$. Consider first the equation of motion for $\\Lauxb(\\ct)$, which is the reduction of \\eqref{chieom} in the de~Sitter  metric \\eqref{dSmetric}:\n\\begin{equation}\n0 = \\Lauxb'' + \\phib''\n\\end{equation}\nEquation \\eqref{R2Rhat} was used to find the contribution of $\\phib$ to the curvature. Recall that a prime denotes the conformal time derivative $\\partial_{\\ct}$. We choose to consider only backgrounds $\\Lauxb(\\ct)$ satisfying\n\\begin{equation}\\label{chisol}\n0 = \\Lauxb' + \\phib'\n\\end{equation}\nThis choice helps facilitate the comparison of cosmological perturbations in 1+1d with those in 3+1d, as will be seen in section \\ref{pertsec}. Eliminating $\\Lauxb'(\\ct)$ via \\eqref{chisol}, the remaining background equations of motion and Hamiltonian constraint become\n\\begin{equation}\\label{cteoms}\n\\begin{aligned}\n0 &= -\\phib'' +  e^{2\\phib}\\bigl[ \\Lambda + (\\gamma^{2}\/4) {\\mathcal{V}}(\\infb) \\bigr] \\\\\n0 &= \\infb'' + {1\\over2} e^{2\\phib}{\\mathcal{V}}_{,\\infsub} (\\infb) \\\\\n0 &= -(\\phib')^{2} + (\\gamma^{2}\/4)(\\infb')^{2} + e^{2\\phib}\\bigl[ \\Lambda + (\\gamma^{2}\/4) {\\mathcal{V}}(\\infb) \\bigr]\n\\end{aligned}\n\\end{equation}\nThese are the Friedmann equations of 2d cosmology \\cite{DaCunha:2003fm}. The first line is the reduction of the $\\phican$ equation of motion \\eqref{phieom}. The second line is the matter equation of motion arising from the action \\eqref{SX}, and the third line is the Hamiltonian constraint \\eqref{constraints}. The momentum constraint is satisfied trivially by spatially homogeneous backgrounds. Of course, only two of the Friedmann equations \\eqref{cteoms} are unique, a situation familiar from 4d. In the absence of matter, the de~Sitter  metric \\eqref{dSmetric} solves the equations with $\\phib(\\ct)$ given by\n\\begin{equation}\n\\sqrt{\\Lambda}\\, e^{\\phib} = -1\/\\ct\n\\end{equation}\n\nThe slow roll expansion is most easily described in coordinate time $\\pht$, defined by\n\\begin{equation}\nd\\pht=a(\\ct)\\, d\\ct~,\\qquad -\\infty<t<\\infty\n\\end{equation}\nThe Friedmann equations \\eqref{cteoms} are expressed in coordinate time as\n\\begin{equation}\\label{phteoms}\n\\begin{aligned}\n0 &= -\\ddot{\\phib}-\\dot{\\phib}^{2} + \\Lambda  + (\\gamma^{2}\/4) {\\mathcal{V}}(\\infb) \\\\\n0 &= \\ddot{\\infb} + \\dot{\\phib}\\dot{\\infb} + {1\\over2}  {\\mathcal{V}}_{,\\infsub}(\\infb) \\\\\n0 &= -\\dot{\\phib}^{2} + \\Lambda  + (\\gamma^{2}\/4) \\bigl[ \\dot{\\infb}^2 \n + {\\mathcal{V}}(\\infb)\\bigr]\n\\end{aligned}\n\\end{equation}\nwhere a dot denotes $\\partial_{\\pht}$. The Hubble parameter is simply $H=\\dot a\/a=\\partial_{t}\\phib$. The slow roll expansion may now be developed following \\cite{Baumann:2009ds}. The dimensionless slow roll parameters are defined as\n\\begin{equation}\\label{SRparams}\n\\epsilon \\equiv \\frac{d}{d\\pht}\\(\\frac{1}{H}\\)~,\\quad\n\\eta \\equiv \\frac{\\dot{\\epsilon}}{H\\epsilon}~,\\quad\n\\kappa \\equiv \\frac{\\dot{\\eta}}{H\\eta}~,\\quad\n\\delta \\equiv -\\frac{\\ddot{\\infb}}{H\\dot{\\infb}}\n\\end{equation}\nCombining the Friedmann equations \\eqref{phteoms} to eliminate ${\\mathcal{V}}(\\inflaton)$ gives\n\\begin{equation}\\label{SRvel}\n\\epsilon  = \\frac{\\gamma^{2}}{4} (\\dot{\\infb}\/\\dot{\\phib})^{2} ~,\\qquad\n\\eta = 2(\\epsilon-\\delta) \n\\end{equation}\nIn the slow roll approximation, both $\\epsilon$ and $|\\delta|$ are assumed to be much less than one --- typically of order a few percent. Then $|\\eta|$ is also small, signifying a long-lived phase of slow roll inflation. The slow roll parameters may also be expressed in terms of the matter potential as\n\\begin{equation}\\label{SRpot}\n\\epsilon = \\frac{1}{\\gamma^{2}} \\( \\frac{{\\mathcal{V}}_{,\\infsub}}{{\\mathcal{V}}} \\)^{2} \\(\\frac{1-\\epsilon}{1-\\delta}\\)^{2}~,\\qquad\n\\eta = \\frac{4}{\\gamma^{2}} \\biggl[ \\frac{{\\mathcal{V}}_{,\\infsub\\infsub}}{{\\mathcal{V}}}-\\(\\frac{{\\mathcal{V}}_{,\\infsub}}{{\\mathcal{V}}}\\)^{2}\\, \\biggr] + \\cdots\n\\end{equation}\nThis is all precisely analogous to the situation in 4d.\n\nIn what follows, explicit slow roll trajectories for the background fields will not be required. Nevertheless, some simple examples are now given for completeness. One exact solution consists of a quadratic potential of frequency $\\omega$, and an inflaton trajectory which is overdamped:\n\\begin{equation}\n\\begin{aligned}\n{\\mathcal{V}}(\\inflaton) &= {1\\over2}\\omega^{2}\\inflaton^{2} + \\frac{2\\omega}{\\gamma}\\sqrt{2\\Lambda+\\omega^{2}}\\,\\inflaton \\\\\n\\infb(\\pht) &= \\frac{\\sqrt{2}}{\\gamma}\\,\\omega \\pht+\\tilde X_{0} \\\\\n\\phib(\\pht) &= -\\frac{1}{4}\\omega^{2}\\pht^{2} - b\\pht + \\phib_{0} \\\\\nb &= \\sqrt{\\Lambda+\\omega^{2}\/2}+\\frac{\\gamma}{\\sqrt{8}}\\omega\\infb_{0}\n\\end{aligned}\n\\end{equation}\nRecall that coordinate time ranges over $-\\infty<t<\\infty$. The scale factor $e^{\\phib}$ expands prior to the moment $t_{c}=-2b\/\\omega^{2}$, after which it contracts. The effective cosmological constant at the minimum of the potential may always be taken positive by adjusting $\\Lambda$. There is an infinite era (in coordinate time $t$) of slow roll inflation when $t<(t_{c}-\\sqrt{2}\/\\omega)$. This era may be cut out and smoothly extended into de~Sitter  space.\n\nAnother example is the exponential potential ${\\mathcal{V}}(\\inflaton)=e^{-\\beta\\inflaton}$. In this case the explicit trajectories for $\\infb$ and $\\phib$ are most easily found from the conformal time equations \\eqref{cteoms}, and the slow roll conditions are satisfied for $|\\beta|\/\\gamma\\ll1$. One may also construct approximate solutions for a quartic potential, following exactly those in \\cite{Linde:1983gd}.\n\n\n\n\n\n\\subsection{Perturbations and gauge transformations}\\label{pertsec}\nCosmological perturbations are now introduced, with discussion of their gauge transformations and effective action. The theory of gauge-invariant perturbations was introduced in \\cite{Bardeen:1980kt}, and reviewed in \\cite{Mukhanov:1990me,Baumann:2009ds}. In 3+1 dimensions, perturbations of the metric are grouped into scalar, vector, and tensor representations of the Lorentz group. Our interest is in the scalar sector, which is precisely related to the perturbative degrees of freedom of extended Liouville theory. When perturbing the metric about a 4d de~Sitter  background, the scalar degrees of freedom are commonly parametrized as \\cite{Baumann:2009ds}\n\\begin{equation}\\label{ds4d}\nds^2_{4d} = a^2 \\Bigl\\{ -(1+2\\Phi)d\\ct^2 + 2B_{,i} dx^i d\\ct + \\bigl[(1-2\\Psi)\\delta_{ij}+2E_{,ij}\\bigr]dx^i dx^j \\Bigr\\}\n\\end{equation}\nThere are four scalar degrees of freedom in the metric ($\\Phi,\\Psi,B$, and $E$), which parametrize the trace component, and curl-free contributions to the shift vector and space-space metric. The vector and tensor sectors --- which are not considered here --- consist of divergence-free vector and transverse traceless tensor contributions, which cannot be expressed as derivatives of scalar functions. The behavior of \\eqref{ds4d} under diffeomorphisms is of interest for constructing gauge-invariant quantities. The scalar sector is closed under diffeomorphisms of the form\n\\begin{equation}\\label{coordtransf4d}\n\\ct\\to \\ct+\\xi^\\ct(\\ct,\\mathbf{x})  ~~, \\quad x^i\\to x^i +\\partial^i\\xi(\\ct,\\mathbf{x})\n\\end{equation}\nwhere $\\xi^{\\ct}$ and $\\xi$ are scalar functions. For each quantity $\\Sigma$ which is split into background plus perturbation as $\\Sigma=\\tilde\\Sigma+\\bm\\sigma$, we are interested in the linearized gauge transformation\n\\begin{equation}\\label{Lie}\n\\bm\\sigma \\to \\bm\\sigma + \\mathsterling_{\\xi}\\bigl( \\tilde\\Sigma+\\bm\\sigma \\bigr)\n\\end{equation}\nIn the case of the scalar sector, $\\bm\\sigma$ ranges over $\\Phi$, $\\Psi$, $B$, and $E$. The entire gauge transformation is assigned to the perturbation, leaving the background invariant. This is appropriate for describing the behavior of perturbations about a fixed background, and coincides with the gauge redundancy of the path integral. In this language, a time-dependent background breaks the gauge symmetry under time reparametrizations. The part of \\eqref{Lie} which is homogeneous in $\\bm\\sigma$ --- that is, the first order Taylor expansion of $\\bm\\sigma(x+\\xi)$ --- is trivial and will be omitted for brevity. The gauge transformation of the scalar perturbations in \\eqref{ds4d} is then \\cite{Mukhanov:1990me,Baumann:2009ds}\n\\begin{equation}\\label{fieldtransf4d}\n\\begin{aligned}\n\\Phi &\\to \\Phi + \\partial_\\ct\\xi^\\ct + \\frac{a'}{a} \\xi^\\ct\\\\\n\\Psi &\\to \\Psi - \\frac{a'}{a} \\xi^\\ct  \\\\\nB &\\to B - \\xi^\\ct + \\xi'\\\\\nE &\\to E + \\xi\n\\end{aligned}\n\\end{equation}\n\nTurning now to extended Liouville theory, we again split all fields into spatially homogeneous backgrounds (denoted by a tilde) plus inhomogeneous perturbations (bold lower case). Using the metric parametrization \\eqref{metparam} and adopting the de~Sitter  background \\eqref{dSmetric}, the perturbative degrees of freedom in the gravitational sector are\n\\begin{equation}\\label{fieldperts}\n\\begin{aligned}\n\\phican  &= \\phib(\\ct) + \\delta\\!\\phican(\\ct,x) \\\\\nN^\\ct  &= 1 + \\delta\\! N^{\\ct}(\\ct,x) \\\\\nN^x  &= \\delta\\! N^{x}(\\ct,x) \\\\\n\\Laux &= \\Lauxb(\\ct) + \\delta\\!\\Laux(\\ct,x) \n\\end{aligned}\n\\end{equation}\nThere are only three metric degrees of freedom in 2d, but the auxiliary field perturbation $\\delta\\!\\Laux$ provides a fourth. For explicit comparison with \\eqref{ds4d}, the perturbed line element is\n\\begin{equation}\\label{ds2d}\nds^{2}_{2d} = e^{2\\phib}\\Bigl\\{ -(1+2\\delta\\!\\phican+2\\delta\\! N^{\\ct})d\\ct^{2} + 2\\delta\\! N^{x} dxd\\ct + (1+2\\delta\\!\\phican)dx^{2} \\Bigr\\}\n\\end{equation}\nThese are all scalar perturbations; the vector and tensor sectors are absent in 2d. Comparison of coefficients in \\eqref{ds2d} and \\eqref{ds4d} suggests an obvious identification between certain linear combinations of metric perturbations. For instance, $\\Phi$ should be identified with $\\delta\\!\\phican + \\delta\\! N^{\\ct}$, and this linear combination merely reflects the differing metric parametrizations adopted in different dimensions. The non-trivial part of the identification concerns the auxiliary field perturbation $\\delta\\!\\Laux$, which acts as the fourth gravitational degree of freedom in extended Liouville theory. To understand this, it helps to consider the 2d analog of the gauge transformations \\eqref{fieldtransf4d}. In 2d, all infinitesimal diffeomorphisms are locally of the form \\eqref{coordtransf4d}, and the linearized gauge transformation of the perturbations is\n\\begin{equation}\\label{fieldtransf}\n\\begin{aligned}\n\\delta\\!\\phican &\\to \\delta\\!\\phican + \\phib'\\xi^\\ct + \\partial_{x}^{2}\\xi\\\\\n\\delta\\! N^{\\ct} &\\to \\delta\\! N^{\\ct} + \\partial_\\ct{\\xi^\\ct}  - \\partial_{x}^{2} \\xi\\\\\n\\delta\\! N^{x} &\\to \\delta\\! N^{x} - \\partial_x\\xi^\\ct + \\partial_{x}\\xi'\\\\\n\\delta\\!\\Laux &\\to \\delta\\!\\Laux - \\phib' \\xi^\\ct\n\\end{aligned}\n\\end{equation}\nFor comparison with \\eqref{fieldtransf4d}, recall that $a=e^{\\phib}$, so $\\phib'=a'\/a$. In the last line of \\eqref{fieldtransf}, the background condition $\\Lauxb'+\\phib'=0$ \\eqref{chisol} allowed us to write the gauge transformation of the auxiliary scalar field ($\\delta\\!\\Laux\\to\\delta\\!\\Laux+\\Lauxb'\\xi^{\\ct}$) in terms of $\\phib'$. In such backgrounds, then, the gauge transformation of $\\delta\\!\\Laux$ is identical to that of $\\Psi$ in 4d, allowing us to identify these two perturbations. The complete map between extended Liouville perturbations and scalar perturbations in 3+1 Einstein gravity is then\n\\begin{equation}\\label{pertmap}\n\\begin{aligned}\n\\delta\\!\\phican &~ \\leftrightarrow~ -\\Psi + \\vec\\partial^{\\,2} E\\\\\n\\delta\\! N^{\\ct} &~\\leftrightarrow~ \\Phi + \\Psi - \\vec\\partial^{\\,2} E \\\\\n\\int^{x}dx'\\, \\delta\\! N^{x} &~ \\leftrightarrow~ B\\\\\n\\delta\\!\\Laux &~ \\leftrightarrow~ \\Psi\n\\end{aligned}\n\\end{equation}\nFluctuations of the scalar inflaton in 2d and 4d are denoted by\n\\begin{equation}\n\\inflaton  = \\infb(\\ct) + \\delta\\!\\inflaton(\\ct,x) \\nonumber\n\\end{equation}\nTheir gauge transformation is\n\\begin{equation}\\label{infgaugetrans}\n\\delta\\!\\inflaton \\to \\delta\\!\\inflaton + \\infb'\\xi^{\\ct}\n\\end{equation}\nand $\\delta\\!\\inflaton_{2d}$ is identified with $\\delta\\!\\inflaton_{4d}$.\n\nGauge-invariant combinations of metric and matter perturbations are constructed in parallel with well-known 4d results. The introduction of the scalar inflaton makes for five scalar degrees of freedom in both dimensions, which may be arranged into three gauge-invariant linear combinations. The Bardeen potentials in 4d are \\cite{Bardeen:1980kt}\n\\begin{equation}\n\\begin{aligned}\n\\Psi_{\\rm B} &= \\Psi + \\frac{a'}{a}(E'-B) \\\\\n\\Phi_{\\rm B} &= \\Phi - \\frac{1}{a}\\, \\partial_{\\ct}\\bigl[ a(E'-B) \\bigr]\n\\end{aligned}\n\\end{equation}\nand their 2d counterparts follow from the map \\eqref{pertmap}. In order to isolate $E$, one must spatially integrate twice the first two lines of \\eqref{pertmap}, reflecting the different parametrization of the space-space metric. The third gauge-invariant variable is a linear combination of $\\Psi$ and the inflaton perturbation $\\delta\\!\\inflaton$, and is known as the Mukhanov-Sasaki variable \\cite{Mukhanov:1988jd,Sasaki:1986hm}. For convenience in writing the quadratic effective action, this variable contains a dimension-dependent power of the scale factor, which has no effect on its gauge transformation. To facilitate a unified presentation in the following discussion, the spacetime dimension will be denoted by $d$. Only the values $d=2$ and $d=4$ are considered:\n\\begin{equation}\nd = 2n \\qquad (n=1,2)\n\\end{equation}\nThe gauge-invariant Mukhanov-Sasaki variable is then\n\\begin{equation}\\label{vdef}\nv = a^{n-1}\\delta\\!\\inflaton + z\\delta\\!\\Laux\n\\end{equation}\nsubject to the identification $\\delta\\!\\Laux\\leftrightarrow\\Psi$. The quantity $z$ is given by\n\\begin{equation}\\label{zdef}\nz(\\ct) = a^{n-1}(\\dot\\infb\/H) = (2\/\\gamma)a^{n-1}\\sqrt{\\epsilon}\n\\end{equation}\nTo interpret the second equality in 4d, recall that $2\/\\gamma$ plays the role of the Planck mass $m_{p}$.\n\n\n\\subsubsection*{Cosmological gauge choices}\nThe identifications \\eqref{pertmap} mean that any gauge choice for the scalar sector in 4d has an analog in extended Liouville theory. Two popular gauges which will be of particular interest in this paper are \\cite{Baumann:2009ds}\n\\begin{align}\n\\qquad \\Psi = 0 = E~&\\leftrightarrow~ \\delta\\!\\Laux = 0 = \\delta\\!\\phican &\\text{(spatially flat gauge)}\\label{sfgauge}\\\\\n\\qquad \\delta\\!\\inflaton = 0 = E~&\\leftrightarrow~ \\delta\\!\\inflaton = 0 = \\delta\\!\\Laux + \\delta\\!\\phican  &\\text{(constant inflaton gauge)}\\label{cfgauge}\n\\end{align}\nBoth of these gauge choices completely fix local diffeomorphisms. In spatially flat gauge in 3+1d, the local coordinates are chosen to set to zero the curvature of spatial slices. Spatial slices in 1+1d don't have curvature, but still the spatial metric is fixed to the identity in spatially flat gauge. Constant inflaton gauge has the same interpretation in both dimensions --- the time-dependent inflaton background is used to set our clock.\n\n\n\n\n\n\n\\subsection{Quadratic fluctuation action}\\label{Mukhanovsec}\nThe structure of the quadratic fluctuation action for extended Liouville theory plus the scalar inflaton follows precisely the familiar Mukhanov-Sasaki action in 3+1 dimensions \\cite{Mukhanov:1988jd,Sasaki:1986hm}. At each order in perturbation theory (quadratic, cubic, etc.), a classical effective action is constructed by adjusting the lapse and shift fluctuations to enforce the constraints.\\footnote{This is not the only possible way to proceed. One may also construct perturbation theory in the Hamiltonian formalism, where the lapse and shift are Lagrange multipliers whose value is arbitrary. The constraints are then solved by adjusting dynamical variables and\/or their momenta. Translating back to the Lagrangian formalism, one finds at quadratic order the lapse and shift appear only as total derivatives. They drop out of the action while remaining arbitrary, leaving the familiar Mukhanov-Sasaki action. In some sense this procedure is more natural than that employed in the text, because it makes clear why the result \\eqref{Meffact} depends only on $v$, and not $\\Phi_{\\rm B}$ or $\\Psi_{\\rm B}$ --- the latter contain arbitrary Lagrange multipliers.} The constrained lapse and shift fluctuations $\\delta\\! N^{\\ct}$ and $\\delta\\! N^{x}$ are functions of the dynamical variables $\\Omega$, whose fluctuations are collectively denoted $\\bm\\omega$. This results in an expansion\n\\begin{equation}\\label{naexp}\n\\bm n^{a}(\\bm\\omega) = \\bm n^{a}_{1}(\\bm\\omega) + \\bm n^{a}_{2}(\\bm\\omega) + \\cdots \\qquad (a=\\ct,x)\n\\end{equation}\nwhere the subscript denotes the order in $\\bm\\omega$. The quadratic action is composed of terms of the form $\\bm\\omega^{2}$, $\\bm\\omega\\, \\bm n_{1}(\\bm\\omega)$, or $\\bm n_{2}(\\bm\\omega)$. The coefficient of the latter is the vanishing background constraints, so we only need to know the first order lapse and shift $\\bm n^{a}_{1}(\\bm\\omega)$.\n\nIn the case of extended Liouville theory plus the scalar inflaton, the Hamiltonian and momentum constraints were given in \\eqref{constraints}. Their expansion to first order in fluctuations is\n\\begin{equation}\\label{constraintpert}\n\\begin{aligned}\n(\\gamma^{2}\/4)\\,\\mathcal{H}_{1} &= \\phib'^{\\,2}(1-\\epsilon) \\bigl(\\bm n^{\\ct}_{1}+\\delta\\!\\phican\\bigr) + \\phib'\\bigl(\\partial_{x}\\bm n^{x}_{1} - \\delta\\!\\phican'\\bigr) + \\partial_{x}^{2}\\delta\\!\\Laux + \\frac{\\gamma^{2}}{4}\\Bigl( \\infb'\\delta\\!\\inflaton' + {1\\over2} e^{2\\phib}{\\mathcal{V}}_{,\\infsub}(\\infb)\\delta\\!\\inflaton \\Bigr) \\\\\n(\\gamma^{2}\/4)\\,\\mathcal{P}_{1} &= - \\phib' \\partial_{x}\\( \\bm n^{\\ct}_{1} + \\delta\\!\\phican + \\delta\\!\\Laux'\/\\phib' - \\frac{\\gamma}{2}\\sqrt\\epsilon\\, \\delta\\!\\inflaton \\)\n\\end{aligned}\n\\end{equation}\nThe equations $\\mathcal{H}_{1}=0=\\mathcal{P}_{1}$ are solved to give\n\\begin{equation}\\label{ntnx}\n\\begin{aligned}\n\\bm n^{\\ct}_{1} &= -\\delta\\!\\phican - \\delta\\!\\Laux'\/\\phib' + \\frac{\\gamma}{2}\\sqrt\\epsilon\\, \\delta\\!\\inflaton \\\\\n\\bm n^{x}_{1} &= \\partial_{x}\\delta\\!\\Laux\/\\phib' + \\psi \n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{psix}\n\\partial_{x}\\psi = \\delta\\!\\phican' + (1-\\epsilon)\\delta\\!\\Laux' - \\frac{\\gamma}{2}\\epsilon\\bigl(\\delta\\!\\inflaton\/\\sqrt\\epsilon\\bigr)'\n\\end{equation}\nThe fact that $\\psi$ is the subject of a differential equation has the effect of introducing spatial nonlocality into the constrained effective action. The quadratic action, however, may be expressed in terms of $\\partial_{x}\\psi$ only, using integration by parts; at cubic order we will see the nonlocality of $\\psi$ is genuine. It is also gauge-invariant, because $\\partial_{x}\\psi$ changes only by a total $x$-derivative under gauge transformations.\n\nTo arrive at the constrained quadratic action, one expands the extended Liouville action \\eqref{Lcovpre} plus scalar inflaton \\eqref{SX} to second order in fluctuations, substitutes \\eqref{ntnx}, and integrates by parts. The final result is the gauge-invariant Mukhanov-Sasaki action \\cite{Mukhanov:1988jd,Sasaki:1986hm}\n\\begin{eqnarray}\n\\label{Meffact}\n{\\mathcal{S}}_{2} = \\frac{1}{2}\\int\\biggl( v'^{\\,2}-(\\partial_x v)^2 + \\frac{z''}{z} v^2 \\biggr)\n\\end{eqnarray}\nThe form of this action is identical in 2d and 4d. The Mukhanov-Sasaki variable $v$ is effectively a free scalar field with time-dependent mass $m^{2}=z''\/z$. In constant inflaton gauge \\eqref{cfgauge}, where $v$ given by \\eqref{vdef} reduces to $-z\\delta\\!\\phican\\sim-\\epsilon^{1\/2}\\delta\\!\\phican$, the action \\eqref{Meffact} is manifestly suppressed by $\\epsilon$. For a more detailed derivation of \\eqref{Meffact} in extended Liouville theory, see \\cite{Martinec:2014uva}.\n\n\n\n\n\n\n\\subsection{Mode functions}\\label{modesec}\nWe now investigate the equation of motion for the Mukhanov-Sasaki action, solving for the spatial momentum mode functions of $v(\\ct,\\mathbf{x})$. We continue to denote the spacetime dimension by $d=2n$. The equation of motion following from the action \\eqref{Meffact} is\n\\begin{equation}\\label{veom}\n\\Bigl( \\partial^{2} + \\frac{z''}{z} \\Bigr) v = 0\n\\end{equation}\nWhen the field $v(\\ct,\\mathbf{x})$ is expanded into spatial momentum modes as\n\\begin{equation}\\label{vexp}\nv(\\ct,\\mathbf{x}) = \\int\\frac{d^{d-1}k}{(2\\pi)^{d-1}} \\Big( a(\\mathbf{k})v_{k}(\\ct)e^{i\\mathbf{k}\\cdot\\mathbf{x}} + \\text{h.c.} \\Big)\n\\end{equation}\nequation \\eqref{veom} becomes the Mukhanov-Sasaki equation for the modes $v_{k}(\\ct)$:\n\\begin{equation}\\label{Mukhanoveq}\nv_{k}'' + \\Bigl( k^{2}-\\frac{z''}{z} \\Bigr) v_{k} = 0 \n\\end{equation}\nFollowing \\cite{Baumann:2009ds}, we proceed to solve this equation by expanding the mass term $z''\/z$ in slow roll. Differentiating $z(\\ct)$ given by \\eqref{zdef} and comparing with the slow roll parameters \\eqref{SRparams}, one finds the exact expressions\n\\begin{equation}\\label{SRz1}\n\\begin{aligned}\n\\frac{z'}{z} & = aH\\Bigl[ n-1 + {1\\over2}\\eta \\Bigr] \\\\\n\\frac{z''}{z} & =(aH)^{2}\\Bigl[ (n-1)(n + \\eta-\\epsilon) + {1\\over2}\\eta\\Bigl( 1 + {1\\over2}\\eta - \\epsilon + \\kappa \\Bigr) \\Bigr]\n\\end{aligned}\n\\end{equation}\nNote that the mass term $z''\/z$ vanishes in 2d at zeroth order in slow roll. This means that in 2d de~Sitter  space, the Mukhanov-Sasaki variable is a massless scalar field. The prefactor $aH=\\phib'$ in \\eqref{SRz1} is expressed in terms of the slow roll parameter $\\epsilon$ as\n\\begin{equation}\n\\frac{1}{aH} = -\\ct(1-\\epsilon)\n\\end{equation}\nwhich satisfies $aH\\to\\infty$ as $\\ct\\to0^{-}$. Substituting in \\eqref{SRz1} gives the exact expressions\n\\begin{equation}\\label{SRz2}\n\\begin{aligned}\n\\frac{z'}{z} & = -\\frac{1}{\\ct(1-\\epsilon)}\\Bigl[ n-1 + {1\\over2}\\eta \\Bigr] \\\\\n\\frac{z''}{z} & =\\frac{1}{\\ct^{2}(1-\\epsilon)^{2}} \\Bigl[ (n-1)(n + \\eta-\\epsilon) + {1\\over2}\\eta\\Bigl( 1 + {1\\over2}\\eta - \\epsilon + \\kappa \\Bigr) \\Bigr]\n\\end{aligned}\n\\end{equation}\nAlthough $z=0$ in de~Sitter  space (where all slow roll parameters vanish), the ratios \\eqref{SRz2} are non-vanishing in de Sitter, except in 2d. The Mukhanov-Sasaki equation \\eqref{Mukhanoveq} may now be written in the form of a Bessel equation:\n\\begin{equation}\\label{Beq}\n v_{k}'' + \\( k^{2}-\\frac{\\nu^{2}-1\/4}{\\ct^{2}} \\) v_{k}  = 0\n\\end{equation}\nThe order $\\nu$, which is constant only to first order in slow roll, is given by\n\\begin{equation}\n\\nu = n - {1\\over2} + (n-1)\\epsilon + {1\\over2}\\eta + \\cdots\n\\end{equation}\nThe ellipsis represents terms of higher order in slow roll; by ignoring time derivatives of $\\nu$ we are already ignoring such terms. The first order solutions to \\eqref{Beq} are\n\\begin{equation}\\label{vksol}\nv_{k}(\\ct) = \\sqrt{-\\ct}\\, H_{\\nu}^{(1,2)}(-k\\ct)\n\\end{equation}\nThe Bunch-Davies condition states that in the far past, when the mode $\\mathbf{k}$ is much smaller than the horizon, the vacuum should be the usual Minkowski vacuum. In other words, if we use the vacuum annihilated by $a(\\mathbf{k})$ in the mode expansion \\eqref{vexp}, then the modes $v_{k}$ must have positive energy according to the Minkowski space representation $E=i\\partial_{t}$. The asymptotic behavior of the Hankel function at early times ($-k\\ct\\to\\infty$) follows from\n\\begin{equation}\nH_{\\nu}^{(1,2)}(x) \\sim \\sqrt{\\frac{2}{\\pi x}} \\,\\exp \\pm i\\Bigl( x-{1\\over2}\\nu\\pi -\\frac{1}{4}\\pi \\Bigr)\\qquad (x\\to\\infty)\n\\end{equation}\nHence the normalized Bunch-Davies solution is given by the Hankel function of the first kind:\n\\begin{equation}\nv_{k}^{(+)}(\\ct) = {1\\over2} e^{+i\\pi(\\nu\/2+1\/4)}\\sqrt{-\\pi \\ct}\\,H_{\\nu}^{(1)}(-k\\ct)  \n\\end{equation}\nThe overall phase is arbitrary and has been chosen for convenience. The Bunch-Davies vacuum will be adopted exclusively from here, and the superscript $(+)$ will be dropped.\n\nIn constant inflaton gauge \\eqref{cfgauge}, the Mukhanov-Sasaki variable \\eqref{vdef} reduces to $v=-z\\delta\\!\\phican \\leftrightarrow z\\Psi$. The comoving curvature perturbation\\footnote{The comoving curvature perturbation in 3+1d describes the curvature of spatial slices in constant inflaton gauge. In 1+1d, spatial slices don't have curvature, so the name ``curvature perturbation'' is something of a misnomer. Perhaps $\\zeta$ is best thought of in 1+1d as a perturbation of the scale factor. For the same reason, spatially flat gauge should perhaps be called ``constant scale factor gauge'' in 1+1d.} $\\zeta$ is defined by $v=-z\\zeta$, or\n\\begin{equation}\n\\zeta_{2d}=\\delta\\!\\phican~,\\qquad \\zeta_{4d}=-\\Psi\n\\end{equation}\nThere are differing sign conventions for $\\zeta$ in the literature; our choice corresponds to a perturbed scale factor $e^{\\phib+\\zeta}$  in constant inflaton gauge. The explicit curvature mode functions at lowest order in slow roll are\n\\begin{align}\n\\zeta_{k}(\\ct) &= - \\frac{1}{z(\\ct)} \\frac{e^{-ik\\ct}}{\\sqrt{2k}} + \\cdots &(d=2) \\label{zetak2d} \\\\\n\\zeta_{k}(\\ct) &=  \\frac{-i}{z(\\ct)(-\\ct)} \\frac{e^{-ik\\ct}}{\\sqrt{2k^{3}}} \\(1+ik\\ct\\) + \\cdots  &(d=4) \\label{zetak4d}\n\\end{align}\nNote that the denominators carry only a weak time dependence, with an exponent proportional to the slow roll parameters. This is because $z(\\ct)$ is given to first order in slow roll by integrating the first order expansion of \\eqref{SRz2}, pretending the slow roll parameters are constant:\n\\begin{equation}\\label{zct0}\nz(\\ct) =z(\\ct_{0}) (\\ct\/\\ct_{0})^{{1\\over2}-\\nu}+\\cdots\n\\end{equation}\nHere $\\ct_{0}$ could be any reference time, so the product $z(\\ct)(-\\ct)^{\\nu-{1\\over2}}=z(\\ct)(-\\ct)^{n-1+\\cdots}$ is constant to first order in slow roll. Note also that the modes \\eqref{zetak2d} and \\eqref{zetak4d} freeze out in physical time $t$ with a dimension-dependent power of $k\/aH=-k\\ct$:\n\\begin{equation}\\label{freezeout}\n\\partial_{t}\\zeta_{k} \\propto (-k\\ct)^{n} \\zeta_{k} \\qquad (-k\\ct\\ll1)\n\\end{equation}\n In particular, $\\partial_{\\ct}\\zeta_{k}=a\\,\\partial_{t}\\zeta_{k}=(-1\/H\\ct)\\partial_{t}\\zeta_{k}$ does not vanish in 2d as $\\ct\\to0^{-}$, so one must be careful to assume only the vanishing of $\\partial_{t}\\zeta_{k}$ on superhorizon scales. This slower rate of freeze out may be traced back to the vanishing in 2d of the effective mass $m^{2}=-z''\/z$ in the Mukhanov-Sasaki equation \\eqref{Mukhanoveq}.\n\n\n\\subsection{Power spectrum}\\label{powersec}\nThe power spectrum of scalar curvature perturbations, $P_{\\zeta}(k)$, is the Fourier transform of the two-point function:\n\\begin{equation}\n\\langle0| \\zeta(\\ct,\\mathbf{x})\\zeta(\\ct,\\mathbf{x}') |0\\rangle = \\int\\frac{d^{d-1}k}{(2\\pi)^{d-1}} e^{i\\mathbf{k}\\cdot(\\mathbf{x}-\\mathbf{x}')} P_{\\zeta}(k)\n\\end{equation}\nExpanding the left side into mode functions, one finds\n\\begin{equation}\nP_{\\zeta}(k) = \\big| \\zeta_{k}(\\ct) \\big|^{2}\n=\\frac{\\pi}{4} \\(\\frac{-\\ct}{z^{2}}\\) \\big|H_{\\nu}^{(1)}(-k\\ct)\\big|^{2}\n\\end{equation}\nTo evaluate this on superhorizon scales ($-k\\ct\\ll1$), we need the late-time behavior of the Hankel function:\n\\begin{equation}\nH_{\\nu}^{(1)}(x) \\sim -(i\/\\pi)\\Gamma(\\nu)(x\/2)^{-\\nu} \\qquad (x\\to0^{+})\n\\end{equation}\nThe power spectrum on superhorizon scales is then\n\\begin{equation}\\label{P1}\nP_{\\zeta}(k) = 2^{2\\nu-2} \\Bigl[z(\\ct)(-\\ct)^{\\nu-{1\\over2}}\\Bigr]^{-2} \\frac{1}{k^{2\\nu}}\n\\qquad (-k\\ct\\ll1)\n\\end{equation}\nThe factor in brackets is constant by \\eqref{zct0}, so $1\/k^{2\\nu}=1\/k^{d-1+\\cdots}$ is the complete momentum dependence of the power spectrum at first order in slow roll. The spectral tilt (the deviation of the exponent from the scale-invariant value $(d-1)$\\,) is\n\\begin{equation}\\label{tilt}\n(n_{s}-1)  = (d-1) - 2\\nu =  -2(n-1)\\epsilon - \\eta + \\cdots\n\\end{equation}\nAnother useful expression for the power spectrum is obtained by specializing the bracket in \\eqref{P1} to the moment $\\ct_{*}(k)$ when the mode $k$ crosses the horizon, given by\n\\begin{equation}\n\\ct_{*}(k) = -1\/k\n\\end{equation}\nEvaluating the constant bracket in \\eqref{P1} at this time (using the definition \\eqref{zdef} of $z(\\ct)$) leads to\n\\begin{equation}\\label{P2}\nP_{\\zeta}(k) = 2^{d-5}\\gamma^{2}\\, \\frac{H_{*}^{d-2}}{\\epsilon_{*}}\\, \\frac{1}{k^{d-1}}\n\\end{equation}\nwhere $H_{*}$ and $\\epsilon_{*}$ are evaluated at horizon crossing. The spectral tilt is now contained in the momentum dependence of $H_{*}$ and $\\epsilon_{*}$. The form \\eqref{P2} of the power spectrum is identical in 2d and 4d (recall that $\\gamma^{2}\/4$ plays the role of $1\/m_{p}^{2}$). A difference is that the spectral tilt \\eqref{tilt} depends only on $\\eta$ in 2d; this traces back to the dimension-dependent factor of $a$ in $z(\\ct)$ given by \\eqref{zdef}.\n\nThis concludes the quantitative comparison of extended Liouville theory with the scalar sector of Einstein gravity at quadratic order. (As a final note, extended Liouville theory has a tensor to scalar ratio of $r=0$\\,! Although, if one wished, it might be possible to mimic the 4d vector and tensor sectors by including extra degrees of freedom in 2d, with a judicious choice of action.) All equations in the remainder of this paper are specialized to $d=2$. To underline this, the degree of freedom in constant inflaton gauge will henceforth be denoted $\\delta\\!\\phican$, although one should keep in mind that this plays the role of $\\zeta$.\n\n\n\n\\subsection[Constancy of zeta outside the horizon]{Constancy of $\\zeta$ outside the horizon}\\label{constancysec}\nBefore turning to the cubic action, let us confirm the constancy of $\\zeta_{2d}=\\delta\\!\\phican$ outside the horizon, to all orders in perturbations. To do this, we begin from the full action \\eqref{Lcovpre} plus \\eqref{SX}, specialize to constant inflaton gauge, and neglect spatial derivatives of the dynamical fields. The key point \\cite{Maldacena:2002vr} is that if the expansion of the superhorizon action in physical time derivatives of $\\delta\\!\\phican$ begins at second order, then the equation of motion for $\\delta\\!\\phican$ will always admit solutions with $\\dot\\delta\\!\\phican=0$. Hence it will be enough to show the zeroth and first order terms in this expansion vanish.\n\nFor ease in applying the constraints to all orders in perturbations, we begin from the Hamiltonian form of the action,\n\\begin{equation}\\label{SHam}\n{\\mathcal{S}} = \\int \\Bigl( \\pi_{\\Laux}\\Laux' + \\pi_{\\phican}\\phican' + \\pi_{\\inflaton}\\inflaton' - N^{\\ct}\\mathcal{H} - N^{x}\\mathcal{P} \\Bigr)\n\\end{equation}\nwhere prime continues to denote $\\partial_{\\ct}$. The conjugate momenta and constraints were given in \\eqref{momenta} and \\eqref{constraints}. In backgrounds satisfying $\\Lauxb'+\\phib'=0$, the constant inflaton gauge condition $\\delta\\!\\Laux+\\delta\\!\\phican=0$ means the full fields (background plus perturbation) obey $\\Laux'+\\phican'=0$. For simplicity we assume that $\\Lambda$ has been absorbed in ${\\mathcal{V}}(\\infb)$; recall that $\\inflaton=\\infb$ is also fixed in constant inflaton gauge. The constraints \\eqref{constraints} now reduce to\n\\begin{equation}\\label{cfconstraints}\n\\begin{aligned}\n(N^{\\ct})^{2}\\, \\mathcal{H} &= \\frac{2}{\\gamma^{2}} \\(-\\phi'^{2} +2\\phi'\\partial_{x} N^{x} \\) + {1\\over2}\\infb'^{2} + {1\\over2} e^{2\\phican}{\\mathcal{V}}(\\infb)(N^{\\ct})^{2} \\\\\n(N^{\\ct})^{2}\\, \\mathcal{P} &= -\\frac{4}{\\gamma^{2}} \\phican' \\partial_{x} N^{\\ct}\n\\end{aligned}\n\\end{equation}\nEnforcing the vanishing of these constraints in the action \\eqref{SHam} in constant inflaton gauge gives outside the horizon\n\\begin{equation}\\label{Scfcon}\n{\\mathcal{S}}_{\\rm constr.} = -\\int  e^{2\\phican}{\\mathcal{V}}(\\infb)N^{\\ct} = -\\int  e^{2\\phib+2\\delta\\!\\phican}{\\mathcal{V}}(\\infb) \\[ 1+ \\delta\\! N^{\\ct}(\\delta\\!\\phican) \\]\n\\end{equation}\nHere $\\delta\\! N^{\\ct}(\\delta\\!\\phican)$ is the solution of the perturbed Hamiltonian constraint. So far we have worked to all orders in field perturbations and time derivatives --- only spatial derivatives have been neglected. We now expand the Hamiltonian constraint \\eqref{cfconstraints} to first order in time derivatives of $\\delta\\!\\phican$, assuming that $\\delta\\! N^{\\ct},\\delta\\! N^{x}$ are themselves of this order. This assumption is consistent with the structure of \\eqref{cfconstraints}, once the background constraints are applied. The perturbed Hamiltonian constraint is solved for $\\delta\\! N^{\\ct}$, giving\n\\begin{equation}\\label{cfnt}\n {\\mathcal{V}}(\\infb)\\delta\\! N^{\\ct} = \\frac{4}{\\gamma^{2}}\\dot\\phib \\( \\dot\\delta\\!\\phican - e^{-\\phib-\\delta\\!\\phican}\\partial_{x}\\delta\\! N^{x} \\) \n\\end{equation}\nPhysical time $t$ has been adopted to allow us to combine terms in the action. The measure $d\\ct$ in the action is replaced by $e^{-\\phib-\\delta\\!\\phican}dt$. Substituting \\eqref{cfnt} in \\eqref{Scfcon} and eliminating the remaining ${\\mathcal{V}}(\\infb)$ by the $\\phib$ equation of motion \\eqref{phteoms}, one finds\n\\begin{equation}\n{\\mathcal{S}}_{\\rm constr.}^{(<2)} = -\\frac{4}{\\gamma^{2}} \\int e^{\\phib+\\delta\\!\\phican}\\( \\ddot\\phib + \\dot\\phib^{2} + \\dot\\phib\\dot\\delta\\!\\phican \\) \n= -\\frac{4}{\\gamma^{2}} \\int \\partial_{t}\\( e^{\\phib+\\delta\\!\\phican} \\dot\\phib \\)\n\\end{equation}\nThis does not contribute to the equation of motion, so indeed the expansion in physical time derivatives begins at second order. There are always solutions with $\\dot\\delta\\!\\phican=0$, so $\\delta\\!\\phican$ freezes out to all orders in perturbation theory. This is consistent with our earlier finding at quadratic order in fields, equation \\eqref{freezeout}. It doesn't tell us about the rate of freeze out to all orders, but this may be expected to follow the pattern at quadratic order, where freeze out was slower in 2d due to the vanishing effective mass in de~Sitter  space.\n\n\n\n\n\n\\section{Cubic fluctuation action}\\label{cubicsec}\n\nThe extended Liouville action \\eqref{Lcovpre} plus scalar inflaton \\eqref{SX} is now expanded to cubic order in fluctuations, constrained, and massaged into a form suitable for computation of the three-point function. Our discussion follows closely that given by Maldacena \\cite{Maldacena:2002vr}. We do not repeat at cubic order the manifestly gauge-invariant structure of the quadratic action \\eqref{Meffact}, in terms of the Mukhanov-Sasaki variable. This would require the use of variables invariant under second-order gauge transformations, the formulation of which is highly complex (see, for example, \\cite{Nakamura:2004rm}). Instead, we follow \\cite{Maldacena:2002vr} by computing the cubic action first in spatially flat gauge \\eqref{sfgauge}, before switching to constant inflaton gauge \\eqref{cfgauge}. A comparison of these two gauges allows us to make clear in each the suppression of the cubic action by two powers of the slow roll parameter $\\epsilon$. A new feature at cubic order is the spatial nonlocality of the constrained action, arising from the solution $\\psi$ of the differential equation \\eqref{psix}. This appears the same in each gauge, because $\\partial_{x}\\psi$ changes only by a total $x$-derivative under gauge transformations.\n\nThe suppression of the cubic action by an additional power of $\\epsilon$ over the quadratic action is the basis for perturbation theory --- the perturbative expansion is an expansion in slow roll parameters. These are assumed to be of order a few percent, hence terms higher than $O(\\epsilon^{2})$ will be freely discarded in the following. \n\nBefore beginning, we mention a helpful simplification regarding the constraints \\cite{Maldacena:2002vr}. The cubic computation continues the perturbative program of ``constrain, then quantize'' begun in sections \\ref{Mukhanovsec} - \\ref{powersec}. Again the perturbed constraints are solved by adjusting the lapse and shift fluctuations, resulting in the order-by-order expansion \\eqref{naexp}. It turns out that the cubic action, like the quadratic action, depends only on the first order lapse and shift fluctuations $\\bm n^{a}_{1}(\\bm\\omega)$ --- the first term in \\eqref{naexp}. This is because the third order term $\\bm n^{a}_{3}(\\bm\\omega)$ is multiplied by the vanishing background constraints, while the second order term $\\bm n^{a}_{2}(\\bm\\omega)$ appears in the cubic action as (schematically)\n\\begin{equation}\n\\(\\fdd{S}{N^{2}}\\Bigr|_{0}\\bm n_{1}(\\bm\\omega)+\\fdd{S}{N\\delta\\Omega}\\Bigr|_{0}\\bm\\omega\\)\\bm n_{2}(\\bm\\omega)\n\\end{equation}\nRecall that $\\Omega$ runs over the dynamical fields, and $\\bm\\omega$ denotes their fluctuations. The brackets contain the first order expansion of the constraints $\\delta S\/\\delta N^{a}$, which vanish by definition of $\\bm n_{1}^{a}(\\bm\\omega)$. Hence we need only know the first order lapse and shift $\\bm n_{1}^{a}(\\bm\\omega)$, which are still given by \\eqref{ntnx}.\n\n\n\n\n\n\\subsection{Spatially flat gauge}\nIt is easiest to exhibit the slow roll suppression of the constrained cubic action in the spatially flat gauge \\eqref{sfgauge}\n\\begin{equation}\n\\delta\\!\\Laux=0=\\delta\\!\\phican \n\\end{equation}\nWe expand the extended Liouville action \\eqref{Lcovpre} plus scalar inflaton \\eqref{SX} to third order in fluctuations, substitute the first order lapse and shift solutions \\eqref{ntnx}, and discard $\\delta\\!\\Laux$ and $\\delta\\!\\phican$ terms. After a number of integrations by parts, one finds\n\\begin{multline}\\label{S3x}\n{\\mathcal{S}}_{3}= \\int -\\frac{\\gamma}{4}\\epsilon^{1\/2} \\bigl[ \\delta\\!\\inflaton\\infpert'^{2}+\\delta\\!\\inflaton(\\partial_{x}\\delta\\!\\inflaton)^{2}\\bigr]  -\\psi\\infpert'\\partial_{x}\\delta\\!\\inflaton \\\\\n+\\(  \\frac{\\gamma}{6}\\epsilon^{3\/2}\\phib''\n + \\frac{\\gamma}{8}\\epsilon^{1\/2}a^{2}{\\mathcal{V}}_{,\\infsub\\infsub}(\\infb) - \\frac{1}{12}a^{2}{\\mathcal{V}}_{,\\infsub\\infsub\\infsub}(\\infb)\n \\) \\delta\\!\\inflaton^{3}\n\\end{multline}\nThis should be compared with the analogous expression in 4d (equation (3.8) of \\cite{Maldacena:2002vr}). Upon wading through the notational differences,\\footnote{The conversion between the notations employed here and in \\cite{Maldacena:2002vr} is: $\\phi$ there $\\leftrightarrow$ $\\infb$ here, $\\varphi$ there $\\leftrightarrow$ $\\delta\\!\\inflaton$ here, $\\rho$ there $\\leftrightarrow$ $\\phib$ here, $\\zeta$ there $\\leftrightarrow$ $\\delta\\!\\phican$ here, $\\partial_{i}\\chi$ there $\\leftrightarrow$ $\\psi$ here.} accounting for the choice of time variable (coordinate time in \\cite{Maldacena:2002vr}, conformal time here), and recalling the expression \\eqref{SRvel} for $\\epsilon$ in terms of the background velocities, one finds that most terms in equation (3.8) of \\cite{Maldacena:2002vr} have precise analogs in \\eqref{S3x}. The remaining terms are simply absent, because there are fewer distinct combinations of spatial derivatives in 1+1d.\n\nNote that $\\psi$ given by \\eqref{psix} reduces in spatially flat gauge to\n\\begin{equation}\\label{psisf}\n\\partial_{x}\\psi = - \\frac{\\gamma}{2}\\epsilon\\bigl(\\delta\\!\\inflaton\/\\sqrt\\epsilon\\bigr)'\n\\end{equation}\nHence the entire first line of \\eqref{S3x} is of order $\\epsilon^{1\/2}\\delta\\!\\inflaton^{3}$. This common slow roll suppression is the advantage of working in spatially flat gauge. The second line of \\eqref{S3x} is of higher order in slow roll; the relations \\eqref{SRpot} imply that derivatives of the matter potential are suppressed as\n\\begin{equation}\n\\(\\pd{}{\\inflaton}\\)^{n} {\\mathcal{V}}(\\infb) \\sim \\epsilon^{n\/2}\n\\end{equation}\nKeeping only the first line of \\eqref{S3x}, we substitute \\eqref{psisf} to arrive at\n\\begin{equation}\\label{S3sf}\n{\\mathcal{S}}_{3} = -\\frac{\\gamma}{4} \\int \\epsilon^{1\/2} \\[ \\delta\\!\\inflaton\\infpert'^{2} +\\delta\\!\\inflaton(\\partial_{x}\\delta\\!\\inflaton)^{2} -2\\delta\\!\\inflaton'\\partial_{x}\\delta\\!\\inflaton\\partial_{x}^{-1}\\delta\\!\\inflaton' \\] \n-\\frac{\\gamma}{4} \\int aH\\epsilon^{1\/2} \\eta\\, \\delta\\!\\inflaton'\\partial_{x}\\delta\\!\\inflaton\\partial_{x}^{-1}\\delta\\!\\inflaton + \\cdots\n\\end{equation}\nAgain, the last term may be discarded at leading order in slow roll. This is our final expression for the cubic action in spatially flat gauge. In the remainder of this section we clarify its order of slow roll suppression.\n\n\n\\subsubsection*{Translating between two gauges}\nThus far we have kept only the leading order slow roll terms, but it is not clear that these have the expected suppression by $\\epsilon^{2}$. This is not surprising --- after all, the quadratic action \\eqref{Meffact} is manifestly of order $\\epsilon$ only when written in terms of $\\zeta=\\delta\\!\\phican$, the degree of freedom in constant inflaton gauge:\n\\begin{equation}\\label{S2cf}\n{\\mathcal{S}}_{2}^{\\rm c.i.} = -\\frac{2}{\\gamma^{2}}\\int \\epsilon\\, (\\partial\\delta\\!\\phican)^{2}\n\\end{equation}\nIn this section we have chosen spatially flat gauge (with degree of freedom $\\delta\\!\\inflaton$), but it turns out to be useful to translate between $\\delta\\!\\inflaton$ and $\\delta\\!\\phican$ \\cite{Maldacena:2002vr}. This is not the same as computing the cubic action in constant inflaton gauge, because \\eqref{S3x} is an effective action where the constraints have been applied in spatially flat gauge. Neither are we performing a diffeomorphism directly on \\eqref{S3sf}. Rather, the following is a change of variable arising from a comparison of the relative order in slow roll of $\\delta\\!\\inflaton$ and $\\delta\\!\\phican$. Of course, $\\delta\\!\\inflaton$ and $\\delta\\!\\phican$ are fluctuations which may have any profile whatsoever; the point is that a given profile of (say) $\\delta\\!\\inflaton$ in spatially flat gauge induces a particular profile of $\\delta\\!\\phican$ in constant inflaton gauge. We will work out this relation to second order in perturbations, which will be of use in the following section. Recall that constant inflaton gauge \\eqref{cfgauge} is\n\\begin{equation}\\label{constx}\n\\delta\\!\\inflaton=0=\\delta\\!\\Laux+\\delta\\!\\phican\n\\end{equation}\nThe condition $\\delta\\!\\Laux+\\delta\\!\\phican=0$ is trivially true in spatially flat gauge ($\\delta\\!\\Laux=0=\\delta\\!\\phican$), and according to the gauge transformations \\eqref{fieldtransf} is preserved under time reparametrizations $\\xi^{\\ct}$. To access the gauge slice $\\delta\\!\\inflaton=0$ beginning from spatially flat gauge, one must make a diffeomorphism $(\\xi^{\\ct},\\xi^{x})$ satisfying\n\\begin{equation}\\label{xiteq}\n\\infb(x^{a}+\\xi^{a})+\\delta\\!\\inflaton_{\\rm s.f.}(x^{a}+\\xi^{a})=\\infb(x^{a})\n\\end{equation}\nwhere $\\delta\\!\\inflaton_{\\rm s.f.}(x^{a})$ is the profile of $\\delta\\!\\inflaton$ in spatially flat gauge. We wish to keep $\\xi^{x}=0$, and solve \\eqref{xiteq} for $\\xi^{\\ct}$ up to second order in $\\delta\\!\\inflaton_{\\rm s.f.}$. The Taylor expansion of $\\delta\\!\\inflaton_{\\rm s.f.}(x^{a}+\\xi^{a})$, which was omitted from the gauge transformation \\eqref{infgaugetrans}, must be included. The solution to \\eqref{xiteq} is then\n\\begin{equation}\\label{xit2}\n\\infb' \\xi^{\\ct}= - \\delta\\!\\inflaton_{\\rm s.f.} + {1\\over2} \\bigl( \\delta\\!\\inflaton_{\\rm s.f.}^{2}\/\\infb' \\bigr)'\n\\end{equation}\nThis is the diffeomorphism which takes us from spatially flat gauge to constant inflaton gauge. The profile of $\\delta\\!\\phican$ in this new gauge is\n\\begin{equation}\\label{chi2}\n\\begin{aligned}\n\\delta\\!\\phican_{\\rm c.i.} &=\\phib'\\xi^{\\ct}+{1\\over2}\\phib''(\\xi^{\\ct})^{2} \\\\\n&= - \\frac{\\phib'}{\\infb'}\\delta\\!\\inflaton_{\\rm s.f.} + \\frac{1}{2\\infb'}\\bigl( \\phib'\\delta\\!\\inflaton_{\\rm s.f.}^{2}\/\\infb' \\bigr)' \n\\end{aligned}\n\\end{equation}\nIn terms of the slow roll parameters,\n\\begin{equation}\\label{phi2x}\n\\delta\\!\\phican_{\\rm c.i.} =  - \\frac{1}{z}\\,\\delta\\!\\inflaton_{\\rm s.f.} \n+ \\frac{1}{2aHz}\\bigl( \\delta\\!\\inflaton_{\\rm s.f.}^{2}\/z \\bigr) '\n\\end{equation}\nThe second order inverse of this relation is\n\\begin{equation}\\label{x2phi}\n\\delta\\!\\inflaton_{\\rm s.f.} = -z\\delta\\!\\phican_{\\rm c.i.} + \\frac{1}{2aH}\\bigl( z\\delta\\!\\phican_{\\rm c.i.}^{2} \\bigr) '\n\\end{equation}\nRecall from \\eqref{zdef} that $z\\propto\\epsilon^{1\/2}$. The first order relation $\\delta\\!\\inflaton_{\\rm s.f.}\\sim\\epsilon^{1\/2}\\delta\\!\\phican_{\\rm c.i.}$ renders the cubic action \\eqref{S3sf} of order $\\epsilon^{2}$ in terms of $\\delta\\!\\phican_{\\rm c.i.}$. To cover all the bases, one must also account for the mixing of quadratic and cubic actions arising from the second order term in \\eqref{x2phi}. This does not spoil the $\\epsilon^{2}$ suppression of the cubic action. An explicit demonstration of this is postponed to the following section, where a similar second order field redefinition arises.\n\n\n\n\n\n\n\\subsection{Constant inflaton gauge}\nWe now rederive the cubic action in constant inflaton gauge \\eqref{cfgauge}, where the degree of freedom is $\\zeta=\\delta\\!\\phican$, the perturbation of the scale factor. This gauge will be advantageous for our computation of the three-point function in section \\ref{3ptsec}, because once $\\delta\\!\\phican$ leaves the horizon it is known to be constant. We can just evaluate the three-point function of $\\delta\\!\\phican$ up to the time of horizon crossing, avoiding the transition from inflation to reheating, where the slow roll parameters are no longer small.\n\nAgain expanding the action \\eqref{Lcovpre} plus \\eqref{SX} to cubic order, substituting the first order lapse and shift \\eqref{ntnx}, imposing the gauge conditions, and integrating by parts, we get\n\\begin{equation}\\label{S3cf}\n{\\mathcal{S}}_{3} = \\frac{2}{\\gamma^{2}}\\int \n \\epsilon\\, \\Bigl\\{ \\delta\\!\\phican\\phipert'^{2} + \\delta\\!\\phican(\\partial_{x}\\delta\\!\\phican)^{2} + \\frac{1}{aH}\\delta\\!\\phican'(\\partial\\delta\\!\\phican)^{2}\n- 2\\psi\\delta\\!\\phican'\\partial_{x}\\delta\\!\\phican \\Bigr\\}\n\\end{equation}\nThe analogous expression in 4d is equation (3.9) of \\cite{Maldacena:2002vr}. In constant inflaton gauge, equation \\eqref{psix} for $\\psi$ reduces to\n\\begin{equation}\\label{psicf}\n\\partial_{x}\\psi = \\epsilon\\delta\\!\\phican'\n\\end{equation}\nAt this point, only the $\\psi$ term in \\eqref{S3cf} is manifestly suppressed by $\\epsilon^{2}$ --- the other terms are suppressed only by $\\epsilon$. Substituting $\\delta\\!\\phican_{\\rm c.i.}=-\\delta\\!\\inflaton_{\\rm s.f.}\/z+\\cdots$ \\eqref{phi2x} would make matters worse, but there is a related field redefinition which happens to be useful. We know from the previous section that the relevant terms in the cubic action \\eqref{S3sf} in spatially flat gauge have a common order of slow roll suppression. It turns out that the action \\eqref{S3cf} adopts a similar form in terms of a new variable $\\overbar{\\delta\\!\\phican}$ satisfying \\cite{Maldacena:2002vr}\n\\begin{equation}\\label{phindef}\n\\overbar{\\delta\\!\\phican}_{\\rm c.i.} = -\\delta\\!\\inflaton_{\\rm s.f.} \/z\n\\end{equation}\nexactly, to all orders in perturbations. Compare this with our previous statement \\eqref{phi2x} relating variables in the two gauges. Using \\eqref{phindef} to eliminate $\\delta\\!\\inflaton_{\\rm s.f.}$ from \\eqref{phi2x}, the relation between $\\delta\\!\\phican$ and $\\overbar{\\delta\\!\\phican}$ to second order is\n\\begin{equation}\\label{phi2phin}\n\\delta\\!\\phican = \\overbar{\\delta\\!\\phican} + \\frac{\\eta}{4}\\,\\overbar{\\delta\\!\\phican}^{2} + \\frac{1}{aH}\\overbar{\\delta\\!\\phican}\\phin'\n\\end{equation}\nThis may be compared with equation (3.10) of \\cite{Maldacena:2002vr}. At first order $\\delta\\!\\phican=\\overbar{\\delta\\!\\phican}$, so the change of variable $\\delta\\!\\phican\\to\\overbar{\\delta\\!\\phican}$ does not alter the quadratic action \\eqref{S2cf}. It does however give rise to additional third order terms:\n\\begin{equation}\n{\\mathcal{S}}_{2}[\\delta\\!\\phican] = {\\mathcal{S}}_{2}[\\overbar{\\delta\\!\\phican}] + \n\\frac{4}{\\gamma^{2}}\\int \\epsilon\\, \\Bigl(  \\frac{\\eta}{4}\\,\\overbar{\\delta\\!\\phican}^{2} + \\frac{1}{aH}\\overbar{\\delta\\!\\phican}\\phin' \\Bigr) \\Bigl( \\partial^{2}\\overbar{\\delta\\!\\phican} - aH\\eta\\overbar{\\delta\\!\\phican}' \\Bigr)\n\\end{equation}\nwhere the last bracket is proportional to the equation of motion $\\delta{\\mathcal{S}}_{2}\/\\delta\\overbar{\\delta\\!\\phican}$. Adding these terms to \\eqref{S3cf} and integrating by parts, the $O(\\epsilon)$ terms in \\eqref{S3cf} are canceled, while new $O(\\epsilon^{2})$ terms are generated. The result is\n\\begin{equation}\\label{S3cffinal}\n{\\mathcal{S}}_{3} = \\frac{2}{\\gamma^{2}}\\int \\epsilon^{2}\\Bigl( \\overbar{\\delta\\!\\phican}\\phin'^{2} + \\overbar{\\delta\\!\\phican}(\\partial_{x}\\overbar{\\delta\\!\\phican})^{2} -2\\overbar{\\delta\\!\\phican}'\\partial_{x}\\overbar{\\delta\\!\\phican}\\partial_{x}^{-1}\\overbar{\\delta\\!\\phican}' \\Bigr)  - \\frac{1}{3\\gamma^{2}}\\int (\\epsilon\\eta')' \\,\\overbar{\\delta\\!\\phican}^{3}\n\\end{equation}\nThis cubic action is correct to all orders in slow roll. It is of a very similar form to the spatially flat action \\eqref{S3sf} --- in fact, the change of variable \\eqref{phindef} shows they are identical up to the non-derivative terms (which are further slow roll suppressed). The point of using the variable $\\overbar{\\var}$ is that each order in the perturbative action is suppressed by an additional factor of $\\epsilon$. The only disadvantage is that $\\overbar{\\var}$ is not constant outside the horizon. \n\nIt is worth noting that the action \\eqref{S3cffinal} could be simplified even further by another change of variable \\cite{Seery:2005wm}. The idea is to integrate by parts, beginning with $\\partial_{x}^{-1}\\overbar{\\delta\\!\\phican}'$ to get $\\partial_{x}^{-2}\\overbar{\\delta\\!\\phican}'$. Whenever second derivatives $\\partial_{x}^{2}\\overbar{\\delta\\!\\phican}$ or $\\overbar{\\delta\\!\\phican}''$ arise, they are eliminated in favor of $\\delta{\\mathcal{S}}_{2}\/\\delta\\overbar{\\delta\\!\\phican}$, plus other terms occurring in the equation of motion. Continuing in this way, the leading cubic action may be expressed as a single vertex plus terms proportional to $\\delta{\\mathcal{S}}_{2}\/\\delta\\overbar{\\delta\\!\\phican}$, which are then removed by a field redefinition. Rather than make this additional field redefinition, we choose to work with the action \\eqref{S3cffinal}.\n\n\n\n\n\n\\section{Three-point function}\\label{3ptsec}\nWe now proceed with the computation of the tree-level three-point function for the curvature perturbation $\\zeta=\\delta\\!\\phican$. The constancy of $\\delta\\!\\phican$ outside the horizon makes it suitable for computing observable signatures at the end of inflation. Following \\cite{Maldacena:2002vr}, the strategy is to begin with the variable $\\overbar{\\delta\\!\\phican}$, in terms of which there is a perturbation expansion governed by the slow roll parameter $\\epsilon$. The three-point function of $\\overbar{\\var}$ is evaluated in section \\ref{phibarsec} using the leading order terms in the cubic action \\eqref{S3cffinal}. After the modes in question exit the horizon, we switch to the variable $\\delta\\!\\phican$, which is then constant. This allows us to infer the three-point function for all subsequent times. The change of variable $\\overbar{\\var}\\to\\delta\\!\\phican$ induces correction terms in the three-point function, which are given in section \\ref{redefsec}. Finally, in section \\ref{consistencysec} we discuss symmetry constraints on the three-point function.\n\nWorking in momentum space, the Fourier transform of $\\delta\\!\\phican(\\ct,\\mathbf{x})$ is written in terms of the mode functions defined in \\eqref{vexp} as\n\\begin{equation}\n\\delta\\!\\phican(\\ct,\\mathbf{k}) =  a(\\mathbf{k})\\delta\\!\\phican_{k}(\\ct) + a^{\\dagger}(-\\mathbf{k})\\delta\\!\\phican_{k}^{*}(\\ct) \n\\end{equation}\nThe spatial momentum $\\mathbf{k}$ in 1+1d is just a real number of either sign; the magnitude of $\\mathbf{k}$ is denoted by $k$. In the interaction picture, the contraction of Fourier components is\n\\begin{equation}\\label{cont}\n\\bigl\\langle0\\bigr| \\delta\\!\\phican(\\ct,\\mathbf{k})\\delta\\!\\phican(\\ct',\\mathbf{p}) \\bigl|0\\bigr\\rangle = \\delta\\!\\phican_{k}(\\ct) \\delta\\!\\phican_{p}^{*}(\\ct') (2\\pi)\\delta(\\mathbf{k}+\\mathbf{p})\n\\end{equation}\nThe Bunch-Davies mode function $\\delta\\!\\phican_{k}(\\ct)$ in the slow roll approximation is equal to the Hankel function \\eqref{vksol}, scaled by $-1\/z$. To evaluate the three-point function to lowest order in slow roll, we need only the lowest order expression for the mode function, equation \\eqref{zetak2d}. The three-point function to be computed is\n\\begin{equation}\n\\lim_{\\ct\\to 0^{-}} \\big\\langle \\delta\\!\\phican(\\ct,\\mathbf{p}_{1})\\delta\\!\\phican(\\ct,\\mathbf{p}_{2})\\delta\\!\\phican(\\ct,\\mathbf{p}_{3}) \\big\\rangle_{\\rm tree}\n\\end{equation}\nThe limit $\\ct\\to 0^{-}$ takes us to the end of inflation; in the following equations we simply specialize to $\\ct=0$. \n\n\n\n\n\\subsection[Three-point function of phi-bar]{Three-point function of $\\overbar{\\var}$}\\label{phibarsec}\nThe three-point function of $\\overbar{\\var}$ is evaluated using the leading terms in the cubic action \\eqref{S3cffinal}. In the in-in formalism, it takes the form of an expectation value along a closed time contour extending from $\\ct=-\\infty$ to $\\ct=0$, and back again \\cite{LimNotes}:\n\\begin{equation}\\label{3pt}\n\\begin{aligned}\n\\big\\langle \\overbar{\\var}(0,\\mathbf{p}_{1}) & \\overbar{\\var}(0,\\mathbf{p}_{2})\\overbar{\\var}(0,\\mathbf{p}_{3}) \\big\\rangle_{\\rm tree} \\\\\n&=\n\\bigl\\langle\\Omega\\bigr| U_{\\rm int}(0,-\\infty)^{-1} \\overbar{\\var}(0,\\mathbf{p}_{1})\\overbar{\\var}(0,\\mathbf{p}_{2})\\overbar{\\var}(0,\\mathbf{p}_{3}) U_{\\rm int}(0,-\\infty) \\bigl|\\Omega\\bigr\\rangle_{\\rm tree}\n\\\\\n&=\\lim_{T\\to-\\infty(1-i\\varepsilon)} 2\\,\\mathrm{Re}\\, \\bigl\\langle0\\bigr|\\, \\overbar{\\var}(0,\\mathbf{p}_{1})\\overbar{\\var}(0,\\mathbf{p}_{2})\\overbar{\\var}(0,\\mathbf{p}_{3}) \\Bigl[ -i\\int_{T}^{0}d\\ct H_{3}(\\ct) \\Bigr] \\bigl|0\\bigr\\rangle + \\cdots \n\\end{aligned}\n\\end{equation}\nAll fields are in the interaction picture, with time dependence determined by the free Hamiltonian. The remaining time dependence is carried by the interacting evolution operator $U_{\\rm int}(\\ct_{\\rm f},\\ct_{\\rm i})$, which is treated perturbatively. The free vacuum $|0\\rangle$ is not an eigenstate of the interacting Hamiltonian, but rather an infinite sum of eigenstates. Of these, the lowest eigenvalue belongs to the interacting vacuum $|\\Omega\\rangle$, which is projected out by an infinitesimal rotation of the time contour, moving the lower terminal from $-\\infty$ to $-\\infty(1-i\\varepsilon)$. Here $\\varepsilon$ is an infinitesimal constant, not to be confused with the slow roll parameter $\\epsilon$.\n\nAt cubic order in $\\overbar{\\var}$ the interacting Hamiltonian is simply $H_{3}(\\pi,\\overbar{\\var})=-L_{3}(\\overbar{\\var}',\\overbar{\\var})$, for the following reason. The conjugate momentum is of the form $\\pi(\\overbar{\\var})=\\epsilon\\overbar{\\var}'+\\mathcal{O}(\\epsilon^{2}\\overbar{\\var}^{2})$, where the nonlinear terms come from the interacting Lagrangian. The Hamiltonian is constructed as the Legendre transform\n\\begin{equation}\\label{Legtrans}\nH = \\pi\\,\\overbar{\\var}'(\\pi)-L_{2}-L_{3}+\\cdots\n\\end{equation}\nThe cubic contributions to the first two terms cancel, and the remaining quadratic terms are identified with the free Hamiltonian. The only other cubic contribution to \\eqref{Legtrans} comes from $L_{3}$, which gives $H_{3}(\\pi,\\overbar{\\var})=-L_{3}(\\overbar{\\var}',\\overbar{\\var})+\\mathcal{O}(\\epsilon^{3}\\overbar{\\var}^{4})$. Hence the cubic Hamiltonian at lowest order in slow roll is\n\\begin{equation}\\label{H3}\nH_{3}(\\ct) = - \\frac{2}{\\gamma^{2}}\\,\\epsilon(\\ct)^{2} \\int dx\\, \\Bigl( \\overbar{\\delta\\!\\phican}\\phin'^{2} + \\overbar{\\delta\\!\\phican}(\\partial_{x}\\overbar{\\delta\\!\\phican})^{2} -2\\overbar{\\delta\\!\\phican}'\\partial_{x}\\overbar{\\delta\\!\\phican}\\partial_{x}^{-1}\\overbar{\\delta\\!\\phican}' \\Bigr) \n\\end{equation}\n\nThe contribution of the first term in \\eqref{H3} (which we denote by $H_{3}^{(1)}$) to the three-point function \\eqref{3pt} will now be evaluated explicitly. Going to momentum space and using the contraction \\eqref{cont}, one finds the expectation value\n\\begin{multline}\\label{3ptstep}\n\\bigl\\langle0\\bigr| \\overbar{\\var}(0,\\mathbf{p}_{1})\\overbar{\\var}(0,\\mathbf{p}_{2})\\overbar{\\var}(0,\\mathbf{p}_{3}) H_{3}^{(1)}(\\ct) \\bigl|0\\bigr\\rangle \n= -\\frac{8\\pi}{\\gamma^{2}}\\, \\epsilon(\\ct)^{2}\\, \n\\delta\\!\\phican_{p_{1}}\\delta\\!\\phican_{p_{2}}\\delta\\!\\phican_{p_{3}}(0)\\, \\delta\\!\\phican_{p_{1}}^{*}\\delta\\!\\phican_{p_{2}}^{*}\\!\\!'\\, \\delta\\!\\phican_{p_{3}}^{*}\\!\\!'(\\ct)\\, \\delta\\(\\sum\\mathbf{p}_{i}\\)   \\\\\n+ (1\\leftrightarrow2) + (1\\leftrightarrow3)\n\\end{multline}\nNote that the mode functions $\\delta\\!\\phican_{k}$ and $\\overbar{\\var}_{k}$ are identical, because they have the same quadratic action ($\\delta\\!\\phican=\\overbar{\\var}$ at lowest order). The momentum-conserving delta function will be implicit in the following expressions for correlation functions. The contribution of $H_{3}^{(1)}$ to the three-point function \\eqref{3pt} is then\n\\begin{equation}\\label{3ptstep2}\n\\big\\langle \\overbar{\\var}(0,\\mathbf{p}_{1})\\overbar{\\var}(0,\\mathbf{p}_{2})\\overbar{\\var}(0,\\mathbf{p}_{3}) \\big\\rangle ' _{\\rm tree} \\supset\n\\frac{16\\pi}{\\gamma^{2}}\\, \\mathrm{Re}\\[ i\\, \\delta\\!\\phican_{p_{1}}\\delta\\!\\phican_{p_{2}}\\delta\\!\\phican_{p_{3}}(0)  \\int_{-\\infty(1-i\\varepsilon)}^{0}d\\ct\\, \\epsilon(\\ct)^{2} \\delta\\!\\phican_{p_{1}}^{*}\\delta\\!\\phican_{p_{2}}^{*}\\!\\!'\\, \\delta\\!\\phican_{p_{3}}^{*}\\!\\!'(\\ct) \\]\n\\end{equation}\nThis should be evaluated to lowest order in slow roll, because we are already neglecting higher terms by restricting to the cubic Hamiltonian. We use the mode functions $\\delta\\!\\phican_{k}=\\zeta_{k}$ given in \\eqref{zetak2d}:\n\\begin{equation}\n\\delta\\!\\phican_{k}(\\ct) = - \\frac{1}{z(\\ct)} \\frac{e^{-ik\\ct}}{\\sqrt{2k}}\n\\end{equation}\nUp to the moment of horizon crossing, this is an excellent approximation to the full Hankel function. At this level of approximation, $z(\\ct)\\propto\\epsilon(\\ct)^{1\/2}$ is a constant, which we take to be $z(\\ct_{*})$, where $\\ct_{*}$ is the moment of horizon crossing for the shortest wavelength mode (the last mode to leave the horizon). Strictly speaking, the upper terminal of the time integral in \\eqref{3ptstep2} should be equal to $\\ct_{*}$, but the additional contribution from the interval $\\ct\\in(\\ct_{*},0)$ is dwarfed by the contribution before horizon crossing, $\\ct<\\ct_{*}$. Hence we can approximate the time integral by\n\\begin{equation}\n\\int_{-\\infty(1-i\\varepsilon)}^{0}d\\ct\\, e^{iP\\ct} = \\frac{1}{iP}\n\\end{equation}\nwhere $P$ denotes the sum of the energies (magnitudes) carried by the three momenta,\n\\begin{equation}\nP=\\sum p_{i}\n\\end{equation}\nThe contribution of $H_{3}^{(1)}$ to the three-point function \\eqref{3pt} is then\n\\begin{equation}\\label{H31contrib}\n\\big\\langle \\overbar{\\var}(0,\\mathbf{p}_{1})\\overbar{\\var}(0,\\mathbf{p}_{2})\\overbar{\\var}(0,\\mathbf{p}_{3}) \\big\\rangle  _{\\rm tree} \\supset\n-\\frac{\\pi\\gamma^{4}}{32\\epsilon_{*}}\\, \\frac{1}{p_{1}p_{2}p_{3}}\\, \\frac{1}{2P} \\sum_{i\\neq j}p_{i}p_{j}\n\\end{equation}\nRecall that the sum of the momenta, $\\sum\\mathbf{p}_{i}$, is constrained to vanish by an implicit momentum-conserving delta function.\n\nThe remaining terms in the Hamiltonian \\eqref{H3} make similar contributions, differing only by the form of the final momentum sum in \\eqref{H31contrib}. The contribution of the second term in \\eqref{H3} is given by replacing in \\eqref{H31contrib}\n\\begin{equation}\n\\sum_{i\\neq j}p_{i}p_{j} \\to \\sum_{i\\neq j}\\mathbf{p}_{i}\\mathbf{p}_{j} = -\\sum_{i}p_{i}^{2} \n\\end{equation}\nwhere momentum conservation was used. The contribution of the final term in \\eqref{H3} is given by replacing\n\\begin{equation}\\label{sum3}\n\\sum_{i\\neq j}p_{i}p_{j} \\to - 2\\Bigl( p_{1}\\mathbf{p}_{2}\\, \\mathrm{sign}(\\mathbf{p}_{3}) +\\text{perms} \\Bigr) \n\\end{equation}\nwhere all six permutations of the momenta are to be included. Again using momentum conservation, this is written in terms of the magnitudes as\n\\begin{equation}\n- 2\\Bigl( p_{1}\\mathbf{p}_{2}\\, \\mathrm{sign}(\\mathbf{p}_{3}) +\\text{perms} \\Bigr)\n= \\frac{p_{1}^{2}}{p_{2}p_{3}}\\(p_{1}^{2}-p_{2}^{2}-p_{3}^{2}\\) + (1\\lr2) + (1\\lr3)\n\\end{equation}\nThe complete three-point function of $\\overbar{\\var}$ is then\n\\begin{equation}\\label{varn3ptfinal}\n\\big\\langle \\overbar{\\var}(0,\\mathbf{p}_{1})\\overbar{\\var}(0,\\mathbf{p}_{2})\\overbar{\\var}(0,\\mathbf{p}_{3}) \\big\\rangle  _{\\rm tree} =\n-\\frac{\\pi\\gamma^{4}}{64\\epsilon_{*}}\\,  F(\\mathbf{p}_{1},\\mathbf{p}_{2},\\mathbf{p}_{3})\n\\end{equation}\nwhere the shape function is\n\\begin{equation}\\label{shape1}\nF(\\mathbf{p}_{1},\\mathbf{p}_{2},\\mathbf{p}_{3}) = \\frac{1}{p_{1}p_{2}p_{3}}\\,\\frac{1}{P} \\biggl[ -\\sum_{i}p_{i}^{2}  + \\sum_{i\\neq j}p_{i}p_{j} \n+ \\frac{p_{1}^{2}}{p_{2}p_{3}}\\(p_{1}^{2}-p_{2}^{2}-p_{3}^{2}\\) + (1\\lr2) + (1\\lr3)\n \\biggr]\n\\end{equation}\nThis may be compared with the local shape function generated by single field inflation in 4d, given explicitly in equation (223) of \\cite{LimNotes}. The first two terms in \\eqref{shape1} are 2d analogs of the first two terms proportional to $\\epsilon$ in \\cite{LimNotes}. The final term is different --- at a superficial level, this stems from the appearance in \\eqref{sum3} of $\\mathrm{sign}(\\mathbf{p}_{i})$. Instead of further massaging \\eqref{shape1} to more closely resemble the 4d result, we prefer to express it in a very compact form, taking advantage of the simplified kinematics of spatial momenta in 1+1d. Imposing momentum conservation explicitly and assuming various kinematic configurations, one finds in every possible case that\n\\begin{equation}\nF(\\mathbf{p}_{1},\\mathbf{p}_{2},\\mathbf{p}_{3})  = F(\\mathbf{p}_{1},\\mathbf{p}_{2},-\\mathbf{p}_{1}-\\mathbf{p}_{2}) = 16\/P^{2}\n\\end{equation}\nHence the three-point function of $\\overbar{\\var}$ has a completely flat shape, depending only on the total energy carried by the three momenta. The change of variable $\\overbar{\\var}\\to\\delta\\!\\phican$ will introduce dependence on the shape of the spatial momentum ``triangle.''\n\n \n\n\n\n\n\\subsection{Field redefinition}\\label{redefsec}\n\nOnce the modes in question are outside the horizon we switch to the variable $\\delta\\!\\phican$, which is then constant to all orders in slow roll (and all orders in perturbations). The inverse of the second order relation \\eqref{phi2phin} between $\\delta\\!\\phican$ and $\\overbar{\\delta\\!\\phican}$ is\n\\begin{equation}\\label{phin2phi}\n\\overbar{\\var} = \\delta\\!\\phican - \\frac{\\eta}{4}\\delta\\!\\phican^{2} - \\frac{1}{H}\\delta\\!\\phican\\dot\\delta\\!\\phican\n\\end{equation}\nRecall that a dot denotes the coordinate time derivative $\\partial_{t}$. The final term in \\eqref{phin2phi} vanishes outside the horizon, so that $\\overbar{\\var}$ has only a weak time dependence inherited from $\\eta$:\n\\begin{equation}\n\\dot\\overbar{\\delta\\!\\phican} = -\\frac{1}{4}\\dot\\eta\\,\\delta\\!\\phican^{2} = -\\frac{1}{4}H\\eta\\kappa\\, \\delta\\!\\phican^{2}\n\\qquad\\text{(superhorizon)}\n\\end{equation}\nwhere $\\kappa$ was defined in \\eqref{SRparams}. Returning to the original relation \\eqref{phi2phin} between $\\delta\\!\\phican$ and $\\overbar{\\delta\\!\\phican}$, the $\\dot\\overbar{\\var}$ term is of higher order in slow roll and may be dropped, leaving\n\\begin{equation}\\label{phi2phinsuper}\n\\delta\\!\\phican = \\overbar{\\delta\\!\\phican} + \\frac{\\eta}{4}\\,\\overbar{\\delta\\!\\phican}^{2}\n\\qquad \\text{(superhorizon)}\n\\end{equation}\nThis leads to the following relation between three-point functions in momentum space:\n\\begin{multline}\\label{3ptcorrec1}\n\\big\\langle \\delta\\!\\phican(\\mathbf{p}_{1})\\delta\\!\\phican(\\mathbf{p}_{2})\\delta\\!\\phican(\\mathbf{p}_{3}) \\big\\rangle \n= \\big\\langle \\overbar{\\var}(\\mathbf{p}_{1})\\overbar{\\var}(\\mathbf{p}_{2})\\overbar{\\var}(\\mathbf{p}_{3}) \\big\\rangle \\\\\n+ \\frac{\\eta}{4} \\int\\frac{d\\mathbf{k}}{2\\pi} \\Bigl[ \\big\\langle\\overbar{\\var}(\\mathbf{p}_{1}-\\mathbf{k})\\delta\\!\\phican(\\mathbf{p}_{2})\\big\\rangle \\big\\langle\\overbar{\\var}(\\mathbf{k})\\delta\\!\\phican(\\mathbf{p}_{3})\\big\\rangle +\\text{perms} \\Bigr] + \\cdots\n\\end{multline}\nThe time arguments of the fields have been suppressed for brevity, but are of course coincident at some time after horizon crossing. The ellipsis represents terms of higher order in slow roll, corresponding to more than one insertion of \\eqref{phi2phinsuper}. To evaluate the corrections \\eqref{3ptcorrec1} to lowest order in slow roll, note that $\\langle\\overbar{\\var}\\delta\\!\\phican\\rangle=\\langle\\delta\\!\\phican\\var\\rangle$ in the free theory, because $\\delta\\!\\phican$ and $\\overbar{\\var}$ share the same quadratic action. Then, for instance, the first correction term on the right hand side of \\eqref{3ptcorrec1} is\n\\begin{equation}\n\\begin{aligned}\n\\int\\frac{d\\mathbf{k}}{2\\pi}  \\big\\langle\\overbar{\\var}(\\mathbf{p}_{1}-\\mathbf{k})\\delta\\!\\phican(\\mathbf{p}_{2})\\big\\rangle \\big\\langle\\overbar{\\var}(\\mathbf{k})\\delta\\!\\phican(\\mathbf{p}_{3})\\big\\rangle\n &= \\delta\\(\\sum \\mathbf{p}_{i}\\) (2\\pi) |\\delta\\!\\phican_{p_{2}}|^{2} |\\delta\\!\\phican_{p_{3}}|^{2} (\\ct_{*}) \\\\\n& =  \\delta\\(\\sum \\mathbf{p}_{i}\\) \\frac{\\pi\\gamma^{4}}{32\\epsilon_{*}^{2}}\\, \\frac{1}{p_{2}p_{3}}\n\\end{aligned}\n\\end{equation}\nThe total relation \\eqref{3ptcorrec1} between three-point functions becomes\n\\begin{equation}\\label{3ptcorrec}\n\\big\\langle \\delta\\!\\phican(\\mathbf{p}_{1})\\delta\\!\\phican(\\mathbf{p}_{2})\\delta\\!\\phican(\\mathbf{p}_{3}) \\big\\rangle \n= \\big\\langle \\overbar{\\var}(\\mathbf{p}_{1})\\overbar{\\var}(\\mathbf{p}_{2})\\overbar{\\var}(\\mathbf{p}_{3}) \\big\\rangle \n-\\frac{\\pi\\gamma^{4}}{64}\\, \\frac{\\eta_{*}}{\\epsilon_{*}^{2}}\\, \\frac{P}{p_{1}p_{2}p_{3}}\n\\end{equation}\nThis is the precise analog of the $\\eta$-dependent term in the 4d shape function, equation (223) of \\cite{LimNotes}. Substituting our result \\eqref{varn3ptfinal} for $\\langle\\overbar{\\var}\\varn\\overbar{\\var}\\rangle$, we find the three-point function for $\\zeta=\\delta\\!\\phican$ at the end of inflation is\n\\begin{equation}\\label{phi3pt}\n\\begin{aligned}\n\\big\\langle \\delta\\!\\phican(0,\\mathbf{p}_{1})\\delta\\!\\phican(0,\\mathbf{p}_{2})\\delta\\!\\phican(0,\\mathbf{p}_{3}) \\big\\rangle _{\\rm tree} &= \n-\\frac{\\pi\\gamma^{4}}{64\\epsilon_{*}}\\biggl(  F(\\mathbf{p}_{1},\\mathbf{p}_{2},\\mathbf{p}_{3}) \n+ \\frac{\\eta_{*}}{\\epsilon_{*}}\\, \\frac{P}{p_{1}p_{2}p_{3}}   \\biggr)    \\\\\n&= -\\frac{\\pi\\gamma^{4}}{64\\epsilon_{*}}\\biggl(  \\frac{16}{P^{2}} \n+ \\frac{\\eta_{*}}{\\epsilon_{*}}\\, \\frac{P}{p_{1}p_{2}p_{3}}   \\biggr) \n\\end{aligned}\n\\end{equation}\nThis is a local shape, which diverges in the squeezed limit (where one of the momenta tends to zero). Hence the greatest non-Gaussianity is associated to squeezed configurations. For a given value of the total energy $P$, the minimum magnitude of \\eqref{phi3pt} (least non-Gaussianity) occurs when two of the momenta are identical, for instance $\\mathbf{p}_{1}=\\mathbf{p}_{2}=-\\mathbf{p}_{3}\/2$. This corresponds to a folded isosceles triangle, which is the closest one can get to an equilateral triangle in one spatial dimension.\n\n\n\n\n\n\n\n\\subsection{Consistency relation}\\label{consistencysec}\n\n\\subsubsection*{Residual diffeomorphisms and global shift symmetry}\nRecently much attention has been devoted to Ward identities for cosmological correlation functions \\cite{Hinterbichler:2013dpa,Berezhiani:2013ewa,Pimentel:2013gza,Creminelli:2012qr}. These identities may be understood to arise from residual diffeomorphism symmetry. Our choice of constant inflaton gauge \\eqref{cfgauge} completely fixes diffeomorphisms which vanish at spatial infinity. Let us define ``physical'' metric and matter fluctuations as those vanishing at spatial infinity. Residual gauge symmetries are the subset of diffeomorphisms not vanishing at infinity, which preserve the gauge choice and the boundary conditions (physicality) of fluctuations. In 4d, the residual symmetries in constant inflaton gauge were classified in \\cite{Hinterbichler:2013dpa}. Following the same arguments in 2d, we begin by looking for diffeomorphisms which preserve the gauge choice \\eqref{cfgauge}:\n\\begin{equation}\\label{cfgauge2}\n\\delta\\!\\inflaton = 0 = \\delta\\!\\Laux + \\delta\\!\\phican\n\\end{equation}\nConsulting the gauge transformations \\eqref{fieldtransf} and \\eqref{infgaugetrans}, we see that a time reparametrization $\\xi^{\\ct}$ breaks the inflaton gauge condition, due to the time-dependent inflaton background. The gauge transformation of $\\delta\\!\\Laux+\\delta\\!\\phican$ is\n\\begin{equation}\\label{chiplusphitrans}\n(\\delta\\!\\Laux+\\delta\\!\\phican) \\to (\\delta\\!\\Laux+\\delta\\!\\phican) + \\partial_{x}^{2}\\xi\n\\end{equation}\nRecall that the spatial reparametrization \\eqref{coordtransf4d} is $x\\to x+\\partial_{x}\\xi$, so there is no meaning to $x$-independent reparametrizations $\\xi$. The gauge condition $\\delta\\!\\Laux+\\delta\\!\\phican=0$ is preserved by spatial translations $\\xi\\sim x$, but these don't give rise to any Ward identities, as our correlation functions already conserve spatial momentum. In 3+1d there are transverse spatial reparametrizations, which preserve the gauge choice, but there is no room for these in 1+1d. This leaves an arbitrary time dependence of $\\xi$ as the only freedom. In summary, the gauge-preserving diffeomorphisms are of the form\n\\begin{equation}\\label{gaugepreserving}\n\\xi^{\\ct}(\\ct,x)=0~,\\qquad \\xi(\\ct,x)=x\\, \\overline{\\xi}(\\ct)\n\\end{equation}\n\nNow let's consider the physicality condition, that the residual gauge symmetry must preserve the vanishing of metric fluctuations at spatial infinity. Diffeomorphisms of the form \\eqref{gaugepreserving} automatically preserve the physicality of $\\delta\\!\\Laux$, $\\delta\\!\\phican$, and $\\delta\\!\\inflaton$. We still have to check the physicality of the lapse and shift fluctuations $\\delta\\! N^{\\ct}$ and $\\delta\\! N^{x}$, which were adjusted to solve the perturbed constraints \\eqref{constraintpert}. The constraints are gauge invariant (by the closure of the constraint algebra), but in solving them we must also choose a boundary condition: the vanishing at spatial infinity of the function $\\psi$, which in constant inflaton gauge is given by \\eqref{psicf}. The variation of $\\psi$ under a gauge-preserving diffeomorphism \\eqref{gaugepreserving} is\n\\begin{equation}\n\\psi \\to \\psi + \\epsilon\\, \\overline{\\xi}\\, '\n\\end{equation}\nHence $\\overline{\\xi}\\, '$ must vanish at spatial infinity; but $\\overline{\\xi}(\\ct)$ is $x$-independent, so it must be a constant. The only diffeomorphism which preserves constant inflaton gauge and the physicality of fluctuations is a spatial translation, and there are no Ward identities!\n\nThis cannot be the whole story, however, because the three-point function \\eqref{phi3pt} does in fact obey Maldacena's consistency relation\\footnote{The sign of this relation differs in some parts of the literature, due to a different convention for the sign of the scalar curvature perturbation.} \\cite{Maldacena:2002vr}\n\\begin{equation}\\label{Maldacena2d}\n\\lim_{q\\to0}\\, \\frac{\\langle\\delta\\!\\phican(\\mathbf{q})\\delta\\!\\phican(\\mathbf{p})\\delta\\!\\phican(-\\mathbf{p}-\\mathbf{q})\\rangle}{P_{\\delta\\!\\phican}(\\mathbf{q})} = 2\\pi(n_{s}-1)_{*}\\, P_{\\delta\\!\\phican}(\\mathbf{p})\n\\end{equation}\nThe sole contribution to the left side in the limit comes from the second term in \\eqref{phi3pt}, which translates between the $\\delta\\!\\phican$ and $\\overbar{\\delta\\!\\phican}$ three-point functions. This term is proportional to $\\eta$, as is the spectral tilt \\eqref{tilt}. Maldacena's consistency relation in 4d is known to arise from a residual gauge symmetry under spatial dilations (see for example \\cite{Hinterbichler:2013dpa}). In 2d a spatial dilation is given by $\\xi\\sim x^{2}$, under which $\\delta\\!\\Laux+\\delta\\!\\phican$ shifts by a constant, according to \\eqref{chiplusphitrans}. The key point is that the gauge condition $\\delta\\!\\Laux+\\delta\\!\\phican=0$ may be restored using the global shift symmetry of $\\Laux$ discussed in section \\ref{Liouvillesec}, and that this is only possible because $\\delta\\!\\Laux+\\delta\\!\\phican$ shifts by a constant under spatial dilations. The shift symmetry of $\\Laux$ arises from the existence of the topological Euler characteristic, and is special to 2d. It is a physical symmetry rather than a gauge symmetry. Spatial diffeomorphisms more complicated than a dilation disturb the gauge condition in ways not related by any symmetry.\n\nA similar shuffling of gauge transformations occurs in 4d, where there is an ambiguity in the parametrization \\eqref{ds4d} of scalar perturbations of the space-space metric: the contribution proportional to $\\delta_{ij}$ may be shuffled between $\\Psi$ and $E$. The ambiguity is resolved by requiring $E$ to be harmonic: $\\vec\\partial^{2}E=0$. Part of the constant inflaton gauge choice is the condition $E=0$, although strictly speaking this should be $E_{,ij}=0$, because only $E_{,ij}$ appears in the metric \\eqref{ds4d}. A typical spatial diffeomorphism $x^{i}\\to x^{i}+\\partial^{i}\\xi$, which sends $E\\to E+\\xi$, will break the gauge condition by a term $\\xi_{,ij}$. If this is proportional to $\\delta_{ij}$ then it may be absorbed in $\\Psi$, so residual diffeomorphisms are required to preserve $E_{,ij}=0$ only up to such terms. Indeed, the spatial dilation $\\xi=(\\lambda\/2) x_{i}x^{i}$ --- which is the source of Maldacena's consistency relation --- falls into this category, inducing $E_{,ij}=\\lambda\\delta_{ij}$. In 2d there is no ambiguity in the parametrization of space-space metric fluctuations, but when $\\partial_{x}^{2}\\xi$ is constant, the global shift symmetry of extended Liouville theory plays a similar role.\n\nFor the sake of completeness, let us mention how the consistency relation \\eqref{Maldacena2d} associated to spatial dilations is arrived at in 2d. It is the leading term in a master consistency relation, similar to those motivated in 4d by examining gauge transformations of the path integral for the quantum effective action \\cite{Berezhiani:2013ewa}. Following an identical argument in 2d, one finds the master consistency relation\n\\begin{equation}\\label{master2d}\n\\frac{1}{2\\pi}\\, \\frac{\\langle\\delta\\!\\phican(\\mathbf{q})\\delta\\!\\phican(\\mathbf{p})\\delta\\!\\phican(-\\mathbf{p}-\\mathbf{q})\\rangle}{P_{\\delta\\!\\phican}(\\mathbf{q})} = -P_{\\delta\\!\\phican}(\\mathbf{p}+\\mathbf{q}) - \\mathbf{p} \\[\\frac{P_{\\delta\\!\\phican}(\\mathbf{p}+\\mathbf{q})-P_{\\delta\\!\\phican}(\\mathbf{p})}{\\mathbf{q}}\\]\n\\end{equation}\nRecall that $\\mathbf{p}$ and $\\mathbf{q}$ are just real numbers in 1+1d. The analogous expression in 3+1d (equation (4.2) of \\cite{Berezhiani:2013ewa}) is proportional to the momentum $q^{i}$ with free spatial index. Maldacena's relation \\eqref{Maldacena2d} is the $q^{0}$ term in the Taylor expansion of \\eqref{master2d} about $\\mathbf{q}=0$, and is associated to spatial dilations. Higher terms in the Taylor expansion are associated to more complicated spatial diffeomorphisms. It was demonstrated above that none of these preserve the gauge condition in 2d, so they are not symmetries of the three-point funcion \\eqref{phi3pt}. If it weren't for this, \\eqref{master2d} would completely determine the three-point function to all orders in $q$. This is in contrast to 3+1d, where there is an infinite hierarchy of consistency relations but still room for model dependence, due to the existence of transverse spatial polarizations.\n\n\n\\subsubsection*{Global Weyl symmetry}\n\nFinally, the shift symmetry of $\\Laux$ may also be thought of as a global Weyl symmetry in constant inflaton gauge, which is related to the familiar local Weyl symmetry of regular Liouville theory. To see this, begin by substituting $g_{ab}=e^{2\\phican}\\hat g_{ab}$ in the action \\eqref{Lcovpre}, upon which it adopts the form \\eqref{Sgravhat}. No gauge choice has yet been assumed if $\\hat g$ is allowed to fluctuate. In backgrounds satisfying $\\Lauxb'+\\phib'=0$, the constant inflaton gauge condition $\\delta\\!\\Laux+\\delta\\!\\phican=0$ makes the full fields (background plus perturbation) satisfy $\\Laux+\\phican=C$, a constant. This constant contributes to the action only as a multiple of the Euler characteristic \\eqref{Euler}, so it may be neglected. We may then impose the gauge condition $\\Laux+\\phican=0$ (ie. $\\Theta=0$) on the action \\eqref{Sgravhat}, reducing it to\n\\begin{equation}\\label{Sci}\n{\\mathcal{S}}_{\\rm c.i.} = \\frac{2}{\\gamma^2} \\int \\sqrt{-\\hat g}\\, \\[(\\hat\\nabla\\phican)^2 + \\hatR\\phican\n-{\\Lambda} e^{2\\phican} \\]\n\\end{equation}\nThis is precisely the action for timelike Liouville theory \\cite{Martinec:2014uva}, but with a different interpretation. The action \\eqref{Sci} is usually studied in conformal gauge, where it describes the dynamics of the Liouville field $\\phican$ in a fixed reference metric $\\hat g$. It is well known to possess a symmetry under local Weyl transformations \\cite{Ginsparg:1993is}\n\\begin{equation}\\label{Weyl}\n\\hat g_{ab}\\to e^{2\\omega}\\hat g_{ab}~, \\qquad\n\\phican \\to \\phican - \\omega\n\\end{equation}\nwhere $\\omega$ is an arbitrary function of space and time. This transformation may appear trivial (as it leaves $g_{ab}=e^{2\\phican}\\hat g_{ab}$ invariant), but it does in fact change the gauge choice. In conformal gauge we exhaust the local diffeomorphism symmetry to fix $\\hat g$, and there is a family of conformal gauges where $\\hat g$ has been fixed to different values. The Weyl transformation \\eqref{Weyl} moves along a slice in this family. This is not a residual gauge symmetry, and there is no a priori reason to expect the gauge-fixed action \\eqref{Sci} to be invariant. In fact, the action changes by\n\\begin{equation}\\label{STLvar}\n{\\mathcal{S}}_{\\rm c.i.}  \\to {\\mathcal{S}}_{\\rm c.i.} - \\frac{2}{\\gamma^2} \\int \\sqrt{-\\hat g}\\, \\hat R\\, \\omega\n\\end{equation}\nThe change depends only on $\\hat g$, which is fixed in conformal gauge. The $\\phican$ equation of motion is unaltered, so the local Weyl transformation \\eqref{Weyl} is a symmetry of Liouville theory in conformal gauge.\n\nIn the present context, we are instead studying \\eqref{Sci} in constant inflaton gauge. Both $\\phican$ and $\\hat g$ are fluctuating, as the gauge freedom has been used to eliminate $\\delta\\!\\Laux$ and fix $\\delta\\!\\inflaton=0$. The variation \\eqref{STLvar} of the gauge-fixed action depends on $\\hat g$, and in general will alter its equation of motion. If $\\omega$ is a constant, however, \\eqref{STLvar} is just a multiple of the Euler characteristic. Hence the action \\eqref{Sci} possesses only a global Weyl symmetry in this context. The consequences of the global Weyl transformation for cosmological perturbation theory are made clear if it is packaged as\n\\begin{equation}\\label{Weylglobal}\n\\phib\\to\\phib+\\omega~,\\qquad \\delta\\!\\phican\\to\\delta\\!\\phican-\\omega\n\\end{equation}\nThis doesn't interfere with the metric parametrization \\eqref{metparam}. Again, it is not a trivial transformation as the gauge has already been fixed. The slow roll expansion of the constrained action depends on $\\phib$ only through the slow roll parameters, which are invariant under \\eqref{Weylglobal} (the transformation of $\\phib$ is just a constant rescaling of the scale factor). Hence the constant shift of $\\delta\\!\\phican$ is a physical symmetry of the correlation functions.\n\n\n\n\n\n\n\\section{Conclusion}\nIn this paper we have applied the framework of cosmological perturbation theory to single field inflation in a generally covariant extension of Liouville theory. Our main results are the unified power spectrum \\eqref{P2}, and the three-point correlation function of scalar curvature perturbations, equation \\eqref{phi3pt}. We have investigated symmetry constraints on the three-point function and explained how Madacena's consistency condition \\eqref{Maldacena2d} arises, despite the absence of residual diffeomorphisms. An important subresult is the proof of freeze out to all orders in perturbation theory, given in section \\ref{constancysec}.\n\nThis work completes the construction of a simplified 2d laboratory for the scalar sector of cosmological perturbations in 4d Einstein gravity. We now possess a precise model of the scalar sector embedded in a renormalizable theory. A future publication will explore renormalized perturbations in cosmological gauges, addressing various aspects including the quantum-corrected constraint algebra, and the classification of physical states via BRST cohomology.\n\n\n\n\\section*{Acknowledgments}\nIt is a pleasure to thank Emil Martinec for suggesting this topic, for discussions, and detailed comments on a draft manuscript. I am also grateful to Wayne Hu, Austin Joyce, and Guilherme Pimentel for discussions. This work was supported in part by DOE grant DE-FG02-13ER41958, and by the US Department of State through a Fulbright Science and Technology Award.\n\n\n\n\n\n\n\n\n\n\n\n\n\\singlespacing\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCauchy's surface area formula is a beautiful and well-known result in integral geometry.  It states that the average area of the projections of a convex body is equal to the surface area of the body, up to a multiplicative constant in the dimension.  That is, let $\\mu_{n-1}$ be $(n-1)$-dimensional Lebesgue measure on ${\\mathbb R}^{n-1}$, and  ${\\mathbb S}^{n-1}$ be the unit sphere in ${\\mathbb R}^n$.  For a convex body $K \\subset {\\mathbb R}^n$ and \n$u \\in {\\mathbb S}^{n-1}$, we denote by $K|u^{\\perp}$ the projection of $K$ onto the $(n-1)$-dimensional subspace of ${\\mathbb R}^n$ perpendicular to $u$.  Then we have:\n\n\\begin{theorem}[Cauchy's surface area formula]\\label{CauchyTheorem}\nSuppose $K \\subset {\\mathbb R}^n$ is a convex body.  Then\n\\begin{equation}\n\\text{S}(K) =  \\frac{1}{\\mu_{n-1}\\left({\\mathbb B}^{n-1}\\right)} \\int_{{\\mathbb S}^{n-1}} \\mu_{n-1}\\left(K|u^\\perp \\right) du.\n\\end{equation}\nwhere $S(K)$ denotes the volume of the surface of $K$.\n\\end{theorem}\n\nWe note that $S(K)$ is succinctly defined as follows:\n\\begin{equation}\\label{SurfaceArea}\nS(K) = \\lim_{\\epsilon \\to 0^+} \\frac{\\mu_n\\left(K+\\epsilon {\\mathbb B}_n\\right)-\\mu_n(K)}{\\epsilon}\n\\end{equation}\nWhere ${\\mathbb B}_n$ is the unit ball in ${\\mathbb R}^n$ and $K+\\epsilon B_n$ indicates the Minkowski sum of $K$ and $\\epsilon B_n$:\n\\begin{equation*}\nK+\\epsilon {\\mathbb B}_n = \\{x+\\epsilon v: x \\in K, v \\in {\\mathbb B}_n\\}.\n\\end{equation*}\n\nCauchy's surface area formula was first proved by Cauchy for $n=2, 3$  in 1841 and 1850 \\cite{Cauchy1}, \\cite{Cauchy2}.  The general formula was proved in work by Kubota \\cite{Kubota1}, and also Minkowski \\cite{Minkowski1}  and Bonnesen \\cite{Bonnesen1}.  Klain and Rota give a lovely proof in Chapter 5 of \\cite{MR1608265}, which uses calculus-like arguments and approximations of convex bodies by polytopes.  (We draw much of our notation from the same text).  In this short note we give an alternative proof, which we believe is new.  This new proof has  the benefit of the fact that the projection is calculated using a Minkowski sum of sets, just as the surface area formula is calculated using a Minkowski sum of sets as in equation \\eqref{SurfaceArea}.  Thus, the two sides of Theorem \\ref{CauchyTheorem} are seen to be calculated using Minkowski sums.  This technique allows  a straightforward extension to intrinsic moment vectors of a convex body (which we will define in Section \\ref{MomentVectors}).\n\n\n\\section{New Proof for Cauchy's Surface Area Formula}\\label{NewProof}\n\nFirst we give our definition, which uses the Minkowski sum of sets as discussed in the Introduction.\n\n\\begin{definition*}\nLet $u, P \\subset {\\mathbb R}^n$. Define\n\\begin{equation}\\label{deriv}\nD_u  (\\mu_n)(P)=\\lim_{\\epsilon \\rightarrow 0^{+}} \\frac{\\mu_n(P+\\epsilon u)-\\mu_n(P)}{\\epsilon}.\n\\end{equation}\n\\end{definition*}\n\nThe special case $D_{\\mathbb{B}^n}  (\\mu_n)(P)$ gives the surface volume:\n\\begin{equation}\\label{minkcontent}\nD_{\\mathbb{B}^n}  (\\mu_n)(P) =S(P)\n\\end{equation}\n\nThe key to our proof is Minkowski's theorem on mixed volumes, which can be found, for example, in  Chapter 5 of Schneider's text \\cite{MR3155183}.  This theorem says that the volume of a Minkowski sum of convex bodies can be written as a polynomial in the coefficients of the Minkowski sum, where the coefficients of the polynomial depend only on the convex bodies.  Specifically:  \n\n\\begin{theorem}\\label{MixedVolumeThm}\nSuppose $K_1, K_2, \\dots, K_m$ are convex bodies in ${\\mathbb R}^n$.  Then\n\\begin{equation*}\n\\mu_n\\left(\\lambda_1K_1+\\lambda_2 K_2+ \\cdots + \\lambda_m K_m\\right) = \\sum \\lambda_{i_1}\\lambda_{i_2} \\cdots \\lambda_{i_n} V\\left(K_{i_1}, K_{i_2}, \\dots, K_{i_n}\\right)\n\\end{equation*}\nwhere the sum on the left-hand side is the Minkowski sum, and the sum on the right-hand side is over all multisets of size $n$ whose elements are in the set $\\{1, 2, \\dots, m\\}$.  The  functions $V$ are nonnegative, symmetric, and depend only on the convex bodies $K_{i_1}, K_{i_2}, \\dots, K_{i_n}$.\n\\end{theorem}\n\nFrom Minkowski's Theorem it follows that if $P$ and $u$ are convex, then $D_u  (\\mu_n)(P)$ is linear in $u$:\n\n\\begin{lemma}\\label{LinearityLemma}\nSuppose $P, u, v \\subset {\\mathbb R}^n$ with $P, u$, and $v$ convex bodies, and suppose $\\alpha, \\beta \\in {\\mathbb R}$.  Then\n\\begin{equation}\nD_{\\alpha u+\\beta v} (\\mu_n)(P) = \\alpha D_u (\\mu_n)(P)+\\beta D_v (\\mu_n)(P)\n\\end{equation}\n\\end{lemma}\n\n\\paragraph{Proof.}\nBy definition we have\n\\begin{equation*}\nD_{\\alpha u+\\beta v}(\\mu_n)(P) = \\lim_{\\epsilon \\to 0^+} \\frac{\\mu_n\\left(P+\\epsilon \\left(\\alpha u+\\beta v\\right)\\right) - \\mu_n(P)}{\\epsilon} =  \\lim_{\\epsilon \\to 0^+} \\frac{\\mu_n\\left(P+\\epsilon \\alpha u+ \\epsilon \\beta v\\right) - \\mu_n(P)}{\\epsilon} \n\\end{equation*}\nFrom Theorem \\ref{MixedVolumeThm}, we can see that\n\\begin{equation*}\nD_{\\alpha u+\\beta v}(\\mu_n)(P) = \\alpha V(\\underbrace{P, P, \\dots, P}_{n-1 \\text{ times}}, u) + \\beta V(\\underbrace{P, P, \\dots, P}_{n-1 \\text{ times}}, v) \n\\end{equation*}\nwhere $V$ represents the function in the statement of Theorem \\ref{MixedVolumeThm}.  Similarly, one can easily see that\n\\begin{align*}\nD_u(\\mu_n)(P) &=  V(\\underbrace{P, P, \\dots, P}_{n-1 \\text{ times}}, u) \\\\\n D_v(\\mu_n)(P) &=  V(\\underbrace{P, P, \\dots, P}_{n-1 \\text{ times}}, v), \n \\end{align*}\nand our Lemma is proved.\n\\proofend\n\n\n For the following Lemma, we prove that the derivative we've defined in equation \\eqref{deriv}  calculates the projection of a convex body in the case where $u$ is a segment of    length 1.  We use the same notation as in the Introduction.\n\n\\begin{lemma}\\label{DerivativeLemma}\nLet $u$ be a segment of length $1$ and $K \\subset {\\mathbb R}^n$ convex. Then\n\\begin{equation}\\label{projeq}\n\\mu_{n-1} \\left(K|u^\\perp\\right) = D_u(\\mu_n)( K).\n\\end{equation}\n\\end{lemma}\n\n\\paragraph{Proof.}\nLet $L_u$ be the set of lines parallel to $u$.  Note that each $l \\in L_u$ corresponds uniquely to a single point in $l^\\perp$, so that $L_u$ is isomorphic to ${\\mathbb R}^{n-1}$ (and thus we can define the measure $\\mu_{n-1}$ on $L_u$).  Then\n\\[\n\\mu_n(K)=\\int_{l \\in L_u} \\mu_1(l \\cap K) \\, d \\mu_{n-1}.\n\\]\nAlso,\n\\[\n\\mu_{n-1}\\left(K|u^\\perp\\right) =\\int_{\\substack{ l \\cap K \\neq \\emptyset \\\\ l \\in L_u }} 1 \\, d \\mu_{n-1}.\n\\]\nFor $\\epsilon >0$, we have\n\\[\n\\mu_n\\left(K+\\epsilon u\\right)-\\mu_n\\left(K\\right)=\\int_{l \\in L_u}\\left( \\mu_1\\left(l \\cap (K+\\epsilon u)\\right)-\\mu_1\\left(l \\cap K\\right) \\right)d\\mu_{n-1}.\n\\]\nConvexity implies that\n\\[\n\\mu_1\\left(l \\cap (K+\\epsilon u)\\right)-\\mu_1\\left(l \\cap K\\right) = \\epsilon\n\\]\nwhenever $l$ intersects $K$, and zero otherwise. Therefore the last integral is equal to\n\\[\n\\int_{\\substack{ l \\cap K \\neq \\emptyset \\\\ l \\in L_u }} \\epsilon \\, d \\mu_{n-1}=\\epsilon\\mu_{n-1} \\left(K|u^\\perp\\right) \n\\]\nand hence,\n\\begin{equation*}\nD_u (\\mu_n)(K) = \\lim_{\\epsilon \\to 0^+}  \\frac{\\mu_n(K+\\epsilon u)-\\mu_n(K)}{\\epsilon} = \\lim_{\\epsilon \\to 0^+} \\frac{\\epsilon\\mu_{n-1} \\left(K|u^\\perp\\right) }{\\epsilon} = \\mu_{n-1} \\left(K|u^\\perp\\right). \n\\end{equation*}\n\\proofend\n\nWe can now give a short proof of Cauchy's surface area formula:\n\n\\paragraph{Proof.} (of Theorem \\ref{CauchyTheorem})\n Using Lemmas \\ref{LinearityLemma},\\ref{DerivativeLemma}, and equation \\eqref{minkcontent}, we have: \n\\begin{equation*}\n \\int_{{\\mathbb S}^{n-1}} \\mu_{n-1}\\left(K|u^\\perp \\right) du\n= \\int_{{\\mathbb S}^{n-1}} D_{[0,u]} (\\mu_n)( K) du\n=  D_{\\int_{{\\mathbb S}^{n-1}}[0,u]du} (\\mu_n)( K)\n=    D_{c(n) {\\mathbb B}^n}  (\\mu_n)(K) \n=  c(n)  S(K). \n\\end{equation*}\n\n\nSpecializing $K$ to be the unit ball ${\\mathbb B}^n$ yields the result.\n\\proofend\n\n\n\\section{Application of This Technique to Moment Vectors}\\label{MomentVectors}\n\n\nThe proof can be extended readily to intrinstic moment vectors of a convex body \\cite[Sec. 5.4]{MR3155183}. We introduce the following notation, modeled on the notation from  \\cite[Sec. 5.4]{MR3155183}.  Let $H^r$ denote the $r$-dimensional Hausdorff measure on ${\\mathbb R}^n$.   For an $r$ dimensional set $K \\subset {\\mathbb R}^n$, let \n\\begin{equation*}\nz_{r+1}(K)=\\int_K x \\, dH^r(x)\n\\end{equation*}\n denote the moment vector of the set $K$. It is related to the familiar centroid $p_0(K)$ via $z_{r+1}(K)=H^r(K)p_0(K)$. We will abuse notation and denote the boundary of $K$ by $S(K)$, when no confusion with the surface area can occur. For a segment $u$, let \n\\[\nK_u= \\lim_{\\epsilon \\rightarrow 0} S(K) \\cap (K+\\epsilon u).\n\\]\nIntuitively, this is the portion of the boundary that is seen by a projection in direction $u$.\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=2in]{ku_ellipse}\n\\caption{The subset $K_u$ of the boundary $\\partial K$ of $K$.}\n\\end{figure}\n\nLet $K$ be an $n$-dimensional subset of ${\\mathbb R}^n$.  The following Theorem now has a very similar flavor to that of the Cauchy surface area formula:\n\\begin{theorem}\\label{MomentTheorem}\nSuppose $K \\subset {\\mathbb R}^n$ is a convex body.  Then\n\\[\nz_n(S (K))=\\frac{1}{\\mu_{n-1}({\\mathbb B}_{n-1})} \\int_{{\\mathbb S}^{n-1}} z_{n}(K_u) du .\n\\]\n\\end{theorem}\n\nBefore we prove this, we need a definition and a few observations.  If $K$ is an  $n$-dimensional subset of ${\\mathbb R}^n$ and $u$ any subset of ${\\mathbb R}^n$, we now define\n\\begin{equation}\\label{MomentDerivative}\nD_u (z_{n+1})(K) = \\lim_{\\epsilon \\to 0^+} \\frac{z_{n+1}(K+\\epsilon u) - z_{n+1}(K)}{\\epsilon}.\n\\end{equation}\nWe note that, as one can find in  \\cite[Sec. 5.4]{MR3155183}, there is a Theorem parallel to Theorem \\ref{MixedVolumeThm} but for the moment vector.  Thus, just as we showed in Lemma \\ref{LinearityLemma}, the operator $D_u$ defined in equation \\eqref{MomentDerivative} is also linear in $u$ (if $K, u, $ and $v$ are convex just as in Lemma \\ref{LinearityLemma}).  We can now proceed with our proof  of Theorem \\ref{MomentTheorem}:\n\n\\paragraph{Proof.} (Proof of Theorem \\ref{MomentTheorem})\nIt is not hard to see from the definition of the moment vector that \n\\[\nz_{n+1}(K+\\epsilon u)=z_{n+1}(K)+z_{n+1}((K+\\epsilon u) \\setminus K)\n\\]\n\n\nTherefore if $u \\in {\\mathbb S}^{n-1}$ and as above we let $[0,u]$ denote the segment from the origin to $u$, we have\n\\[\nD_{[0,u]}(z_{n+1})(K)=\\lim_{\\epsilon \\rightarrow 0^+} \\frac{z_{n+1}((K+\\epsilon u) \\setminus K)}{\\epsilon}=z_n(K_u).\n\\]\nwhich gives \n\\[\n\\int_{{\\mathbb S}^{n-1}} D_{[0,u]}(z_{n+1})(K) du=\\int_{{\\mathbb S}^{n-1}} z_n(K_u) du.\n\\]\nSimilarly, we note that\n\\begin{equation*}\nz_{n+1}(K+\\epsilon B^n) = z_{n+1}(K)+z_{n+1}((K+\\epsilon {\\mathbb B}_n) \\backslash K)\n\\end{equation*}\nso that\n\\begin{equation}\\label{DerivativeMomentBall}\nD_{{\\mathbb B}^n} (z_{n+1})(K) = \\lim_{\\epsilon \\rightarrow 0^+} \\frac{z_{n+1}((K+ \\epsilon {\\mathbb B}_n) \\backslash K}{\\epsilon} = z_n(S(K)).\n\\end{equation}\n\nNote that $K$ and $[0,u]$ are convex bodies so that we can use the linearity of the $D$ operator defined in \\eqref{MomentDerivative}.  Using this linearity, our calculations from the proof of Theorem \\ref{CauchyTheorem}, and equation \\eqref{DerivativeMomentBall}, we now have\n\\begin{multline*}\n\\int_{{\\mathbb S}^{n-1}} D_{[0,u]}(z_{n+1})(K)  \\,du=D_{\\int_{ {\\mathbb S}^{n-1}}  [0,u] du} (z_{n+1})(K)\n= \\mu_{n-1}({\\mathbb B}^{n-1})D_{{\\mathbb B}^n}(z_{n+1})(K) = \\mu_{n-1}({\\mathbb B}^{n-1})z_n(S(K)).\n\\end{multline*}\nThus the Theorem is proved.\n\\proofend\n\n\n\n\n\\bigskip\n{\\bf Acknowledgments}.  \nThe authors would like to thank Alexander Barvinok for his comments and Sergei Tabachnikov for his advice and encouragement.\n\nThis material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE 1106400. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors(s) and do not necessarily reflect the views of the National Science Foundation.\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe theory of large deviations (see e.g.\\ \nthe books by Dembo and Zeitouni~\\cite{DemboZeitouni10ldta} or Rassoul-Agha and Sepp\\\"al\\\"ainen~\\cite{RassoulaghaSeppelainen14cldigm})\n studies the probabilities of rare events on the exponential scale. \nThis is formally captured by the following definition.\n\n\\begin{definition}\\label{df:LDP}\n\tA sequence of Borel probability measures $(\\mu_n)_{n\\in \\I N}$ on a \n\ttopological space $X$ satisfies a \\emph{large deviation principle} (\\emph{LDP} for short) with \\emph{speed} \n\t$s:\\I N\\to(0,\\infty)$\n\tand \\emph{rate function} $I:X\\to[0,\\infty]$ if\n\t\\begin{itemize}[nosep]\n\t\t\\item \n\t$s(n)\\to\\infty$ as $n\\to\\infty$, \n\t\\item the function $I$ is \\emph{lower semi-continuous} (\\emph{lsc} for short), that is, for each $\\rho\\in\\I R$\n\tthe level set $\\{I\\le \\rho\\}:=\\{x\\in X\\mid I(x)\\le \\rho\\}$ is a closed subset of~$X$,\n\t\\item the following \\emph{lower bound} holds:\n\t\\beq{eq:lowerGen}\n\t\t\\liminf_{n\\to\\infty} \\frac1{s(n)}\\,{\\log\\, \\mu_n(G)} \n\t\t\\ge -\\inf_{x\\in G} I(x),\\quad \\mbox{for every open $G\\subseteq X$,}\n\t\t\\end{equation}\n\t\t\\item the following \\emph{upper bound} holds:\n\t\t\\beq{eq:upperGen}\\limsup_{n\\to\\infty} \\frac1{s(n)}\\,{ \\log\\,\\mu_n(F)} \\le  -\\inf_{x\\in F} I(x),\\quad \\mbox{for every closed $F\\subseteq X$.}\n\\end{equation}\n\\end{itemize}\n\\end{definition}\n\nAs it is well-known (see e.g.\\ \\RS{Lemma~2.11}), if~\\eqref{eq:lowerGen} and~\\eqref{eq:upperGen} hold for some (not necessarily lsc) function $I:X\\to [0,\\infty]$ then we can replace $I$ without violating these bounds by its \\emph{lower semi-continuous\n\tregularization} \n\\beq{eq:lscR}\n I_{\\mathrm{lsc}}(x):=\\sup\\left\\{\\inf_{y\\in G} I(y)\\mid G\\ni x\\mbox{ and $G\\subseteq X$ is open}\\right\\},\\quad x\\in X;\n\\end{equation} \nfurthermore (see e.g.\\ \\RS{Lemma~2.8}), \n $I_{\\mathrm{lsc}}$ is lower semi-continuous  and, in fact, $I_{\\mathrm{lsc}}$ is the largest lsc function with $I_{\\mathrm{lsc}}\\le I$. \nIf $X$ is a regular topological space then there can be at most one lower semi-continuous rate function satisfying Definition~\\ref{df:LDP} (see e.g.\\ \\DZ{Lemma 4.1.4} or \\RS{Theorem~2.13}). This (as well as some other results, such as Lemma~\\ref{lm:LDP} below) motivates the requirement that $I$ is lsc in Definition~\\ref{df:LDP}. \n\n\nLarge deviations for various models of random graphs have been receiving much attention in the recent years; see e.g.\\ the survey by Chatterjee~\\cite{Chatterjee16bams}, or \\cite[Section~1.7]{BCGPS} for references to some more recent results. A basic but central model\nis the \\emph{binomial random graph} $\\I G(n,p)$, where the vertex set is $[n]:=\\{1,\\dots,n\\}$ and \neach pair of vertices is an edge with probability~$p$, independently of other pairs. A large deviation principle for $\\I G(n,p)$ for constant $p\\in (0,1)$ was established in a ground-breaking paper of\nChatterjee and Varadhan~\\cite{ChatterjeeVaradhan11} as follows. (See also the exposition of this proof in  Chatterjee's book~\\cite{Chatterjee17ldrg}.) \n\nAs it turns out, the ``correct'' setting is to consider \\emph{graphons}, that is,  measurable symmetric functions $[0,1]^2\\to [0,1]$. On the set $\\C W$  of all {graphons}, one can define the so-called \\emph{cut-distance} $\\delta_\\Box$, which is a pseudo-metric on $\\C W$ (see Section~\\ref{graphons} for all missing definitions related to graphons). Consider the factor space \n$$\n\\widetilde{\\mathcal{W}}:=\\{\\T U: U\\in\\C W\\},\n$$ \nwhere $\\T U:=\\{V\\in\\C W\\mid \\delta_\\Box(U,V)=0\\}$ consists of all graphons \\emph{weakly isomorphic} to~$U$.\n The space $(\\widetilde{\\mathcal{W}},\\delta_\\Box)$ naturally appears in the limit theory of dense graphs, see e.g.\\ the book by\nLov\\'asz~\\cite{Lovasz:lngl}. In particular, a graph $G$ on $[n]$ can be identified with the graphon $\\f{G}$ where we partition $[0,1]$ into intervals of length $1\/n$ each and let $\\f{G}$ be the $\\{0,1\\}$-valued step function that encodes the adjacency relation.\nThis way, $\\I G(n,p)$ gives a (discrete) probability measure $\\widetilde{\\mathbb{P}}_{n,p}$ on $(\\widetilde{\\mathcal{W}},\\delta_\\Box)$, where $\\widetilde{\\mathbb{P}}_{n,p}(A)$ for $A\\subseteq \\widetilde{\\mathcal{W}}$ is the probability that the sampled graph,\nwhen viewed as a graphon up to weak isomorphism, belongs to the set~$A$. \n\nAlso, recall that, for $p\\in [0,1]$,  the \\emph{relative entropy} $h_p:[0,1]\\to [0,\\infty]$ is defined by\n\\beq{eq:hp}\n h_p(\\rho):=\\rho\\log\\left(\\frac{\\rho}{p}\\right)+(1-\\rho)\\log\\left(\\frac{1-\\rho}{1-p}\\right),\\quad \\rho\\in[0,1].\n\\end{equation}\n\n\n\n\\begin{theorem}[Chatterjee and Varadhan~\\cite{ChatterjeeVaradhan11}]\n\t\\label{th:CV}\n\tLet $p\\in [0,1]$. The function $I_p:\\C W\\to [0,\\infty]$ defined by \n\t\\beq{eq:IpCV}\n\t I_p(U):=\\frac12 \\int_{[0,1]^2} h_p(U(x,y))\\,\\mathrm{d} x\\,\\mathrm{d} y,\\quad U\\in\\C W,\n\t \\end{equation}\n\t gives a well-defined function $\\widetilde{\\mathcal{W}}\\to [0,\\infty]$ (that is, $I_p$ assumes the same value at any two graphons at $\\delta_\\Box$-distance 0) which is lower semi-continuous on $(\\widetilde{\\mathcal{W}},\\delta_\\Box)$. Moreover,\n\t the sequence of measures $(\\widetilde{\\mathbb{P}}_{n,p})_{n\\in\\I N}$ on $(\\widetilde{\\mathcal{W}},\\delta_\\Box)$ satisfies an LDP with speed $n^2$ and rate function~$I_p$.\n\t \\end{theorem}\n\nBorgs, Chayes, Gaudio, Petti and Sen~\\cite{BCGPS} extended this result to $k$-block stochastic models as follows. \nLet $k\\ge 1$ be a fixed integer. Let $\\V p=(p_{i,j})_{i,j\\in [k]}\\in [0,1]^{k\\times k}$ be a symmetric $k\\times k$ matrix with entries in~$[0,1]$. For an integer vector $\\V a=(a_1,\\dots,a_k)\\in\\I N_{\\ge 0}^k$, where \n$$\n\\I N_{\\ge 0}:=\\{0,1,2,\\ldots\\}\n$$\n denotes the set of non-negative integers, let $\\widetilde{\\mathbb{P}}_{\\V a,\\V p}$ be the probability distribution on $\\widetilde{\\mathcal{W}}$ defined as follows.  Set $n$ to be $\\|\\V a\\|_1=a_1+\\dots+a_k$ and  let $(A_1,\\dots, A_k)$ be the partition of $[n]$ into $k$ consecutive intervals with $A_i$ having $a_i$ elements. The random graph $\\I G(\\V a,\\V p)$ on $[n]$ is produced by making each pair $\\{x,y\\}$ of $[n]$ an edge with probability $p_{i,j}$ where $i,j\\in [k]$ are the indices with $x\\in A_i$ and $y\\in A_j$, with all choices made independently of each other. Output the weak isomorphism class $\\tf{G}\\in\\widetilde{\\mathcal{W}}$ of the graphon $\\f{G}$ corresponding to the generated graph~$G\\sim \\I G(\\V a,\\V p)$ on~$[n]$. Informally speaking, we take $k$ blocks consisting of exactly $a_1,\\dots,a_k$ vertices respectively, make pairs into edges with the probabilities given by the $k\\times k$ matrix~$\\V p$, and them forget the block structure.\n\nNext, we define a rate function for a given non-zero real $k$-vector $\\V\\alpha=(\\alpha_1,\\dots,\\alpha_k)\\in[0,\\infty)^k$. \n Let $(\\cI{\\V\\alpha}{1},\\dots,\\cI{\\V\\alpha}{k})$ denote the partition of $[0,1]$ into consecutive intervals such that each interval $\\cI{\\V\\alpha}{i}$ has length $\\alpha_i\/\\|\\V\\alpha\\|_1$. \n Define the function $J_{\\V\\alpha,\\V p}:\\widetilde{\\mathcal{W}}\\to [0,\\infty)$ by\n \\beq{eq:JAlphaP}\nJ_{\\V\\alpha,\\V p}(\\T U):=\\inf_{V\\in \\T U}\\,  \\frac12\\sum_{i,j\\in [k]} \\int_{\\cI{\\V\\alpha}{i}\\times \\cI{\\V\\alpha}{j}} h_{p_{i,j}}(V(x,y))\\,\\mathrm{d} x\\,\\mathrm{d} y,\\quad U\\in {\\C W}.\n \\end{equation}\n Note that the function $J_{\\V\\alpha,\\V p}$ will not change if we multiply the vector $\\V\\alpha$ by any positive scalar.\n \nIn the above notation, the LDP of Borgs et al~\\cite[Theorem 1 and Remark~2]{BCGPS} states the following.\n \n \\begin{theorem}[Borgs, Chayes, Gaudio, Petti and Sen~\\cite{BCGPS}]\n \t\\label{th:BCGPS}\n \tLet $\\V\\alpha\\in \\I N_{\\ge 0}^k$ be a non-zero integer $k$-vector and let $\\V p\\in [0,1]^{k\\times k}$ be a symmetric $k\\times k$ matrix. Then the sequence of measures $(\\widetilde{\\mathbb{P}}_{n\\V\\alpha,\\V p})_{n\\in\\I N}$ on $(\\widetilde{\\mathcal{W}},\\delta_\\Box)$ satisfies an LDP with speed $(n\\,\\|\\V\\alpha\\|_1)^2$ and rate function~$(J_{\\V\\alpha,\\V p})_{\\mathrm{lsc}}$.\n \n \t\\end{theorem}\n \n Note that the special case $k=1$ and $\\V\\alpha=(1)$ of Theorem~\\ref{th:BCGPS} and the assumption that\n the function $J_{\\V\\alpha,\\V p}:\\widetilde{\\mathcal{W}}\\to [0,\\infty]$  is lower semi-continuous give the second part of Theorem~\\ref{th:CV}.\n \n Our contribution is as follows. \n \n \nFirst, we prove that the function $J_{\\V\\alpha,\\V p}:\\widetilde{\\mathcal{W}}\\to [0,\\infty]$ is  lower semi-continuous (so, in particular, there is no need to take\n the lower semi-continuous\n regularization in Theorem~\\ref{th:BCGPS}):\n \n \\begin{theorem}\\label{th:JLSC}\n For every symmetric matrix $\\B p\\in [0,1]^{k\\times k}$ and every non-zero real $k$-vector $\\V \\alpha\\in [0,\\infty)^k$, the function $J_{\\V\\alpha,\\V p}:\\widetilde{\\mathcal{W}}\\to[0,\\infty]$ is lower semi-continuous with respect to the metric~$\\delta_\\Box$.\n \\end{theorem}\n\n\nSecond, we extend Theorem~\\ref{th:BCGPS} by allowing the fraction of vertices assigned to a part to depend on $n$ as long as it converges to any finite (possibly irrational) limit.\n\n\n\\begin{theorem}\\label{th:GenLDP}\n\tFix \n\n\tany symmetric $k\\times k$ matrix $\\V p\\in [0,1]^{k\\times k}$ and a non-zero real $k$-vector $\\V\\alpha=(\\alpha_1,\\dots,\\alpha_k)\\in [0,\\infty)^k$.  Let\n\t$$\n\t\\V a_n=(a_{n,1},\\dots,a_{n,k})\\in \\I N_{\\ge 0}^k,\\quad \\mbox{for $n\\in\\I N$},\n\t$$ \n\tbe arbitrary non-zero integer $k$-vectors  such that $\\lim_{n\\to\\infty} a_{n,i}\/n=\\alpha_i$ for each $i\\in [k]$. \n\tThen the sequence of measures $(\\widetilde{\\mathbb{P}}_{\\V a_n,\\V p})_{n\\in\\I N}$ on $(\\widetilde{\\mathcal{W}},\\delta_\\Box)$ satisfies an LDP with speed $\\|\\V a_n\\|_1^2$ and rate function~$J_{\\V\\alpha,\\V p}$.\n\\end{theorem}\n \n \nOne application of Theorem~\\ref{th:GenLDP}\nis as follows. Each graphon $W\\in\\C W$ gives rise to the following inhomogeneous random graph model. Namely, the \\emph{random $W$-graph} $\\I G(n,W)$ is generated by first sampling\nuniform elements $x_1,\\dots,x_n\\in [0,1]$ and then making each pair $\\{i,j\\}$ an edge with probability $W(x_i,x_j)$, where all choices are independent of each other. Let $\\widetilde{\\mathbb{R}}_{n,W}$ be the corresponding (discrete) measure on~$\\widetilde{\\mathcal{W}}$ where we take the equivalence class $\\tf{G}\\in\\widetilde{\\mathcal{W}}$ of the sampled graph~$G$. When $W$ is the constant function $p$, we get exactly the binomial random graph $\\I G(n,p)$ and $\\widetilde{\\mathbb{R}}_{n,W}=\\widetilde{\\mathbb{P}}_{n,p}$.\n\nThe authors showed in~\\cite{GrebikPikhurko:LDP} that, for any graphon $W\\in\\C W$, the only ``interesting'' speeds for the sequence of measures $(\\widetilde{\\mathbb{R}}_{n,W})_{n\\in\\I N}$ are $\\Theta(n)$ and $\\Theta(n^2)$, and established a general LDP for speed~$n$. The case when speed is $n^2$ seems rather difficult. Here (in Theorem~\\ref{th:ourLDP}) we prove an LDP for speed $n^2$\nwhen $W$ is a \\emph{$k$-step graphon}, that is, there is a measurable partition $[0,1]=A_1\\cup\\dots\\cup A_k$ such that $W$ is a constant $p_{i,j}$ on each product $A_i\\times A_j$, $i,j\\in [k]$. We can assume that each $A_i$ has positive measure, since changing the values of $W$ on a null subset of $[0,1]^2$ does not affect the distribution of~$\\I G(n,W)$. \n\nBefore stating our LDP, let us point out the difference between the random graphs $\\I G((a_1,\\dots,a_k),(p_{i,j})_{i,j\\in [k]})$ and $\\I G(a_1+\\dots+a_k,W)$ when $W$ and $(p_{i,j})_{i,j\\in [k]}$ are as above.\nIn the former model, we have exactly $a_i$ vertices in the $i$-th block for each $i\\in [k]$. In the latter model, each vertex is put into one of the $k$ blocks with the probabilities given by the measures of $A_1,\\dots,A_k$, independently of the other vertices; thus the number\nof vertices in each block is binomially distributed.\n\\hide{\nBefore stating our LDP, let us point out that some differences and relations between the random graphs $\\I G(\\V a,\\V p)$ and $\\I C(n,W)$ when $W$ and $\\V p$ are as in the previous paragraph, $\\V a=(a_1,\\dots,a_k)$, and $n=a_1+\\dots+a_k$.\nIn $\\I G(\\V a,\\V p)$ we take exactly $a_i$ vertices from the $i$-th block. On the other hand, in $\\I C(n,W)$ each vertex is put into one the blocks independently of the other vertices with probabilities equal to the measures\n$\\lambda(A_1),\\dots,\\lambda(A_k)$ of the parts; thus the total number of vertices in the $i$-th block has the binomial distribution with parameters $(n,\\lambda(A_i))$. Once the vertices are assigned to blocks, the rule for generating the edges is same in both models: a pair connecting the $i$-th and $j$-th block is made an edge with probability $p_{i,j}$. Thus if we condition $\\I C(n,W)$ on the blocks being consecutive intervals of $[n]$ of lengths respectively $a_1,\\dots,a_k$, then the conditional distribution will be exactly the distribution of~$\\I G(\\V a,\\V p)$.}%\nIt comes as no surprise that if we consider large deviations for $(\\widetilde{\\mathbb{R}}_{n,W})_{n\\in\\I N}$ at speed \n$n^2$ then the rate function depends only on $(p_{i,j})_{i,j\\in [k]}$ but not on the (non-zero) measures of the parts $A_i$ since, informally speaking,  the price we ``pay'' to get any desired distribution of vertices per parts is multiplicative ${\\mathrm e}^{-O(n)}$, which is negligible for speed~$n^2$.\n\n\\begin{theorem}\\label{th:ourLDP}\n\tLet $W$ be a $k$-step graphon with $k$ non-null parts whose values are encoded by a symmetric $k\\times k$ matrix $\\V p\\in [0,1]^{k\\times k}$. Define\n\t\\beq{eq:R}\n\tR_{\\V p}(\\T U):=\\inf_{\\V\\alpha\\in [0,1]^k\\atop \\alpha_1+\\ldots+\\alpha_k=1} J_{\\V\\alpha,\\V p}(\\T U),\\quad U\\in\\C W.\n\t\\end{equation}\n\tThen the function $R_{\\V p}:\\widetilde{\\mathcal{W}}\\to [0,\\infty]$ is lower semi-continuous with respect to the metric $\\delta_\\Box$. Moreover, the sequence of measures $(\\widetilde{\\mathbb{R}}_{n,W})_{n\\in\\I N}$ on $(\\widetilde{\\mathcal{W}},\\delta_\\Box)$ satisfies an LDP with speed $n^2$ and rate function~$R_{\\V p}$.\n\t\\end{theorem}\n \n\nFor $k=1$, we recover the LDP result of Chatterjee and Varadhan~\\cite{ChatterjeeVaradhan11} (that is, Theorem~\\ref{th:CV}). \n\nInitially, we proved Theorem~\\ref{th:ourLDP} independently of the work by Borgs et al~\\cite{BCGPS}, by first proving an LDP for what we call \\emph{$k$-coloured graphons} (that are defined in Section~\\ref{ColGraphons}). Since our original proof of Theorem~\\ref{th:ourLDP} is quite long and shares many common steps with the proof from~\\cite{BCGPS} (with both being built upon the method of\nChatterjee and Varadhan~\\cite{ChatterjeeVaradhan11}), we decided to derive Theorem~\\ref{th:ourLDP} from the results in~\\cite{BCGPS} with a rather short proof, also strengthening the LDP of Borgs et al~\\cite{BCGPS} in the process.\n\n\\medskip This paper is organised as follows. In Section~\\ref{prelim} we give further definitions (repeating some definitions from the Introduction) and provide some standard or easy results that we will need later. Section~\\ref{ColGraphons} introduces $k$-coloured graphons and proves a compactness result. This result is used in Section~\\ref{Rate} to prove that the functions $J_{\\V a,\\V p}$ and $R_{\\V p}$ are lower semi-continuous. The large deviation principles stated in Theorems~\\ref{th:GenLDP} and~\\ref{th:ourLDP} are proved in Section~\\ref{GenLDP} and~\\ref{ourLDP} respectively.\n\n\\section{Preliminaries}\\label{prelim}\n\nRecall that the relative entropy $h_p$ was defined in~\\eqref{eq:hp} and observe the conventions that $0\\log(0)=0\\log\\left(\\frac{0}{0}\\right)=0$, $h_0(\\rho)=+\\infty$ whenever $\\rho\\not=0$ and $h_1(\\rho)=+\\infty$ whenever $\\rho\\not =1$.\nThe \\emph{indicator function} $\\I 1_X$ of a set $X$ assumes value $1$ for every $x\\in X$ and 0 otherwise.\n\nA measurable space $(\\Omega,\\C A)$ is called \\emph{standard}\nif there is a Polish topology on $\\Omega$ whose Borel $\\sigma$-algebra is equal to~$\\C A$. Given a measure\n$\\mu$ on $(\\Omega,\\C A)$, we call a subset of $\\Omega$ \\emph{measurable} if it belongs to the \\emph{completion} of $\\C A$\nby $\\mu$, that is, the $\\sigma$-algebra generated by $\\C A$ and $\\mu$-null sets.\nWe will usually omit $\\sigma$-algebras from our notation.\n\nUnless specified otherwise, the interval $[0,1]$ of reals is always equipped with the Lebesgue measure, denoted by~$\\lambda$. By $\\lambda^{\\oplus k}$ we denote the completion of the $k$-th power of $\\lambda$, that is, $\\lambda^{\\oplus k}$ is the Lebesgue measure on~$[0,1]^k$\n\nDenote as ${\\bf A}^{(k)}$ the set of all ordered partitions of $[0,1]$ into $k$ measurable sets.\nFor a non-zero vector $\\V\\alpha=(\\alpha_1,\\dots,\\alpha_k)\\in [0,\\infty)^k$, we let ${\\bf A}^{(\\V\\alpha)}\\subseteq{\\bf A}^{(k)}$ to be the set of all ordered partitions of $[0,1]$ into $k$ measurable sets such that the $i$-th set has Lebesgue measure exactly~$\\alpha_i\/\\|\\V\\alpha\\|_1$. Recall that $(\\cI{\\V\\alpha}{1},\\dots,\\cI{\\V\\alpha}{k})\\in{\\bf A}^{(\\V\\alpha)}$ denotes the partition of $[0,1]$ into consecutive intervals whose lengths are given by~$\\V\\alpha\/\\|\\V\\alpha\\|_1$  (where each dividing point is assigned  to e.g.\\ its right interval for definiteness).\n\nWe will also need the following result. \n\\hide{ (whose proof can be found in e.g.~\\cite[Theorem 3.4.23]{Srivastava98cbs}).\n\n\\begin{theorem}[Isomorphism Theorem for Measure Spaces]\\label{th:ISMS}\n\tFor every two atomless standard probability spaces, \n\n\tthere is a measure-preserving Borel isomorphism between them.\n\\end{theorem}\t\n}%\n\n\n\\begin{theorem}\\label{th:ISMS}\n\tFor every two atomless standard measure spaces \n$(\\Omega,\\mu)$ and $(\\Omega',\\mu')$, \nand Borel subsets $A\\subseteq \\Omega$ and $A'\\subseteq \\Omega'$ with $0<\\mu(A)=\\mu'(A')<\\infty$,\n\tthere is a measure-preserving Borel isomorphism between $A$ and~$A'$.\n\\end{theorem}\n\n\\begin{proof} The case when $A=\\Omega$ and $A'=\\Omega'$ amounts to the Isomorphism Theorem for Measure Spaces, whose proof can be found in e.g.~\\cite[Theorem 3.4.23]{Srivastava98cbs}. The general case follows by restricting everything to $A$ and $A'$, and noting that the obtained measure spaces are standard by e.g.\\ \\cite[Theorem 3.2.4]{Srivastava98cbs}.\\end{proof}\t\n\n\n\n\n\\subsection{Graphons}\\label{graphons}\n\n\nA \\emph{graphon} $U$ is a function  $U:[0,1]^2\\to[0,1]$ which is \\emph{symmetric} (that is,\n$U(x,y)=U(y,x)$ for all $x,y\\in [0,1]$) and \\emph{measurable} (that is, for every $a\\in \\I R$, the level set $\\{W\\le a\\}$ is a (Lebesgue) measurable subset of $[0,1]^2$). \nRecall that we denote the set of all graphons by~$\\mathcal{W}$.\nWe define the \\emph{cut-norm} $d_\\Box:\\C W^2\\to [0,1]$ by \n \\beq{eq:CutNorm}\nd_\\Box(U,V):=\\sup_{A,B\\subseteq [0,1]} \\left|\\int_{A\\times B}\\left(U-V\\right) \\,\\mathrm{d}\\lambda^{\\oplus 2}\\right|,\\quad U,V\\in \\mathcal{W},\n\\end{equation}\nwhere the supremum is taken over all pairs of measurable subsets of $[0,1]$.\nFor a function $\\phi:[0,1]\\to [0,1]$ and a graphon $U$, \nthe \\emph{pull-back} $U^\\phi$ of $U$ along $\\phi$ is defined \nby \n$$\n U^{\\phi}(x,y):=U(\\phi(x),\\phi(y)),\\quad x,y\\in [0,1].\n $$\nThe \\emph{cut-distance} $\\delta_\\Box:\\C W^2\\to [0,1]$ can be defined as\n\\beq{eq:CutDistance}\n\\delta_{\\Box}(U,V):=\\inf_{\\phi,\\psi} d_\\Box (U^\\phi,V^\\psi),\n\\end{equation}\nwhere the infimum is taken over all measure-preserving maps $\\phi,\\psi:[0,1]\\to [0,1]$ (then the pull-backs $U^\\phi$ and $V^\\psi$ are necessarily measurable functions).\nSee \\Lo{Section~8.2} for more details and, in particular, \\Lo{Theorem 8.13} for some alternative definitions that give the same distance. It can be easily verified that $\\delta_\\Box$ is a pseudo-metric on~$\\C W$.\nRecall that we denote the metric quotient by $\\widetilde{\\mathcal{W}}$ and  the equivalence class of $U\\in \\mathcal{W}$ by~$\\widetilde{U}$. For $U\\in\\C W$ and $\\eta\\ge 0$, we will denote the closed radius-$\\eta$ ball around $\\T U$ in \n$\\widetilde{\\mathcal{W}}$ by\n$$\n\tS(\\widetilde U,\\eta):= \\left\\{\\T V\\in \\T{\\C W}\\mid \\delta_\\Box(U,V)\\le\\eta\\right\\}.\n$$\n\n\\begin{remark} Without affecting $(\\widetilde{\\mathcal{W}},\\delta_\\Box)$, we could have defined a graphon as a Borel symmetric function $[0,1]^2\\to [0,1]$ and required that the measure-preserving maps in the definition of $\\delta_\\Box$ are Borel. Then some parts could be simplified (for example, the first claim of Lemma~\\ref{lm:AlmostMP} would not be necessary as the function $U^\\phi$ would be Borel for every Borel~$\\phi$). However, \n\t\twe prefer to use the (now standard) conventions from Lov\\'asz' book~\\cite{Lovasz:lngl}.\t\t\t\n\\end{remark}\n\n\t\nWe will need the following auxiliary result.\n\n\n\n\\begin{lemma}\\label{lm:AlmostMP}\n\tLet $U$ be a graphon and $\\phi:[0,1]\\to [0,1]$ be a measurable function such that the \\emph{push-forward measure} $\\phi_*\\lambda$ (defined by $(\\phi_*\\lambda)(X):=\\lambda(\\phi^{-1}(X))$ for measurable $X\\subseteq [0,1]$) satisfies $\\phi_*\\lambda\\ll \\lambda$, that is, \n is absolutely continuous with respect to the Lebesgue measure~$\\lambda$. Then  $U^\\phi$ is a graphon. Moreover, \n\tif the Radon-Nikodym derivative $D:=\\frac{\\,\\mathrm{d} (\\phi_*\\lambda)}{\\,\\mathrm{d} \\lambda}$ satisfies $D(x)\\le 1+\\varepsilon$ for a.e.\\ \n\t$x\\in[0,1]$ then\n\t\\beq{eq:AlmostMP}\n\t\\delta_\\Box(U,U^\\phi)\\le 2\\varepsilon.\n\t\\end{equation}\n\\end{lemma}\n\n\\begin{proof} \n\t\tThe function $U^\\phi:[0,1]^2\\to [0,1]$ is clearly symmetric so we have to show that it is measurable.\n\tSince the pre-image under $\\phi$ of any $\\lambda$-null set is again $\\lambda$-null by our assumption $\\phi_*\\lambda\\ll \\lambda$, there is a Borel map  $\\psi:[0,1]\\to[0,1]$  such that the set \n\t$$X:=\\{x\\in [0,1]:\\psi(x)\\not=\\phi(x)\\}$$ \n\tis $\\lambda$-null. (For a proof, see e.g.~\\cite[Proposition~2.2.5]{Cohn13mt}.)\nTake any $\\rho\\in\\I R$. The set \n$$\n A:=\\{(x,y)\\in [0,1]^2\\mid U(x,y)\\le \\rho\\}\n $$ \n is measurable so by e.g.~\\cite[Proposition~1.5.2]{Cohn13mt} there are $B,N\\subseteq [0,1]^2$ such that $B$ is Borel, $N$ is $\\lambda^{\\oplus 2}$-null and $A\\bigtriangleup B\\subseteq N$, where $A\\bigtriangleup B:=(A\\setminus B)\\cup (B\\setminus A)$ denotes the \\emph{symmetric difference} of the sets $A$ and $B$. The pre-image of $N$ under the Borel map $\\psi^{\\oplus2}(x,y):=(\\psi(x),\\psi(y))$ is also $\\lambda^{\\oplus 2}$-null for otherwise\n this would contradict the absolute continuity $(\\phi_*\\lambda)^{\\oplus 2}\\ll \\lambda^{\\oplus 2}$ (which follows from\n$\\phi_*\\lambda\\ll \\lambda$ by the Fubini-Tonelli Theorem for Complete Measures).\n Thus the level set \n $\\{U^\\phi\\le \\rho\\}$\n\n is Lebesgue measurable since its symmetric difference with the Borel set $(\\psi^{\\oplus2})^{-1}(B)$ is a subset of the null set $(\\psi^{\\oplus2})^{-1}(N)\\cup (X\\times [0,1])\\cup ([0,1]\\times X)$. As $\\rho\\in\\I R$ was arbitrary, $U^\\phi$ is a measurable function and thus a graphon.\n\n\nFor the second part, it will be convenient to use the following generalisation of a graphon (which will not used anywhere else in this paper except this proof).\nNamely, by\na \\emph{generalised graphon} we mean a triple $(V,\\Omega,\\mu)$ where $(\\Omega,\\mu)$ is an atomless standard probability space  and $V:(\\Omega^2,\\mu^{\\oplus 2})\\to [0,1]$ is a symmetric measurable function. In the special case $(\\Omega,\\mu)=([0,1],\\lambda)$ we get our notion of a graphon from the Introduction. Most definitions\nand results extend with obvious modifications from graphons to generalised graphons  (see \\Lo{Chapter~13.1} for details).\nIn particular, we will need the facts that if $(V,\\Omega,\\mu)$ is a generalised graphon and\n$\\phi: (\\Omega',\\mu')\\to(\\Omega,\\mu)$ is a measure-preserving  map between standard probability spaces,\nthen the function $V^\\phi$ is measurable (which can be proved by adapting the proof of the first part of the lemma) and\n\\beq{eq:51}\n\\delta_\\Box((V,\\Omega,\\mu),(V^\\phi, \\Omega',\\mu'))=0,\n\\end{equation}\nwhere we define $V^\\phi(x,y):=V(\\phi(x),\\phi(y))$ for $x,y\\in \\Omega'$ and $\\delta_\\Box$ is the extension of the cut-distance to generalised graphons via the obvious analogues of~\\eqref{eq:CutNorm} and~\\eqref{eq:CutDistance}.  \n\n\nLet us return to the proof of the second part of the lemma.\n\tWe can assume that the set $\\{x\\in [0,1]\\mid D(x)\\not=1\\}$ has positive measure for otherwise $\\phi$ is a measure-preserving map and $\\delta_\\Box(U,U^\\phi)=0$ by~\\eqref{eq:51}, as required. \n\tLet $(V,[0,1]^2,\\lambda^{\\oplus 2})$ be the generalised graphon defined by $V((x,y),(x',y')):=U(x,x')$, for $x,y,x',y'\\in [0,1]$. Thus $V=U^\\pi$, where\n\t$\\pi:[0,1]^2\\to [0,1]$ is the (measure-preserving) projection on the first coordinate. \n\tBy~\\eqref{eq:51}, it holds that\n\t$$\n\t\\delta_\\Box ((V,[0,1]^2,\\lambda^{\\oplus 2}),U)=0.\n\t$$\n\t\n\tBy changing $D$ on a $\\lambda$-null set, we can make it a Borel function with $D(x)\\le 1+\\varepsilon$ for every $x\\in [0,1]$. Then\n\t$$\n\t\\Omega:=\\{(x,y)\\in [0,1]\\times \\I R\\mid 0\\le y\\le D(x)\\}\n\t$$ \n\tis a Borel subset of $[0,1]\\times \\I R$ (see e.g.\\ \\cite[Example 5.3.1]{Cohn13mt}) and thus induces a standard measurable space. Let $\\mu$ be the restriction of the Lebesgue measure on $\\I R^2$ to~$\\Omega$. Define $W:\\Omega^2\\to [0,1]$ by $W((x,y),(x',y')):=U(x,x')$ for $(x,y),(x',y')\\in \\Omega$. Thus $U^\\phi$ and $(W,\\Omega,\\mu)$ are measure-preserving pull-backs of the generalised graphon $(U,[0,1],\\phi_*\\mu)$ \nalong respectively the map $\\phi$ and the projection $\\Omega\\to[0,1]$ on the first coordinate.\n\t Therefore we have by~\\eqref{eq:51} and the Triangle Inequality for $\\delta_\\Box$ that\n\t$$\n\\delta_\\Box(U^\\phi,(W,\\Omega,\\mu))\\le \\delta_\\Box(U^\\phi,(U,[0,1],\\phi_*\\mu))+\\delta_\\Box((U,[0,1],\\phi_*\\mu),(W,\\Omega,\\mu))=0.\n$$  \n\t\n\tThus it suffices to show that the cut-distance $\\delta_\\Box$ between $(V,[0,1]^2,\\lambda^{\\oplus 2})$ and $(W,\\Omega,\\mu)$ is at most~$2\\varepsilon$.\n\tThe functions $V$ and $W$ and the measures $\\lambda^{\\oplus 2}$ and $\\mu$\n\tcoincide on~$X^2$, where $X:=[0,1]^2\\cap \\Omega$. \n\tSince $D\\not= 1$ on a set of positive measure, it holds that $\\mu(X)<1$. \nThe Borel subsets $[0,1]^2\\setminus X$ and $\\Omega\\setminus X$ of $\\I R^2$ have the same positive Lebesgue measure and thus,\n\n\tby Theorem~\\ref{th:ISMS}, there is a Borel measure-preserving bijection $\\psi$ between them. By letting $\\psi$ be the identity function on $X$, we get a Borel measure-preserving bijection $\\psi:\\Omega\\to [0,1]^2$. \n\tSince $D\\le 1+\\varepsilon$, we have that $\\Omega\\setminus X\\subseteq [0,1]\\times [1,1+\\varepsilon]$ has measure at most~$\\varepsilon$. \t\nThe function $W$ and the pull-back $V^\\psi$, as maps $\\Omega^2\\to [0,1]$, coincide on the set $X^2$ of measure at least $(1-\\varepsilon)^2\\ge 1-2\\varepsilon$. It follows that the $d_\\Box$-distance between them at most $2\\varepsilon$. (Indeed, when we compute it via the analogue of~\\eqref{eq:CutNorm}, the integrand is bounded by 1 in absolute value and is non-zero on a set of measure at most~$2\\varepsilon$.) This finishes the proof of the lemma.\\end{proof}\n\nInformally speaking, the following result states that if we delete a small subset of $[0,1]$ and stretch the rest of a graphon uniformly then the new graphon is close to the original one.\n\n\\begin{lemma}\\label{lm:Delete01}\n\tLet $U\\in \\C W$ be a graphon, $s\\in (0,1]$ be a non-zero real and $\\phi:[0,1]\\to [0,1]$ be the map that sends $x$ to~$sx$. \n\tThen $U^\\phi$ is a graphon and $\\delta_\\Box(U,U^\\phi)\\le 2(\\frac1s -1)$.\n\\end{lemma}\n\\begin{proof}  Clearly, the push-forward $\\phi_*\\lambda$ is the uniform probability measure on $[0,s]$ so the Radon-Nikodym derivative $\\frac{\\,\\mathrm{d}( \\phi_*\\lambda)}{\\,\\mathrm{d}\\lambda}$ is a.e.\\ $1\/s$ on~$[0,s]$ and $0$ on~$[s,1]$. The result now follows from Lemma~\\ref{lm:AlmostMP}. \n\\end{proof} \n\n\n\n\n\n\n\n\n\\subsection{Large deviations for compact metric spaces}\\label{LDP}\n\nRecall that the definition of a large deviation principle was given in Definition~\\ref{df:LDP}. Since we will be dealing with LDPs for compact metric spaces only, we may use the following alternative characterisation that follows from \\DZ{Theorems 4.1.11 and 4.1.18} (see also~\\RS{Exercise~2.24}). \n\n\\begin{lemma}\\label{lm:LDP}\n\tLet $(X,d)$ be  a compact metric space, $s:\\I N\\to (0,\\infty)$ satisfy $s(n)\\to\\infty$, and $I:X\\to [0,\\infty]$ \n\tbe a lower semi-continuous function on~$(X,d)$. Then\n\ta sequence of Borel probability measures $(\\mu_n)_{n\\in \\I N}$ on $(X,d)$ satisfies an LDP \n\twith speed $s$\n\tand rate function $I$\n\tif and only \n\t\\begin{eqnarray}\n\t\t\\lim_{\\eta\\to 0}\\liminf_{n\\to\\infty} \\frac1{s(n)}\\,{\\log\\Big(\\mu_n\\big(\n\t\t\t\\{y\\in X\\mid d(x,y)\\le \\eta\\}\n\t\t\t\\big)\\Big)} \n\t\t&\\ge& -I(x),\\quad\\mbox{for every $x\\in X$,}\\label{eq:lower}\\\\\n\t\t\\lim_{\\eta\\to 0}\\limsup_{n\\to\\infty} \\frac1{s(n)}\\,{ \\log\\Big(\\mu_n\\big(\n\t\t\t\\{y\\in X\\mid d(x,y)\\le \\eta\\}\n\t\t\t\\big)\\Big)} &\\le & -I(x),\\quad\\mbox{for every $x\\in X$}.\\label{eq:upper}\n\t\\end{eqnarray}\n\\end{lemma}\n\t\n\tIn fact, under the assumptions of Lemma~\\ref{lm:LDP}, the bounds~\\eqref{eq:lowerGen} and~\\eqref{eq:lower} (resp.\\ \\eqref{eq:upperGen} and~\\eqref{eq:upper}) are equivalent to each other. So we will also refer to \n\t\\eqref{eq:lower} and~\\eqref{eq:upper}\n\tas the \\emph{lower bound} and the \\emph{upper bound} respectively.\n\n\n\n\n\n\n\n\n\n\n\\section{Coloured graphons}\\label{ColGraphons}\n\nThe definitions and results of this section are needed in order to establish the lower semi-continuity of the functions $J_{\\V \\alpha,\\V p}$ and~$R_{\\V p}$.\n\nFix $k\\in \\mathbb{N}$.\nBy a \\emph{$k$-coloured graphon} we mean a pair $(W,\\mathcal{A})$ where $W\\in \\mathcal{W}$ and $\\mathcal{A}\\in {\\bf A}^{(k)}$. (One can view the partition $\\mathcal{A}$ as a $k$-colouring of $[0,1]$.)\nWrite $\\mathcal{W}^{(k)}:=\\C W\\times {\\bf A}^{(k)}$ for the space of all $k$-coloured graphons.\nWe define the pseudo-metric $d^{(k)}_\\Box$ (the analogue of the cut norm $d_\\Box$) on $\\mathcal{W}^{(k)}$ as\n\\begin{equation*}\nd^{(k)}_\\Box((U,\\mathcal{A}),(V,\\mathcal{B})):=  \n\\sup_{C,D\\subseteq [0,1]} \\sum_{i,j\\in [k]}\\left| \\int_{C\\times D} ({\\I 1}_{A_i\\times A_j}U-{\\I 1}_{B_i\\times B_j}V) \\,\\mathrm{d}\\lambda^{\\oplus 2}\\right|\n + \\sum_{i\\in [k]}\\lambda(A_i\\triangle B_i),\n\\end{equation*}\n for $(U,\\mathcal{A}),(V,\\mathcal{B})\\in \\mathcal{W}^{(k)}$, where $\\C A=(A_1,\\dots,A_k)$ and $\\C B=(B_1,\\dots,B_k)$.\n Informally speaking, two $k$-coloured graphons are close to each other in $d^{(k)}$ if\n they have similar distributions of coloured edges across cuts, where an edge is coloured by the colours of its endpoints. The second term is added so that e.g.\\ we can distinguish two constant-0 graphons with different part measures.\n \nThe \\emph{cut distance} for coloured graphons is then defined as\n\\beq{eq:DeltaK}\n\\delta^{(k)}_\\Box((U,\\mathcal{A}),(V,\\mathcal{B})):=\\inf_{\\phi,\\psi} d^{(k)}_\\Box((U,\\mathcal{A})^\\phi,(V,\\mathcal{B})^\\psi),\n\\end{equation}\nwhere the infimum is taken over measure-preserving maps $\\phi,\\psi:[0,1]\\to [0,1]$ and we denote\n$(U,\\mathcal{A})^\\phi:=(U^\\phi,\\C A^\\phi)$ and $\\C A^\\phi:=(\\phi^{-1}(A_1),\\dots,\\phi^{-1}(A_k))$ \nwith $(A_1,\\dots,A_k)$ being the parts of~$\\C A$.\nAs in the graphon case (compare with e.g.\\ \\Lo{Theorem 8.13}), some other definitions give the same distance (e.g.\\ it is enough to take the identity function for $\\psi$). We chose this definition as it is immediately clear from it that the function $\\delta^{(k)}_\\Box$ is symmetric and defines a pseudo-metric.\nWe write $\\widetilde{\\mathcal{W}}^{(k)}$ for the corresponding quotient, where we identify two $k$-coloured graphons \nat $\\delta^{(k)}_\\Box$-distance~0.\n\n\\begin{theorem}\\label{th:compact colorod graphon}\nThe metric space $(\\widetilde{\\mathcal{W}}^{(k)},\\delta^{(k)}_\\Box)$ is compact. \n\\end{theorem}\n\\begin{proof}\nThe proof is obtained by the obvious adaptation of the proof of Lov\\'asz and Szegedy~\\cite[Theorem~5.1]{LovaszSzegedy07gafa} (see also~\\cite[Theorem 9.23]{Lovasz:lngl}) that the space $(\\widetilde{\\mathcal{W}},\\delta_\\Box)$ is compact.\n\nLet $(W_n,\\mathcal A_n)_{n\\in\\mathbb N}$ be an arbitrary sequence of elements of~$\\mathcal{W}^{(k)}$. We have to find a subsequence \nthat converges to some element of $\\mathcal{W}^{(k)}$ with respect to~$\\delta_\\Box^{(k)}$. \n\nWhen dealing with the elements of $\\mathcal{W}^{(k)}$,\nwe can ignore null subsets of $[0,1]$; thus all relevant statements, e.g.\\ that one partition refines another, are meant to hold almost everywhere.\n\nFor $n\\in\\mathbb N$, let the parts of $\\mathcal A_n$ be $(A_{n,1},\\dots,A_{n,k})$ and, by applying a measure-preserving bijection to $(W_n,\\mathcal A_n)$, assume by Theorem~\\ref{th:ISMS} that the colour classes $A_{n,1},\\dots,A_{n,k}\\subseteq [0,1]$ are all intervals, \ncoming in this order. By passing to a subsequence, assume that, for each $i\\in [k]$, the length of $A_{n,i}$ converges to some $\\alpha_i\\in [0,1]$ as $n\\to\\infty$. With $\\V\\alpha:=(\\alpha_1,\\dots,\\alpha_k)$, this gives rise to the ``limiting'' partition \n$$\n\\mathcal A\n:=(\\cI{\\V\\alpha}{1},\\dots,\\cI{\\V\\alpha}{k})\n\\in{\\bf A}^{(\\V\\alpha)}\n$$ of $[0,1]$ into intervals. \n\n\nLet $m_1:=k$ and inductively for $\\ell=2,3,\\ldots$ let $m_\\ell$ be sufficiently large such that for every graphon $W$ and a measurable partition $\\mathcal A'$ of $[0,1]$ with $|\\mathcal A'|\\le m_{\\ell-1}$ there is a measurable partition $\\mathcal P=(P_1,\\dots,P_m)$ of $[0,1]$ refining $\\mathcal A'$ such that $m\\le m_\\ell$ and $d_{\\Box}(W,W_{\\mathcal P})\\le 1\/\\ell$. Here $W_{\\mathcal P}$ denotes the projection of $W$ to the space of $\\C P$-step graphons; namely, for every $i,j\\in [m]$ with $P_i\\times P_j$ non-null in $\\lambda^{\\oplus 2}$, $W_{\\C P}$ assumes the constant value $\\frac1{\\lambda(P_i)\\lambda(P_j)}\\int_{P_i\\times P_j} W\\,\\mathrm{d}\\lambda^{\\oplus 2}$ on $P_i\\times P_j$ (and, say, $W_{\\C P}$ is defined to be 0 on all $\\lambda^{\\oplus 2}$-null products $P_i\\times P_j$).\nSuch a number $m_\\ell$\nexists by~\\cite[Lemma 9.15]{Lovasz:lngl}, a version of the Weak Regularity Lemma for graphons.\n \nFor each $n\\in\\mathbb N$, we do the following. Let $\\mathcal P_{n,1}:=\\mathcal A_n$ and, inductively on $\\ell=2,3,\\dots$, let $\\mathcal P_{n,\\ell}$ be the partition with at most $m_{\\ell}$ parts obtained by applying the above Weak Regularity Lemma to $(W_n,\\mathcal P_{n,\\ell-1})$. By adding empty parts to $\\mathcal P_{n,\\ell}$, for each $\\ell\\ge 1$, we can assume that it has the same number of parts (namely, $m_\\ell$) of each colour, that is, we can denote its parts as $(P_{n,\\ell,i,j})_{i\\in [k], j\\in [m_\\ell]}$, so that $P_{n,\\ell,i,j}\\subseteq A_{n,i}$ for all $(i,j)\\in [k]\\times [m_\\ell]$.\nAlso, define $W_{n,\\ell}:=(W_n)_{\\mathcal P_{n,\\ell}}$ to be the projection of the graphon $W_n$ on the space of $\\mathcal P_{n,\\ell}$-step graphons.\n\n\nThen, iteratively for $\\ell=2,3,\\dots$, repeat the following. Find a measure-preserving bijection $\\phi:[0,1]\\to[0,1]$ such that $(\\mathcal P_{n,\\ell})^\\phi$ is a partition into intervals and $\\phi$ preserves the previous partitions\n$\\mathcal P_{n,1},\\dots,\\mathcal P_{n,\\ell-1}$  (each of which is a partition into intervals by induction on~$\\ell$). Then, for each $m\\ge \\ell$, replace $(W_{n,m},\\mathcal P_{n,m})$ by $(W_{n,m},\\mathcal P_{n,m})^\\phi$. When we are done with this step, the following properties hold for each integer~$\\ell\\ge 2$:\n\\begin{itemize}\n\t\\item\\label{it:P1} $\\delta_\\Box^{(k)}((W_{n,\\ell},\\mathcal A_n),(W_n,\\mathcal A_n))\\le 1\/\\ell$;\n\t\\item\\label{it:P3}\nThe partition $\\mathcal P_{n,\\ell}$ refines $\\mathcal P_{n,\\ell-1}$ (and, inductively, also refines $\\mathcal A_{n}=\\mathcal P_{n,1}$);\n\t\\item\\label{it:P2} $|\\mathcal P_{n,\\ell}|= k m_\\ell$ with exactly $m_\\ell$ parts assigned to each colour class of $\\mathcal A_n$.\n\\end{itemize}\n\n\n\nNext, iteratively for $\\ell=1,2,\\dots$, we pass to a subsequence of $n$ so that for every $(i,j)\\in [k]\\times [m_\\ell]$, the length of the interval $P_{n,\\ell,i,j}$ converges, and for every pair $(i,j),(i',j')\\in [k]\\times [m_\\ell]$, the common value of the step-graphon $W_{n,\\ell}$ on $P_{n,\\ell,i,j}\\times P_{n,\\ell,i',j'}$ converges.\n\tIt follows that the sequence $W_{n,\\ell}$ converges pointwise to some graphon $U_\\ell$ which is itself a step-function with $km_\\ell$ parts that are intervals. We use diagonalisation to find a subsequence of $n$ so that, for each $\\ell\\in\\mathbb N$, $W_{n,\\ell}$ converges to some step-graphon $U_\\ell$ a.e.\\ as $n\\to\\infty$, with the step partition $\\mathcal P_\\ell$ of  $U_\\ell$  consisting of $km_\\ell$ intervals and refining the partition~$\\C A$. \n\\hide{Moreover, \twe can denote the parts of $\\mathcal P_\\ell$ as\n\t$(P_{\\ell,i,j})_{i\\in[k], m\\in[m_\\ell]}$, so that $P_{\\ell,i,1},\\dots,P_{\\ell,i,m_\\ell}$ form a partition of~$\\cI{\\V\\alpha}{i}$ for each $i\\in [k]$.\n}\t\n\t\n\t\nIt follows that, for all $s<t$ in $\\I N$, the partition $\\mathcal P_{t}$ is a refinement of $\\mathcal P_s$ and, moreover,  $U_s$ is the conditional expectation $\\mathbb E[U_t|\\mathcal P_s]$ a.e. \nAs observed in the proof of Lov\\'asz and Szegedy~\\cite[Theorem~5.1]{LovaszSzegedy07gafa},\nthis (and $0\\le U_t\\le 1$ a.e.) implies by the Martingale Convergence Theorem that $U_\\ell$ converge a.e.\\ to some graphon $U$ as $\\ell\\to\\infty$. By the Dominated Convergence Theorem, $U_\\ell\\to U$ also in the $L^1$-distance.\n\nWe claim that $\\delta_\\Box((W_n,\\mathcal A_n),(U,\\mathcal A))\\to 0$ as $n\\to\\infty$ (after we passed to the subsequence defined as above). Take any $\\varepsilon>0$. Fix an integer $\\ell>4\/\\varepsilon$ such that $\\|U-U_\\ell\\|_1\\le \\varepsilon\/4$. Given $\\ell$, fix $n_0$ such that  for all $n\\ge n_0$ we have $\\|U_\\ell-W_{n,\\ell}\\|_1\\le \\varepsilon\/4$ and $\\sum_{i=1}^k \\lambda(\\cI{\\V\\alpha}{i}\\bigtriangleup A_{n,i})\\le \\varepsilon\/4$. Then, for every $n\\ge n_0$ we have\n\\begin{eqnarray*}\n\\delta_\\Box^{(k)}((U,\\mathcal A),(W_n,\\mathcal A_n))&\\le &\nd_\\Box^{(k)}((U,\\mathcal A),(U_\\ell,\\mathcal A))\n+ d_\\Box^{(k)}((U_\\ell,\\mathcal A),(W_{n,\\ell},\\mathcal A_n))\\\\\n&+& \\delta_\\Box^{(k)}((W_{n,\\ell},\\mathcal A_n),(W_n,\\mathcal A_n))\\\\\n&\\le& \\|U-U_\\ell\\|_1 + \\|U_\\ell-W_{n,\\ell}\\|_1 + \\sum_{i=1}^k \\lambda(\\cI{\\V\\alpha}{i}\\bigtriangleup A_{n,i}) + 1\/\\ell\\\\\n&\\le& \\varepsilon\/4+\\varepsilon\/4+\\varepsilon\/4+\\varepsilon\/4\\ =\\ \\varepsilon.\n\\end{eqnarray*}\n Since $\\varepsilon>0$ was arbitrary, the claim is proved. Thus the metric space $(\\widetilde{\\mathcal{W}}^{(k)},\\delta^{(k)}_\\Box)$ is indeed compact.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The lower semi-continuity of $J_{\\V\\alpha,\\V p}$ and $R_{\\V p}$}\\label{Rate}\n\nFor this section we fix an integer $k\\ge 1$, a symmetric $k\\times k$ matrix $\\V p=(p_{i,j})_{i,j\\in [k]}\\in [0,1]^{k\\times k}$ and a non-zero real vector $\\V\\alpha\\in [0,\\infty)^k$.\nWe show that the functions $J_{\\V\\alpha,\\V p}$ and  $R_{\\V p}$\nare lower semi-continuous functions from $(\\widetilde{\\mathcal{W}},\\delta_\\Box)$ to $[0,+\\infty]$.\n\nLet $\\Gamma:\\mathcal{W}^{(k)}\\to \\C W$ be the map that forgets the colouring, i.e., $\\Gamma(U,\\mathcal{A}):=U$\nfor $(U,\\C A)\\in \\mathcal{W}^{(k)}$.\nFor $i,j\\in [k]$, let the map $\\Gamma_{i,j}:\\mathcal{W}^{(k)}\\to \\C W$ send $(U,(A_1,\\dots,A_k))\\in \\mathcal{W}^{(k)}$  to the graphon $V$ defined as\n$$\nV(x,y):=\\left\\{\\begin{array}{ll} \n\tU(x,y),& (x,y)\\in (A_i\\times A_j)\\cup (A_j\\times A_i),\\\\\n\tp_{i,j},& \\mbox{otherwise,}\n\\end{array}\n\\right.\\quad \\mbox{for $x,y\\in [0,1]$.}\n$$\n\n\\begin{lemma}\\label{lm:Lipschitz}\n\tThe maps $\\Gamma$ and $\\Gamma_{i,j}$, for ${i,j\\in [k]}$, are $1$-Lipschitz maps from $(\\mathcal{W}^{(k)},d_\\Box^{(k)})$ to $(\\C W,d_\\Box)$.\n\\end{lemma}\n\\begin{proof}\n\tFirst, consider $\\Gamma:\\mathcal{W}^{(k)}\\to \\mathcal{W}$. Take arbitrary $(U,\\mathcal{A}),(V,\\mathcal{B})\\in \\mathcal{W}^{(k)}$.\n\tLet $\\C A=(A_1,\\dots,A_k)$ and $\\mathcal{B}=(B_1,\\dots,B_k)$. Clearly, the pairwise products $A_i\\times A_j$ (resp.\\ $B_i\\times B_j$) for $i,j\\in [k]$ partition $[0,1]^2$. Thus\n\twe have\n\t\\begin{eqnarray}\t\t\n\t\t\td_\\Box(\\Gamma(U,\\mathcal{A}),\\Gamma(V,\\mathcal{B})) &= & d_\\Box(U,V) \\\n\t\t\t= \\ \\sup_{C,D\\subseteq [0,1]} \\left |\\int_{C\\times D} (U-V) \\,\\mathrm{d}\\lambda^{\\oplus 2}\\right |\\nonumber \\\\\n\t\t\t&\\le & \\sup_{C,D\\subseteq [0,1]}\\sum_{i,j\\in [k]}\\left|\\int_{C\\times D} (U\\,{\\I 1}_{A_i\\times A_j}-V\\,\\I 1_{B_i\\times B_j}) \\,\\mathrm{d}\\lambda^{\\oplus 2}\\right| \\nonumber\\\\\n\t\t\t&\\le &  d_{\\Box}^{(k)}((U,\\mathcal{A}),(V,\\mathcal{B})).\\label{eq:Ga}\n\t\t\t\\end{eqnarray}\n\tThus the function $\\Gamma$ is indeed $1$-Lipschitz.\n\n\t\n\tThe claim about $\\Gamma_{i,j}$ follows by observing that\n\t$$d_\\Box(\\Gamma_{i,j}(U,\\mathcal{A}),\\Gamma_{i,j}(V,\\mathcal{B}))\\le d_\\Box(\\Gamma(U,\\mathcal{A}),\\Gamma(V,\\mathcal{B}))$$\n\tfor every $(U,\\mathcal{A}),(V,\\mathcal{B})\\in\\mathcal{W}^{(k)}$.\\end{proof}\n\n\n\n\\begin{lemma}\\label{lm:Cont}\n\tLet $F$ be $\\Gamma$ or $\\Gamma_{i,j}$ for some $i,j\\in [k]$.\n\tThen $F$  gives rise to a well-defined function $\\widetilde{\\mathcal{W}}^{(k)}\\to\\widetilde{\\mathcal{W}}$ which, moreover, is $1$-Lipschitz\n\tas a function from $(\\widetilde{\\mathcal{W}}^{(k)},\\delta_\\Box^{(k)})$ to~$(\\widetilde{\\mathcal{W}},\\delta_\\Box)$.\n\\end{lemma}\n\\begin{proof} \n\t\n\tTake any $(U,\\C A),(V,\\C B)\\in\\mathcal{W}^{(k)}$. Let $\\varepsilon>0$ be arbitrary. Fix measure-preserving maps $\\phi,\\psi:[0,1]\\to [0,1]$ with $d_\\Box^{(k)}((U,\\C A)^\\psi,(V,\\C B)^\\phi)<\\delta_\\Box^{(k)}((U,\\C A),(V,\\C B))+\\varepsilon$. By Lemma~\\ref{lm:Lipschitz}, \nwe have \n\\begin{eqnarray*}\n\t\\delta_\\Box\\left(F(U,\\C A),F(V,\\C B)\\right)&\\le& \n\td_\\Box\\left((F(U,\\C A))^\\psi,(F(V,\\C B))^\\phi\\right)\\ =\\\n\td_\\Box\\left(F(U^\\psi,\\C A^\\psi),F(V^\\phi,\\C B^\\phi)\\right)\\\\\n\t&\\le& d_\\Box^{(k)}((U^\\psi,\\C A^\\psi),(V^\\phi,\\C B^\\phi))\n\n\t\\ <\\ \\delta_\\Box^{(k)}\\left((U,\\C A),(V,\\C B)\\right)+\\varepsilon.\n\\end{eqnarray*}\nThis implies both claims about $F$ as $\\varepsilon>0$ was arbitrary.\n\\end{proof}\n\n\n\nFor $(U,(A_1,\\dots,A_k))\\in \\mathcal{W}^{(k)}$, define\n\\beq{eq:DefIkp}\nI^{(k)}_{\\V p}(U,(A_1,\\dots,A_k)):=\\frac12\\sum_{i,j\\in [k]} \\int_{A_i\\times A_j} I_{p_{i,j}}(U) \\,\\mathrm{d}\\lambda^{\\oplus 2}.\n\\end{equation}\n\n\tIn the special case of~\\eqref{eq:DefIkp}  when $k=1$ and $p_{1,1}=p$ (and we ignore the second component since ${\\bf A}^{(1)}$ consists of just the trivial partition of $[0,1]$ into one part), we get the function\n$I_p:\\C W\\to[0,\\infty]$ of Chatterjee and Varadhan defined in~\\eqref{eq:IpCV}.\n\n\\begin{lemma}\\label{lm:lsc}\n\tThe function $I^{(k)}_{\\V p}$ gives a well-defined function $\\widetilde{\\mathcal{W}}^{(k)}\\to [0,+\\infty]$ which, moreover, is lower semi-continuous as a function on $(\\widetilde{\\mathcal{W}}^{(k)},\\delta^{(k)}_\\Box)$.\n\\end{lemma}\n\\begin{proof}\nNote that we can write\n\t\\beq{eq:IkWCV}\n\tI^{(k)}_{\\V p}(U,\\mathcal{A})=\\sum_{1\\le i\\le j\\le k} I_{p_{i,j}}(\\Gamma_{i,j}(U,\\mathcal{A})),\\quad (U,\\C A)\\in\\mathcal{W}^{(k)},\n\t\\end{equation}\n\tbecause $\\Gamma_{i,j}(U,\\C A)$ assumes value $p_{i,j}$ outside of $(A_i\\times A_j)\\cup (A_j\\times A_i)$ while $I_p(p)=0$ for any $p\\in [0,1]$. Recall that, by Theorem~\\ref{th:CV},\n\n\t$I_p$ gives a well-defined function $\\widetilde{\\mathcal{W}}\\to[0,\\infty]$ for every $p\\in [0,1]$. \n\tThus, by Lemma~\\ref{lm:Cont}, the right-hand side of~\\eqref{eq:IkWCV} does not change if we replace $(U,\\C A)$ by any other element of $\\mathcal{W}^{(k)}$ at $\\delta_\\Box^{(k)}$-distance 0. We conclude that $I_{\\V p}^{(k)}$ gives a well-defined function on~$\\widetilde{\\mathcal{W}}^{(k)}$.\n\t\n\tEach composition $I_p\\circ \\Gamma_{i,j}$ \n\n\tis lsc as a function $(\\widetilde{\\mathcal{W}}^{(k)},\\delta_{\\Box}^{(k)})\\to [0,\\infty]$\t\n\tbecause, for every $\\rho\\in\\I R$, the level set $\\{I_p\\circ \\Gamma_{ij}\\le \\rho\\}$ is closed as the pre-image under the continuous function $\\Gamma_{i,j}:\\widetilde{\\mathcal{W}}^{(k)}\\to\\widetilde{\\mathcal{W}}$ of the closed \n\tset~$\\{I_p\\le \\rho\\}$.\n\n\t(Recall that the function $I_p:\\widetilde{\\mathcal{W}}\\to [0,\\infty]$ is lsc by Theorem~\\ref{th:CV}.)\n\tThus $I^{(k)}_{\\V p}:\\widetilde{\\mathcal{W}}^{(k)}\\to[0,\\infty]$ is lsc by~\\eqref{eq:IkWCV}, as a finite sum of lsc functions.\n\\end{proof}\n\n\nNow we are ready to show that $J_{\\V\\alpha,\\V p}$ and $R_{\\V p}$ are lsc (in particular, proving Theorem~\\ref{th:JLSC}). The argument showing the lower semi-continuity of these functions is motivated by the Contraction Principle (see e.g.\\ \\DZ{Theorem 4.2.1} or \\RS{Section~3.1}). \n\n\\begin{corollary}\\label{cr:lsc}\n\tFor every symmetric matrix $\\V p\\in [0,1]^{k\\times k}$ and every non-zero real vector $\\V\\alpha\\in [0,\\infty)^k$, the functions $J_{\\V\\alpha,\\V p}$ and $R_{\\V p}$ are lower semi-continuous on $(\\widetilde{\\mathcal{W}},\\delta_\\Box)$.\t\n\\end{corollary}\n\n\\begin{proof}\n\tNote that $J_{\\V\\alpha,\\V p}(\\T U)$ for any $U\\in\\C W$ is equal to the infimum of $I_{\\V p}^{(k)}(V,\\C A)$ over all $(V,\\C A)\\in \\mathcal{W}\\times {\\bf A}^{(\\V\\alpha)}$ such that $\\Gamma(V,\\C A)=V$ belongs to~$\\T U$. Indeed, for any partition $\\C A\\in {\\bf A}^{(\\V\\alpha)}$ of $[0,1]$ one can find by \n\tTheorem~\\ref{th:ISMS}\n\n\ta measure-preserving Borel bijection\t$\\phi$ of $[0,1]$ such that $\\C A^\\phi$ is equal a.e.\\ to $(\\cI{\\V\\alpha}{1},\\dots,\\cI{\\V\\alpha}{k})$\n\t\n\t\n\tIn the rest of the proof, let us view $\\Gamma$ and $I^{(k)}_{\\V p}$ as functions on $\\widetilde{\\mathcal{W}}^{(k)}$ (by Lemmas~\\ref{lm:Cont} and~\\ref{lm:lsc}). Thus we have\n\t\\begin{equation}\\label{eq:Jap2}\n\tJ_{\\V\\alpha,\\V p}(\\T U)=\\inf_{\\Gamma^{-1}(\\T U)\\cap\\widetilde{\\mathcal{W}}^{(\\V\\alpha)}}\\, I_{\\V p}^{(k)},\n\\quad \\mbox{for each $ U\\in\\C W$},\n\t\\end{equation}\n where $\\widetilde{\\mathcal{W}}^{(\\V\\alpha)}\\subseteq \\widetilde{\\mathcal{W}}^{(k)}$ denotes the set of all $\\delta_\\Box^{(k)}$-equivalence classes that intersect $\\mathcal{W}\\times {\\bf A}^{(\\V\\alpha)}$ (equivalently, lie entirely inside $\\mathcal{W}\\times {\\bf A}^{(\\V\\alpha)}$).\n \nTake any graphon $U\\in\\C W$. The pre-image $\\Gamma^{-1}(\\T U)$ is a closed subset of $\\widetilde{\\mathcal{W}}^{(k)}$ by the continuity of $\\Gamma$ (Lemma~\\ref{lm:Cont}). Also, $\\widetilde{\\mathcal{W}}^{(\\V\\alpha)}$\n\tis a closed subset of $\\widetilde{\\mathcal{W}}^{(k)}$: if \n\t$(V,(B_1,\\dots,B_k))\\in \\mathcal{W}^{(k)}$ is not in $\\mathcal{W}\\times {\\bf A}^{(\\V\\alpha)}$, \n\n\tthen the $\\delta_{\\Box}^{(k)}$-ball of radius e.g.\\ \n\t$$\n\t \\frac12\\,\\left\\|\\,(\\lambda(B_i))_{i\\in [k]} - \\frac1{\\|\\V\\alpha\\|_1} \\V\\alpha\\,\\right\\|_1>0\n\t $$\n\t  around it is disjoint from~$\\mathcal{W}\\times {\\bf A}^{(\\V\\alpha)}$. Recall that the space $\\widetilde{\\mathcal{W}}^{(k)}$ is compact by Theorem~\\ref{th:compact colorod graphon}. Thus the infimum in~\\eqref{eq:Jap2} is taken over a (non-empty) compact set.  As any lsc function attains its infimum on any non-empty compact set and\n\t$I^{(k)}_{\\V p}:\\widetilde{\\mathcal{W}}^{(k)}\\to [0,+\\infty]$ is lsc by Lemma~\\ref{lm:lsc}, \n\n\tthere is $(V,\\C A)\\in \\mathcal{W}\\times {\\bf A}^{(\\V\\alpha)}$ such that $V\\in\\T U$ and $I_{\\V p}^{(k)}(\\Tk{V,\\C A})=J_{\\V\\alpha,\\V p}(\\T U)$,\n\twhere $\\Tk{V,\\C A}$ denotes the $\\delta_\\Box^{(k)}$-equivalence class of $(V,\\C A)$.\n\t\n\tThus for any $\\rho\\in\\I R$ the level set \n\t$\\{J_{\\V\\alpha,\\V p}\\le \\rho\\}$ \n\tis equal to the image of $\\{I_{\\V p}^{(k)}\\le \\rho\\}\\cap\\widetilde{\\mathcal{W}}^{(\\V\\alpha)}$ under~$\\Gamma$. Since the function $I_{\\V p}^{(k)}$ is lsc by Lemma~\\ref{lm:lsc}, the level set $\\{I_{\\V p}^{(k)}\\le \\rho\\}$ is a closed and thus compact subset of $\\widetilde{\\mathcal{W}}^{(k)}$. Thus the set $\\{I_{\\V p}^{(k)}\\le \\rho\\}\\cap\\widetilde{\\mathcal{W}}^{(\\V\\alpha)}$ is compact. Its image $\\{J_{\\V\\alpha,\\V p}\\le \\rho\\}$  under the continuous map $\\Gamma:\\widetilde{\\mathcal{W}}^{(k)}\\to \\widetilde{\\mathcal{W}}$ is compact and thus closed. Since $\\rho\\in\\I R$ was arbitrary, the function $J_{\\V\\alpha,\\V p}:\\widetilde{\\mathcal{W}}\\to [0,\\infty]$ is lsc. \n\t\n\t\n\tSince $R_{\\V p}(\\T U)$ is equal to the infimum of $I_{\\V p}^{(k)}$ over $\\Gamma^{-1}(\\T U)$, the same argument (except we do not need to intersect $\\Gamma^{-1}(\\T U)$ with $\\widetilde{\\mathcal{W}}^{(\\V\\alpha)}$ anywhere) also works for $R_{\\V p}$. \n\\end{proof}\n\n\n\\section{Proof of Theorem~\\ref{th:GenLDP}}\\label{GenLDP}\n\n\nFirst we prove two auxiliary lemmas. The first one states, informally speaking, that the measure $\\widetilde{\\mathbb{P}}_{\\V a,\\V p}$ is ``uniformly continuous'' in~$\\V a$.\n\n\\begin{lemma}\\label{lm:DifferentRatios}\n\tFor every symmetric\tmatrix $\\V p\\in [0,1]^{k\\times k}$, real $\\varepsilon\\in (0,1)$ and non-zero integer vectors $\\V a,\\V b\\in\\I N^k$, if \n\t$$\\|\\V b-\\V a\\|_1\\le \\varepsilon\\, \\min\\left\\{\\,\\|\\V a\\|_1,\\|\\V b\\|_1\\,\\right\\}$$ \n\tthen there is a (discrete) measure $\\T{\\I C}$ on $\\widetilde{\\mathcal{W}}\\times \\widetilde{\\mathcal{W}}$ which gives a coupling between $\\widetilde{\\mathbb{P}}_{\\V a,\\V p}$ and $\\widetilde{\\mathbb{P}}_{\\V b,\\V p}$ such that for every $(\\T U,\\T V)$ in the support of $\\T{\\I C}$ we have $\\delta_\\Box(U,V)\\le 4\\varepsilon\/(1-\\varepsilon)$.\n\t\\end{lemma}\n\n\\begin{proof}\n\tLet $m:=\\|\\V a\\|_1$ and $n:=\\|\\V b\\|_1$. \n\t\tLet $[m]=A_1\\cup\\dots\\cup A_k$ (resp.\\ $[n]=B_1\\cup\\dots\\cup B_k$) be the partition into consecutive intervals with $a_1,\\dots,a_k$ (resp.\\ $b_1,\\dots,b_k$) elements. For each $i\\in [k]$ fix some subsets $A_i'\\subseteq A_i$ and $B_i'\\subseteq B_i$ of size $\\min(a_i,b_i)$. Define $A':=\\cup_{i=1}^k A_i'$ and\n\t$B':=\\cup_{i=1}^k B_i'$. Fix any bijection $h:A'\\to B'$ that sends each $A_i'$ to $B_i'$. We have\n\t\\beq{eq:m-A'}\n\t\\left|\\,[m]\\setminus A'\\,\\right|\\le\\sum_{i=1}^k \\max(0,a_i-b_i)\\le \\|\\V a-\\V b\\|_1\\le \\varepsilon m\n\t\\end{equation}\n\tand similarly $\\left|\\,[n]\\setminus B'\\,\\right|\\le \\varepsilon n$. \n\t\nWe can couple random graphs $G\\sim \\I G(\\V a,\\V p)$ and $H\\sim\\I G(\\V b,\\V p)$ so that every pair $\\{x,y\\}$ in $A'$ is an edge in $G$ if and only if $\\{h(x),h(y)\\}$ is an edge in~$H$.\nThis is possible because, with $i,j\\in [k]$ satisfying $(x,y)\\in A_i'\\times A_j'$, the probability of $\\{x,y\\}\\in E(G)$ is $p_{i,j}$, the same as the probability with which $h(x)\\in B_i$ and $h(y)\\in B_j$ are made adjacent in~$H$ (so we can just use the same coin flip for both pairs). By making all edge choices to be mutually independent otherwise, we get a probability measure $\\I C$ on pairs of graphs which is a coupling between $\\I G(\\V a,\\V p)$ and $\\I G(\\V b,\\V p)$.\n The corresponding measure $\\T{\\I C}$ on $\\widetilde{\\mathcal{W}}\\times \\widetilde{\\mathcal{W}}$ gives  a coupling between $\\widetilde{\\mathbb{P}}_{\\V a,\\V p}$ and $\\widetilde{\\mathbb{P}}_{\\V b,\\V p}$. \n\nLet us show that this coupling $\\I C$ satisfies the distance requirement. Take any pair of graphs $(G,H)$ in the support of~$\\I C$. Let $G':=G[A']$ be obtained from $G$ by removing all vertices in $[m]\\setminus A'$. We remove at most $\\varepsilon$ fraction of vertices by~\\eqref{eq:m-A'}. By relabelling the vertices of~$G$, we can assume that all removed vertices come at the very end. Then the union of the intervals in the graphon $\\f{G}$ corresponding to the vertices of $G'$ is an initial segment of $[0,1]$ of length $s\\ge 1-\\varepsilon$ and the graphon of $G'$ is the pull-back of $\\f{G}$ under the map $x\\mapsto sx$.\nBy Lemma~\\ref{lm:Delete01}, we have $\\delta_\\Box(\\f{G},\\f{G'})\\le 2(\\frac{1}{1-\\varepsilon}-1)= \\frac{2\\varepsilon}{1-\\varepsilon}$. By symmetry, the same estimate applies to the pair $(H,H')$, where $H'$ is obtained from $H$ by deleting all vertices from~$[n]\\setminus B'$. By the definition of our coupling, the function $h:V(G')\\to V(H')$ is (deterministically) an isomorphism between $G'$ and~$H'$. Thus the graphons of $G'$ and $H'$ are weakly isomorphic. \nThis gives by the Triangle Inequality the required upper bound on the cut-distance between $\\tf{G}$ and $\\tf{H}$. \n\\end{proof}\n\n\nThe function $J_{\\V\\alpha,\\V p}(\\T V)$ is not in general continuous in $\\V\\alpha$ even when $\\V p$ and $\\T V$ are fixed. (For example, if $k=2$, $V=1-\\f{K_2}$ is the limit of $K_n\\sqcup K_n$, the disjoint union of two cliques of order $n$ each, and $\\V p$ is the identity $2\\times 2$ matrix then $J_{\\V\\alpha,\\V p}(\\T V)$ is $0$ for $\\V\\alpha=(\\frac12,\\frac12)$ and $\\infty$ otherwise.) However, the following version of ``uniform semi-continuity'' in $\\V\\alpha$ will suffice for our purposes.\n\n\n\n\\begin{lemma}\\label{lm:ContOfJ}\n\tFix any symmetric $\\V p\\in [0,1]^{k\\times k}$.\n\tThen for every $\\eta>0$ there is $\\varepsilon=\\varepsilon(\\eta,\\V p)>0$ such that if  vectors ${\\V\\gamma},\\V\\kappa\\in [0,\\infty)^k$ with $\\|\\V\\gamma\\|_1=\\|\\V\\kappa\\|_1=1$ satisfy\n\t\\begin{equation}\\label{eq:cond}\n\t\t\\kappa_i\\le (1+\\varepsilon)\\gamma_i,\\quad\\mbox{for every $i\\in [k]$},\n\t\n\t\t\\end{equation}\n\t then\n\tfor every $U\\in {\\mathcal{W}}$ there is ${V}\\in {\\C W}$ with $\\delta_\\Box(U,{V})\\le \\eta$ and\n\t$$J_{\\V\\kappa,\\V p}(\\T{V})\\le J_{{\\V\\gamma},\\V p}(\\T U) +\\eta.$$\n\\end{lemma}\n\\begin{proof}\n\tFor every fixed $p\\in (0,1)$, the relative entropy function $h_p:[0,1]\\to [0,\\infty]$ is bounded (in fact, by  $\\max\\{h_p(0),h_p(1)\\}<\\infty$). Thus we can find a constant $C$ that depends on $\\V p$ only such that for every $x\\in [0,1]$ and $i,j\\in [k]$, $h_{p_{i,j}}(x)$ is either $\\infty$ or at most~$C$. \n\t\t\t\n\t\t\t\nLet us show that any positive $\\varepsilon\\le \\min\\{\\,\\frac{\\eta}{3(2C+\\eta)},\\,\\frac{\\eta}2\\,\\}$ works. Let ${\\V\\gamma}$, $\\V\\kappa$, and $U$ be as in the lemma. \tAssume that $J_{{\\V\\gamma},\\V p}(\\T U)<\\infty$ for otherwise  we can trivially take ${V}:=U$. By the choice of $C$, we have that $J_{{\\V\\gamma},\\V p}(\\T U)\\le C$. \n\nBy replacing $U\\in\\C W$ with a weakly isomorphic graphon, assume that, for the partition $(\\cI{{\\V\\gamma}}{i})_{i\\in [k]}$ of $[0,1]$ into intervals, \nwe have\n\t$$\\frac{1}{2}\\sum_{i,j\\in [k]} \\int_{\\cI{{\\V\\gamma}}{i}\\times \\cI{{\\V\\gamma}}{j}} I_{p_{i,j}}(U) \\,\\mathrm{d}\\lambda^{\\oplus 2}<J_{{\\V\\gamma},\\V p}(\\T U)+\\frac{\\eta}{2}.$$\n\t\nLet $\\phi:[0,1]\\to [0,1]$ be the a.e.\\ defined function  which  on each interval $\\cI{{\\V\\kappa}}{i}$ of positive length is the increasing linear function that bijectively maps this interval onto $\\cI{{\\V\\gamma}}{i}$. \nThe \nRadon-Nikodym derivative $D:=\\frac{\\,\\mathrm{d} \\mu}{\\,\\mathrm{d}\\lambda}$ of the push-forward $\\mu:=\\phi_*\\lambda$ of the Lebesgue measure $\\lambda$ along $\\phi$ assumes value\n$\\kappa_i\/\\gamma_i$ a.e.\\ on~$\\cI{{\\V\\gamma}}{i}$. (The union of $\\cI{\\V\\gamma}{i}$ with $\\gamma_i=0$ \nis a countable and thus null set,\nand can be ignored from the point of view of~$D$.)\n\tLet $V:=U^\\phi$. It is a graphon by the first part of Lemma~\\ref{lm:AlmostMP}. \nWe have by the definition of $J_{\\V\\kappa,\\V p}$ that\n\t\\begin{equation*}\n\t\t\tJ_{\\V\\kappa,\\V p}(\\T{V})\\le  \\ \\frac{1}{2}\\sum_{i,j\\in [k]} \\int_{\\cI{{\\V\\kappa}}{i}\\times \\cI{{\\V\\kappa}}{j}} I_{p_{i,j}}(U^\\phi) \\ \\,\\mathrm{d} \\lambda^{\\oplus 2} \n\t\t\t=  \\ \\frac{1}{2}\\sum_{i,j\\in [k]} \\frac{\\kappa_i\\kappa_j}{\\gamma_i\\gamma_j}\\int_{\\cI{{\\V\\gamma}}{i}\\times \\cI{{\\V\\gamma}}{j}} I_{p_{i,j}}(U) \\,\\mathrm{d}\\lambda^{\\oplus 2}.\n\t\\end{equation*}\n By $\\frac{\\kappa_i\\kappa_j}{\\gamma_i\\gamma_j}\\le (1+\\varepsilon)^2\\le 1+3\\varepsilon$, this  is at most\n $$\n (1+3\\varepsilon)\\,\\frac1{2}\\sum_{i,j\\in [k]} \\int_{\\cI{{\\V\\gamma}}{i}\\times \\cI{{\\V\\gamma}}{j}} I_{p_{i,j}}(U) \\,\\mathrm{d}\\lambda^{\\oplus 2} \n\\le (1+3\\varepsilon)  \\left(J_{{\\V\\gamma},\\V p}(\\T U)+\\frac{\\eta}2 \\right).\n $$\t\n By $J_{{\\V\\gamma},\\V p}(\\T U)\\le C$ and our choice of $\\varepsilon$, this in turn is at most\n$$\nJ_{{\\V\\gamma},\\V p}(\\T U) + 3\\varepsilon C+(1+3\\varepsilon)\\frac{\\eta}2\\le  \\ J_{{\\V\\gamma},\\V p}(\\T U)+ \\eta.\n$$\n\n\t\nThus it remains to estimate the cut-distance between $U$ and~${V}=U^\\phi$. Since the Radon-Nikodym derivative $D=\\frac{\\,\\mathrm{d} \\phi_*\\lambda}{\\,\\mathrm{d}\\lambda}$ satisfies $D(x)\\le 1+\\varepsilon$ for a.e.\\ $x\\in [0,1]$ by~\\eqref{eq:cond}, we have by Lemma~\\ref{lm:AlmostMP} that $\\delta_\\Box(U,{V})\\le 2\\varepsilon\\le \\eta$. Thus $V$ is as desired.\\end{proof}\n\nNow we are ready to prove Theorem~\\ref{th:GenLDP}.\n\n\\begin{proof}[Proof of Theorem~\\ref{th:GenLDP}]\nRecall that we have to establish an LDP for the sequence $(\\widetilde{\\mathbb{P}}_{\\V a_n,\\V p})_{n\\in\\I N}$ of measures, where the integer vectors $\\V a_n$, after scaling by $1\/n$, converge to a non-zero vector $\\V\\alpha\\in[0,\\infty)^k$. Since the underlying metric space $(\\widetilde{\\mathcal{W}},\\delta_{\\Box})$ is compact and the proposed rate function $J_{\\V\\alpha,\\V p}$ is lsc by Theorem~\\ref{th:JLSC}, it suffices to prove the bounds in~\\eqref{eq:lower} and~\\eqref{eq:upper} of Lemma~\\ref{lm:LDP} for any given graphon~$\\T U\\in \\widetilde{\\mathcal{W}}$. \n\n\nLet us show the upper bound first, that is, that\n\\beq{eq:red9}\n \\lim_{\\eta\\to 0}\\limsup_{n\\to\\infty} \\frac{1}{(\\|\\V a_n\\|_1)^2}\\log \\widetilde{\\mathbb{P}}_{\\V a_n,\\V p}(S(\\widetilde U,\\eta))\\le -J_{\\V\\alpha,\\V p}(\\T U).\n\\end{equation}\n\n Take any $\\eta>0$. Assume that, for example, $\\eta<1\/9$. Let $m\\in\\I N$ be sufficiently large. Define\n\\beq{eq:b}\n \\V b:=(a_{m,1}\\I 1_{\\alpha_1>0},\\ldots,a_{m,k}\\I 1_{\\alpha_k>0})\\in\\I Z^k.\n \\end{equation}\n In other words, we let $\\V b$ to be $\\V a_m$ except we set its $i$-th entry $b_i$ to be~0  for each $i\\in [k]$ with $\\alpha_i=0$.\n By Theorem~\\ref{th:BCGPS} (the LDP by Borgs et al~\\cite{BCGPS}) and Theorem~\\ref{th:JLSC}, we have that\n \\beq{eq:BCGPSApplied1}\n \\limsup_{n\\to\\infty} \\frac{1}{(n\\,\\|\\V b\\|_1)^2}\\log\\widetilde{\\mathbb{P}}_{n\\V b,\\V p}(S(\\widetilde U,\\eta))\\le \n\n -\\inf_{S(\\widetilde U,\\eta)} J_{\\V b,\\V p}.\n \\end{equation}\n \n \nSince $m$ is sufficiently large, we can assume that $\\|\\frac1n\\, \\V a_n-\\V\\alpha\\|_1\\le \\xi\\,\\|\\V\\alpha\\|_1$ for every $n\\ge m$, where e.g.~$\\xi:=\\eta\/40$. \nIn particular, we have \n$\\|\\V a_m-\\V b\\|_1\n\\le \\|\\V a_m-m\\V\\alpha\\|_1\n\\le \\xi m\\|\\V\\alpha\\|_1$ from which it follows that \n$$\n \\|\\V b -m\\V\\alpha\\|_1\\le \\|\\V b -\\V a_m\\|_1+\\| \\V a_m - m\\V\\alpha\\|_1\\le 2\\xi m\\,\\|\\V\\alpha\\|_1.\n$$ \nLet $n$ be sufficiently large (in particular, $n\\ge m$) and let $n':= \\floor{n\/m}$. By above, we have\n\\hide{\\begin{eqnarray}\n\\|n'\\V b-\\V a_n\\|_1&\\le& \\textstyle \n \\|\\V b\\|_1+\\|\\frac{n}{m}\\V b-n\\V\\alpha\\|_1+\\|n\\V\\alpha-\\V a_n\\|_1\\\n\\le\\ \\|\\V b\\|_1+3\\xi n\\|\\V\\alpha\\|_1\\nonumber\\\\\n &\\le &\\textstyle\n \\|\\V b\\|_1 + 4\\xi \\min\\{\\,\\|\\frac{n}{m}\\V b\\|_1,\\,\\| \\V a_n\\|_1\\,\\}\\ \\le\\ \\frac{\\eta}9\\,\\min\\{\\,\\|n'\\V b\\|_1,\\,\\| \\V a_n\\|_1\\,\\}.\\label{eq:n'b-an}\n\\end{eqnarray}\n}\n\\begin{eqnarray}\n\\|n'\\V b-\\V a_n\\|_1&\\le& \\textstyle \n \\|\\V b\\|_1+\\|\\frac{n}{m}\\V b-n\\V\\alpha\\|_1+\\|n\\V\\alpha-\\V a_n\\|_1\\nonumber\\\\\n&\\le& \\|\\V b\\|_1+3\\xi n\\|\\V\\alpha\\|_1\\nonumber\\\\\n &\\le &\\textstyle\n \\|\\V b\\|_1 + 4\\xi \\min\\{\\,\\|\\frac{n}{m}\\V b\\|_1,\\,\\| \\V a_n\\|_1\\,\\}\\nonumber\\\\ \n&\\le& \\textstyle\\frac{\\eta}9\\,\\min\\{\\,\\|n'\\V b\\|_1,\\,\\| \\V a_n\\|_1\\,\\}.\\label{eq:n'b-an}\n\\end{eqnarray}\n Thus, by Lemma~\\ref{lm:DifferentRatios}, there is a coupling $\\T{\\I C}$ between $\\widetilde{\\mathbb{P}}_{n'\\V b,\\V p}$ and $\\widetilde{\\mathbb{P}}_{\\V a_n,\\V p}$ such that for every $(\\widetilde{V},\\widetilde{W})$ in the support of $\\T{\\I C}$ we have $\\delta_\\Box(V,{W})\\le \\eta\/2$. Thus, if $\\widetilde{W}\\sim\\widetilde{\\mathbb{P}}_{\\V a_n,\\V p}$ lands in $S(\\widetilde{ U},\\eta\/2)$ then necessarily $\\widetilde{V}\\sim \\widetilde{\\mathbb{P}}_{n'\\V b,\\V p}$ lands in $S(\\widetilde{U},\\eta)$. This gives that\n $$\n \\widetilde{\\mathbb{P}}_{\\V a_n,\\V p}(S(\\widetilde U,\\eta\/2))\\le \\widetilde{\\mathbb{P}}_{n'\\V b,\\V p}(S(\\widetilde U,\\eta)).\n $$\n Since this is true for every sufficiently large $n$ and it holds by~\\eqref{eq:n'b-an} that, for example, \n $\\|\\V a_n\\|_1\/(n'\\,\\|\\V b\\|_1) \\ge 1-\\eta\/9\\ge \\sqrt{1-\\eta}$, we have that\n \\beq{eq:red1}\n \\limsup_{n\\to\\infty} \\frac{1-\\eta}{(\\|\\V a_n\\|_1)^2}\\log \\widetilde{\\mathbb{P}}_{\\V a_n,\\V p}(S(\\widetilde U,\\eta\/2))\\le \\limsup_{n'\\to\\infty}  \\frac{1}{(n'\\,\\|\\V b\\|_1)^2}\\log \\widetilde{\\mathbb{P}}_{n'\\V b,\\V p}(S(\\widetilde U,\\eta)).\n \\end{equation}\n \n \nLet us turn our attention to the right-hand side of~\\eqref{eq:BCGPSApplied1}. \nPick some $\\T V\\in S(\\T U,\\eta)$ with $J_{\\V b,\\V p}(\\T V)\\le \\inf_{S(\\T U,\\eta)} J_{\\V b,\\V p}+\\eta$. (In fact, by the lower semi-continuity of $J_{\\V b,\\V p}$ and the compactness of $S(\\T U,\\eta)$, we could have required\nthat $J_{\\V b,\\V p}(\\T V)= \\inf_{S(\\T U,\\eta)} J_{\\V b,\\V p}$.) \nSince $\\V a_n\/\\|\\V a_n\\|_1$ converges to $\\V\\alpha\/\\|\\V\\alpha\\|_1$ as $n\\to\\infty$ and we chose $m$ to be sufficiently large,\nwe can assume that $b_i=0$ if and only if $\\alpha_i=0$ and that \nLemma~\\ref{lm:ContOfJ} applies for $\\V\\gamma:=\\V b\/\\|\\V b\\|_1$ and $\\V\\kappa:=\\V\\alpha\/\\|\\V\\alpha\\|_1$ (and our~$\\eta$). \nThe lemma gives that \nthere is $\\T{W}\\inS(\\T V,\\eta)$ such that $J_{\\V\\alpha,\\V p}(\\T{W})-\\eta\\le J_{\\V b,\\V p}(\\T{V})$. Thus we get the following upper bound on the right-hand side of~\\eqref{eq:BCGPSApplied1}:\n\\beq{eq:red2}\n-\\inf_{S(\\T U,\\eta)} J_{\\V b,\\V p}\\le -J_{\\V b,\\V p}(\\T V)+\\eta\\le -J_{\\V\\alpha,\\V p}(\\T{W})+2\\eta\\le -\\inf_{S(\\T U,2\\eta)} J_{\\V \\alpha,\\V p}+2\\eta.\n\\end{equation}\n By putting~\\eqref{eq:BCGPSApplied1}, \\eqref{eq:red1} and~\\eqref{eq:red2} together we get that, for every $\\eta\\in (0,1\/9)$,\n $$\n \\limsup_{n\\to\\infty} \\frac{1-\\eta}{(\\|\\V a_n\\|_1)^2}\\log \\widetilde{\\mathbb{P}}_{\\V a_n,\\V p}(S(\\widetilde U,\\eta\/2))\\le\n-\\inf_{S(\\T U,2\\eta)} J_{\\V \\alpha,\\V p}+2\\eta.\n$$ \n If we take here the limit as $\\eta\\to 0$ then the infimum in the right-hand side converges to~$J_{\\V \\alpha,\\V p}(\\T U)$ by the lower semi-continuity of $J_{\\V \\alpha,\\V p}$ (established in Theorem~\\ref{th:JLSC}), giving the claimed upper bound~\\eqref{eq:red9}.\n \n \nLet us turn to the lower bound, i.e.\\ we prove~\\eqref{eq:lower} for $\\T U\\in\\widetilde{\\mathcal{W}}$. As before, take any sufficiently small $\\eta>0$, then sufficiently large $m\\in\\I N$ and define $\\V b$ by~\\eqref{eq:b}. By Theorems~\\ref{th:BCGPS} and~\\ref{th:JLSC} applied to the open ball around $\\T U$ of radius $2\\eta$, we have\n \\beq{eq:red4}\n\\liminf_{n\\to\\infty} \\frac{1}{(n\\,\\|\\V b\\|_1)^2}\\log\\widetilde{\\mathbb{P}}_{n\\V b,\\V p}(S(\\widetilde U,2\\eta))\\ge \n-\\inf_{S(\\T U,\\eta)} J_{\\V b,\\V p}.\n \\end{equation}\n Similarly as for the upper bound, the left-hand side can be upper bounded via Lemma~\\ref{lm:ContOfJ} by, for example, \n \\beq{eq:red3}\n \\liminf_{n\\to\\infty} \\frac{1+\\eta}{(\\|\\V a_n\\|_1)^2}\\log \\widetilde{\\mathbb{P}}_{\\V a_n,\\V p}(S(\\widetilde U,3\\eta)). \n \\end{equation}\n Since $m$ is sufficiently large,\n we can apply Lemma~\\ref{lm:ContOfJ} to $\\V\\kappa:=\\B b\/\\|\\V b\\|_1$, $\\V\\gamma:=\\V\\alpha\/\\|\\V\\alpha\\|_1$, the given graphon $U$ and our chosen $\\eta$ \n\n to find $\\T{V}\\in S(\\T U, \\eta)$ such that $J_{\\V b,\\V p}(\\T{V})\\le J_{\\V\\alpha,\\V p}(\\T U)+\\eta$. Thus\n $$\n -\\inf_{S(\\T U,\\eta)} J_{\\V b,\\V p}\\ge -J_{\\V b,\\V p}(\\T{V})\\ge -J_{\\V\\alpha,\\V p}(\\T U)-\\eta.\n $$\n By~\\eqref{eq:red4}, this is a lower bound on the expression in~\\eqref{eq:red3}. Taking the limit as $\\eta\\to 0$ we get the required LDP lower bound~\\eqref{eq:lower}. This finishes the proof of Theorem~\\ref{th:GenLDP}.\n \\end{proof}\n \n\n\n\n \n\\section{Proof of Theorem~\\ref{th:ourLDP}}\\label{ourLDP}\n\n\nRecall that $W$ is a $k$-step graphon with non-null parts whose values are encoded by a symmetric $k\\times k$ matrix $\\V p\\in [0,1]^{k\\times k}$. We consider the $W$-random graph $\\I G(n,W)$ where we first sample $n$ independent uniform points $x_1,\\dots,x_n\\in [0,1]$ and then make each pair $\\{i,j\\}\\subseteq [n]$ an edge with probability $W(x_i,x_j)$. We have to prove an LDP for the corresponding sequence $(\\widetilde{\\mathbb{R}}_{n,W})_{n\\in\\I N}$ of measures on the metric space $(\\widetilde{\\mathcal{W}},\\delta_\\Box)$ with speed $n^2$ and the rate function $R_{\\V p}$ that was defined in~\\eqref{eq:R}. Recall that the lower semi-continuity of $R_{\\V p}$ was established in Corollary~\\ref{cr:lsc}.\n\n\n\n\nLet us show the lower bound. Since the underlying space $(\\widetilde{\\mathcal{W}},\\delta_\\Box)$ is compact, it is enough to prove the bound in~\\eqref{eq:lower} of Lemma~\\ref{lm:LDP} for any given $\\T U\\in \\T{\\C W}$, that is, that\n\\beq{eq:L1}\n\\lim_{\\eta\\to 0}\\liminf_{n\\to\\infty} \\frac{1}{n^2} \\log \\widetilde{\\mathbb{R}}_{n,W}(S(\\widetilde U,\\eta))\\ge - R_{\\V p}(\\T U).\n\\end{equation}\nTake any $\\varepsilon>0$. By the definition of $R_{\\V p}$, we can fix a vector $\\V\\alpha=(\\alpha_1,\\dots,\\alpha_k)\\in [0,1]^k$ \nsuch that $\\|\\V\\alpha\\|_1=1$ and \n $$\n R_{\\V p}(\\T U)\\ge J_{\\V \\alpha,\\V p}(\\T U)-\\varepsilon.\n$$\nLet  $m$ be the number of non-zero entries of $\\V \\alpha$. As $\\V \\alpha$ is non-zero, we have $m\\ge 1$. We can assume by symmetry that $\\alpha_1,\\dots,\\alpha_m$ are the non-zero entries. Define $c:=\\frac12 \\min_{i\\in [m]} \\alpha_i>0$. \n\nFor each $n\\in\\I N$, take any integer vector $\\V a_n=(a_{n,1},\\dots,a_{n,k})\\in \\I N_{\\ge 0}^k$ such that $\\|\\V a_n\\|_1=n$ and $\\|\\V a_{n}-n\\,\\V\\alpha\\|_\\infty<1$ (in particular, we have $a_{n,i}=0$ if $\\alpha_i=0$). \n\nLet $n$ be sufficiently large. In particular, for every $i\\in [m]$ we have that $a_{n,i}\/n\\ge c$. \nWhen we generate $G\\sim \\I G(n,W)$ by choosing first random $x_1,\\dots,x_n\\in [0,1]$, \n it holds with probability at least, very roughly, $c^{-n}$ that for each $i\\in [k]$ the number of $x_j$'s that belong to the $i$-th part of the step graphon~$W$ is exactly~$a_{n,i}$. Conditioned on this event of positive measure, the resulting graphon\n $\\tf{G}$ is distributed according to $\\widetilde{\\mathbb{P}}_{\\V a_n,\\V p}$. Thus \n $$\n \\widetilde{\\mathbb{R}}_{n,W}(S)\\ge c^{-n}\\,\\widetilde{\\mathbb{P}}_{\\V a_n,\\V p}(S),\\quad\\mbox{for every $S\\subseteq \\widetilde{\\mathcal{W}}$}.\n $$\n This and the new LDP result (Theorem~\\ref{th:GenLDP}) give that, for every $\\eta>0$,\n \\begin{eqnarray*}\n \\limsup_{n\\to\\infty} \\frac{1}{n^2} \\log \\widetilde{\\mathbb{R}}_{n,W}(S(\\widetilde U,\\eta))&\\ge& \\limsup_{n\\to\\infty} \\frac{1}{n^2} \\log \\widetilde{\\mathbb{P}}_{\\V a_n,\\V p}(S(\\widetilde U,\\eta))\\\\\n &\\ge& -\\inf_{S(\\T U,\\eta\/2)} J_{\\V\\alpha,\\V p}\n \\ \\ge\\ -J_{\\V\\alpha,\\V p}(\\T U)\\ \\ge\\ -R_{\\V p}(\\T U)-\\varepsilon.\n \\end{eqnarray*}\n Taking the limit as $\\eta\\to 0$, we conclude that the LDP lower bound~\\eqref{eq:L1} holds within additive error~$\\varepsilon$. As $\\varepsilon>0$ was arbitrary, the lower bound\n\n holds.\n \n Let us show the upper bound~\\eqref{eq:upperGen} of Definition~\\ref{df:LDP} for any closed set $F\\subseteq \\widetilde{\\mathcal{W}}$, that is, that\n  \\beq{eq:U3}\n \\limsup_{n\\to\\infty} \\frac1{n^2} \\log \\widetilde{\\mathbb{R}}_{n,W}(F)\\le -\\inf_F R_{\\V p}.\n \\end{equation}\n \n \n For each $n\\in\\I N$, we can write $\\widetilde{\\mathbb{R}}_{n,W}(F)$ as the sum over all  $\\V a\\in\\I N_{\\ge 0}^k$ with $\\|\\V a\\|_1=n$ of the probability that the distribution of random independent $x_1,\\dots,x_n\\in [0,1]$ per $k$ steps of $W$ is given by the vector $\\V a$ times the probability conditioned on $\\V a$ to hit the set~$F$. This conditional probability is exactly $\\widetilde{\\mathbb{P}}_{\\V a,\\V p}(F)$. Thus $\\widetilde{\\mathbb{R}}_{n,W}(F)$ is a convex combination of the reals $\\widetilde{\\mathbb{P}}_{\\V a,\\V p}(F)$ and there is a vector $\\V a_n\\in \\I N_{\\ge 0}^k$ with $\\|\\V a_n\\|_1=n$ such that\n \\beq{eq:an}\n \\widetilde{\\mathbb{R}}_{n,W}(F)\\le \\widetilde{\\mathbb{P}}_{\\V a_n,\\V p}(F).\n \\end{equation}\n Fix one such vector~$\\V a_n$ for each $n\\in\\I N$.\n \n Since the set of all real vectors $\\V\\alpha\\in [0,1]^k$ with $\\|\\V\\alpha\\|_1=1$ is compact, we can find an increasing sequence $(n_i)_{i\\in\\I N}$ of integers such that \n \\beq{eq:U1}\n \\limsup_{n\\to\\infty} \\frac1{n^2} \\log \\widetilde{\\mathbb{R}}_{n,W}(F)=\\lim_{i\\to\\infty} \\frac1{n_i^2} \\log \\widetilde{\\mathbb{R}}_{n_i,W}(F),\n \\end{equation}\n and the scaled vectors $\\frac1{n_i}\\,\\V a_{n_i}$ converge to some real vector $\\V\\alpha\\in [0,1]^k$.\n \n Let $(\\V b_n)_{n\\in\\I N}$ be obtained by filling the gaps in $(\\V a_{n_i})_{n\\in\\I N}$, meaning that if $n=n_i$ for some  $i\\in\\I N$ then we let $\\V b_n:=\\V a_{n_i}$; otherwise we pick any $\\V b_n\\in\\I N_{\\ge 0}^k$ that satisfies $\\|\\V b_n\\|_1=n$ and $\\|\\V b_n-n\\,\\V\\alpha\\|_\\infty<1$. Since the normalised vectors $\\V b_{n}\/\\|\\V b_{n}\\|_1$ converge to the same limiting vector~$\\V\\alpha$, we have by Theorem~\\ref{th:GenLDP} that\n \\beq{eq:U2}\n \\limsup_{i\\to\\infty} \\frac1{n_i^2} \\log \\widetilde{\\mathbb{P}}_{\\V a_{n_i},\\V p}(F)\\le \\limsup_{n\\to\\infty} \\frac1{n^2} \\log \\widetilde{\\mathbb{P}}_{\\V b_{n},\\V p}(F)\\le -\\inf_F J_{\\V\\alpha,\\V p}.\n\n \\end{equation}\n Putting~\\eqref{eq:an}, \\eqref{eq:U1} and~\\eqref{eq:U2} together with the trivial consequence $J_{\\V\\alpha,\\V p}\\ge R_{\\V p}$ of the definition of $R_{\\V p}$, we get the desired upper bound~\\eqref{eq:U3}. This finishes the proof of Theorem~\\ref{th:ourLDP}.\n \n\n\\section*{Acknowledgements}\n\nJan Greb\\'\\i k and Oleg Pikhurko were supported by \nLeverhulme Research Project Grant RPG-2018-424.\n\n\n\\begin{bibdiv}\n\\begin{biblist}\n\n\\bib{BCGPS}{unpublished}{\n      author={Borgs, C.},\n      author={Chayes, J.},\n      author={Gaudio, J.},\n      author={Petti, S.},\n      author={Sen, S.},\n       title={A large deviation principle for block models},\n        date={2020},\n        note={E-print arxiv:2007.14508},\n}\n\n\\bib{Chatterjee16bams}{article}{\n      author={Chatterjee, S.},\n       title={An introduction to large deviations for random graphs},\n        date={2016},\n     journal={Bull. Amer. Math. Soc. (N.S.)},\n      volume={53},\n       pages={617\\ndash 642},\n}\n\n\\bib{Chatterjee17ldrg}{book}{\n      author={Chatterjee, S.},\n       title={Large deviations for random graphs},\n      series={Lecture Notes in Mathematics},\n   publisher={Springer, Cham},\n        date={2017},\n      volume={2197},\n        note={Lecture notes from the 45th Probability Summer School held in\n  Saint-Flour, June 2015},\n}\n\n\\bib{ChatterjeeVaradhan11}{article}{\n      author={Chatterjee, S.},\n      author={Varadhan, S. R.~S.},\n       title={The large deviation principle for the {Erd\\H{o}s-R\\'enyi} random\n  graph},\n        date={2011},\n     journal={Europ.\\ J.\\ Combin.},\n      volume={32},\n       pages={1000\\ndash 1017},\n}\n\n\\bib{Cohn13mt}{book}{\n      author={Cohn, D.~L.},\n       title={Measure theory},\n     edition={Second},\n      series={Birkh\\\"{a}user Advanced Texts: Basel Textbooks},\n   publisher={Birkh\\\"{a}user\/Springer, New York},\n        date={2013},\n}\n\n\\bib{DemboZeitouni10ldta}{book}{\n      author={Dembo, A.},\n      author={Zeitouni, O.},\n       title={Large deviations techniques and applications},\n      series={Stochastic Modelling and Applied Probability},\n   publisher={Springer-Verlag, Berlin},\n        date={2010},\n      volume={38},\n        note={Corrected reprint of the second (1998) edition},\n}\n\n\\bib{GrebikPikhurko:LDP}{unpublished}{\n      author={Greb{\\'\\i}k, J.},\n      author={Pikhurko, O.},\n       title={Large deviation for graphon sampling},\n        date={2021},\n        note={In preparation},\n}\n\n\\bib{Kechris:cdst}{book}{\n      author={Kechris, A.~S.},\n       title={Classical descriptive set theory},\n      series={Graduate Texts in Mathematics},\n   publisher={Springer-Verlag, New York},\n        date={1995},\n      volume={156},\n}\n\n\\bib{Lovasz:lngl}{book}{\n      author={{Lov\\'asz}, L.},\n       title={Large networks and graph limits},\n      series={Colloquium Publications},\n   publisher={Amer.\\ Math.\\ Soc.},\n        date={2012},\n}\n\n\\bib{LovaszSzegedy07gafa}{article}{\n      author={Lov{\\'a}sz, L.},\n      author={Szegedy, B.},\n       title={{Szemer\\'edi's} lemma for the analyst},\n        date={2007},\n     journal={Geom.\\ Func.\\ Analysis},\n      volume={17},\n       pages={252\\ndash 270},\n}\n\n\\bib{RassoulaghaSeppelainen14cldigm}{book}{\n      author={Rassoul-Agha, F.},\n      author={Sepp\\\"{a}l\\\"{a}inen, T.},\n       title={A course on large deviations with an introduction to {G}ibbs\n  measures},\n      series={Graduate Studies in Mathematics},\n   publisher={American Mathematical Society, Providence, RI},\n        date={2015},\n      volume={162},\n}\n\n\\bib{Srivastava98cbs}{book}{\n      author={Srivastava, S.~M.},\n       title={A course on {B}orel sets},\n      series={Graduate Texts in Mathematics},\n   publisher={Springer-Verlag, New York},\n        date={1998},\n      volume={180},\n}\n\n\\end{biblist}\n\\end{bibdiv}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Proof of \\Cref{prop: vanish_incoherence}}\nWe first prove the item 1. If $\\epsilon_i = 0$ for certain $i$, we have that\n\\begin{align}\n\\sup_{\\theta \\in \\Theta^*}   \\ell_i(\\theta) =  \\ell_i^\\ast. \\nonumber\n\\end{align}\nSince $\\ell_{i}^*$ is the global minimum of $\\ell_i$, we conclude from the above equality that $\\Theta^* \\subset \\Theta_i^*$. \n\nNext, we prove item 2. If $\\epsilon_i = 0$ for  all $i$, by item 1 we know that $\\Theta^* \\subset \\Theta_i^*$ for all $i$, and hence $\\Theta^* \\subset \\bigcap_{i=1}^n \\Theta_i^*$. Now suppose there exists $\\theta \\in \\bigcap_{i=1}^n \\Theta_i^* \\setminus \\Theta^*$. Then, $\\theta$ simultaneously minimizes all the sample losses and must be a minimizer of the total loss, i.e., $\\theta\\in \\Theta^*$, contradiction.\n\n\\section{Connection between Minimizer Incoherence and other Loss Conditions}\\label{subsec: compare_condition}\nThe notion of minimizer incoherence is related to other loss conditions that have been studied in the existing literature. We outline their connections in this section. \n\n$\\blacktriangleright$ \\textbf{Bounded variance}  \\shaocong{\\cite{ghadimi2013stochastic}}:\nIn stochastic optimization, it is standard to assume  that the variance of the stochastic gradients is bounded, i.e., for all $\\theta\\in \\mathbb{R}^d$,\n\\begin{align}\n\t\\mathbb{E}_{\\xi}\\|\\nabla \\ell_\\xi(\\theta) - \\nabla f(\\theta) \\|^2 \\le \\sigma^2. \\label{eq: variance}\n\\end{align}\nIn particular, when the total loss $f$ has a unique minimizer $\\theta^*$ and all sample losses $\\{\\ell_i\\}_{i=1}^n$ are $1$-gradient dominated\\footnote{$\\ell$ is called 1-gradient dominated if $\\ell(\\theta) - \\ell^* \\le \\|\\nabla \\ell(\\theta)\\|^2$.}, the stochastic gradient variance at  $\\theta^*$ satisfies\n\\begin{align}\n\t\\mathbb{E}_{\\xi}\\|\\nabla \\ell_\\xi(\\theta^*) - \\nabla f(\\theta^*) \\|^2 \n\n\t&\\ge \\frac{1}{n}\\sum_{i=1}^{n} (\\ell_i(\\theta^*) - \\ell_i^*) \\nonumber.\n\\end{align}\nin which the right hand side corresponds to the average minimizer incoherence $\\frac{1}{n}\\sum_{i=1}^{n} \\epsilon_i$. Therefore, minimizer incoherence provides an estimate of the stochastic gradient variance at the global minimum, and is weaker than the uniformly-bounded variance condition in \\cref{eq: variance}.\n\n$\\blacktriangleright$ \\textbf{Second moment condition} \\cite{bottou2018optimization}: This condition generalizes the previous bounded variance condition as: for some $C\\ge 1$ and all $\\theta\\in \\mathbb{R}^d$,\n\\begin{align}\n\t\\mathbb{E}_{\\xi}  \\| \\nabla \\ell_{\\xi}(\\theta)\\|^2 \\leq \\sigma^2 + C\\| \\nabla f(\\theta) \\|^2. \\label{eq: second_m}\n\\end{align} \nIn particular, the bounded variance condition corresponds to the second moment condition with $C=1$. In the special case that $\\sigma^2=0$ and all the sample losses are convex, the second moment condition implies that $\\nabla \\ell_i(\\theta^*)=0$ for all $i$ and all $\\theta^*\\in \\Theta^*$, i.e.,  every global minimizer of the total loss also minimizes all the sample losses, which further implies full minimizer coherence.\n\n$\\blacktriangleright$ \\textbf{Interpolation}  \\cite{ma2017power}: This condition assumes that the total loss $f$ has a unique minimizer $\\theta^\\ast$ such that\n\\begin{align*}\n\t\\ell_i(\\theta^\\ast ) = \\ell_i^* ~\\text{for all}~i=1,...,n.\n\\end{align*}\nIt can be viewed a special case of the full minimizer coherence, in which the sample losses can share multiple minimizers.\n\n\n$\\blacktriangleright$ \\textbf{Growth condition:} In \\cite{tseng1998incremental,schmidt2013fast}, the authors considered a  strong growth condition: for some $C \\geq 1$ and all $\\theta \\in \\mathbb{R}^d$,\n\\begin{align}\n\t\\max_i \\| \\nabla \\ell_i(\\theta) \\| \\leq C \\| \\nabla f(\\theta) \\|. \\label{eq: strong_grow}\n\\end{align}  \nWhen all the sample losses are convex, the above condition implies full minimizer coherence. A relaxed version of this condition has been proposed in \\cite{vaswani2018fast} as the weak growth condition, which relaxes the $\\max_i$ in \\cref{eq: strong_grow} to $\\mathbb{E}_i$. \n\n$\\blacktriangleright$ \\textbf{Expected smoothness} \\cite{gower2019}: This condition generalizes the weak growth condition as: \nfor some $L> 0$ all $\\theta \\in \\mathbb{R}^d$,\n\\begin{align}\n\t\\mathbb{E}_{\\xi} \\big[\\| \\nabla \\ell_{\\xi}(\\theta) - \\nabla \\ell_{\\xi}(\\theta^*)\\|^2 \\big] \\le L\\big(f(\\theta) - f^* \\big),\n\\end{align}\nwhere $\\theta^*$ is the unique minimizer of $f$. In the case of full minimizer coherence, \\cite{gower2019} proved that expected smoothness implies the weak growth condition.\n\n\n\\section{Proof of \\Cref{lemma: 1}}\nConsider the $k$-th iteration with sample $\\xi(k)$. By smoothness of $\\ell_{\\xi(k)}$, we obtain that\n$$\\ell_{\\xi(k)}(\\theta_{k+1} )\\leq \\ell_{\\xi(k)} (\\theta_k) + \\langle \\theta_{k+1}-\\theta_k , \\nabla \\ell_{\\xi(k)}(\\theta_k) \\rangle + \\frac{L}{2}\\|\\theta_{k+1}  - \\theta_k \\|^2. $$\nOn the other hand, by restricted convexity of $\\ell_{\\xi(k)}$, we have: for all $\\omega \\in \\Theta_{\\xi(k)}^*$,\n$$\\ell_{\\xi(k)}(\\omega) \\geq \\ell_{\\xi(k)}(\\theta_k) + \\langle \\omega - \\theta_k, \\nabla \\ell_{\\xi(k)}(\\theta_k) \\rangle.$$\nCombining the above two inequalities yields that\n\\begin{align*}\n\\ell_{\\xi(k)}(\\theta_{k+1} )&\\leq   { \\ell_{\\xi(k)}(\\omega)  + \\langle  \\theta_k -\\omega, \\nabla \\ell_{\\xi(k)}(\\theta_k)\\rangle } + \\langle \\theta_{k+1}-\\theta_k , \\nabla \\ell_{\\xi(k)}(\\theta_k) \\rangle + \\frac{L}{2}\\|\\theta_{k+1}  - \\theta_k \\|^2 \\\\\n&= \\ell_{\\xi(k)}^*  +  \\langle \\theta_{k+1}-\\omega ,  \\nabla \\ell_{\\xi(k)}(\\theta_k)   \\rangle + \\frac{L}{2}\\|\\theta_{k+1}  - \\theta_k \\|^2  \\\\\n&= \\ell_{\\xi(k)}^*  +  \\langle \\theta_{k+1}-\\omega , {-\\frac{1}{\\eta}(\\theta_{k+1} - \\theta_k)} \\rangle + \\frac{L}{2}\\|\\theta_{k+1}  - \\theta_k \\|^2 \\\\ \n&= \\ell_{\\xi(k)}^*  + { {\\frac{1}{\\shaocong{2}\\eta}\\left[ \\| \\theta_{k} - \\omega \\|^2 - \\| \\theta_{k+1} -\\omega \\|^2 - \\| \\theta_{k+1} - \\theta_{k} \\|^2 \\right]}}  + \\frac{L}{2}\\|\\theta_{k+1}  - \\theta_k \\|^2 \\\\\n&= \\ell_{\\xi(k)}^*  +  \\frac{1}{\\shaocong{2}\\eta}[ \\| \\theta_{k} - \\omega \\|^2 - \\| \\theta_{k+1} -\\omega \\|^2 ] - \\Big(\\frac{1}{\\shaocong{2}\\eta} - \\frac{L}{2} \\Big)\\|\\theta_{k+1}  - \\theta_k \\|^2.\n\\end{align*} \nRearranging the above inequality further yields that: for all $\\omega \\in \\Theta_{\\xi(k)}^*$,\n\\begin{align}\n\\| \\theta_{k+1} -\\omega \\|^2 \\leq \\| \\theta_{k} - \\omega \\|^2- \\shaocong{2}\\eta[ \\ell_{\\xi(k)}(\\theta_{k+1} ) - \\ell_{\\xi(k)}^*  ]  - \\big(1 - \\shaocong{\\eta L} \\big)\\|\\theta_{k+1}  - \\theta_k \\|^2. \\label{eq: lemma1}\n\\end{align}   \nChoose \\shaocong{$\\eta\\le \\frac{1}{L}$}, we conclude that for all $\\omega \\in \\Theta_{\\xi(k)}^*$,\n\\begin{align*}\n\\| \\theta_{k+1} -\\omega \\|^2 \\leq \\| \\theta_{k} - \\omega \\|^2- \\shaocong{2}\\eta\\big( \\ell_{\\xi(k)}(\\theta_{k+1} ) - \\ell_{\\xi(k)}^* \\big). \n\\end{align*}   \n\n\n\n\\section{Proof of \\Cref{lemma: 3}}\n\n{\n\t\n\tNote that by Lemma 1, we have that for all $\\omega\\in \\Theta_{\\xi(k)}^*$,\n\t\\begin{align*}\n\t\\|\\theta_{k+1} - \\omega\\|^2 &\\le \\|\\theta_{k} -\\omega \\|^2  - \\eta \\big( \\ell_{\\xi(k)}(\\theta_{k+1}) -\\ell_{\\xi(k)}^* \\big) \\nonumber\\\\\n\t&\\le \\|\\theta_{k} -\\omega \\|^2. \\nonumber\n\t\\end{align*} \n\tIn the case of full minimizer coherence, we have $\\Theta^*\\subset \\Theta_{\\xi(k)}^*$. Therefore,  the above result further implies that: for all $k$ and any fixed $\\omega \\in \\Theta^*$,\n\t\\begin{align*}\n\t\\|\\theta_{k+1} - \\omega\\| \\le \\|\\theta_{k} -\\omega \\| \\le \\cdots \\le \\|\\theta_{0} -\\omega\\|<+\\infty, \n\t\\end{align*} \n\twhere we have used the fact that both $\\Theta^*$ and $\\theta_{0}$ are bounded. Further notice that $\\|\\theta_{k+1}\\|\\le \\|\\omega\\|+\\|\\theta_{k+1}-\\omega\\|$, we conclude that the entire trajectory $\\{\\theta_{k}\\}_k$ is bounded. \n}\n\n\n\n\\section{Proof of \\Cref{prop: 2}}\n  We first prove item 1. Note that by \\Cref{prop: vanish_incoherence} we have $\\Theta^* = \\bigcap_{i=1}^n \\Theta_i^*$. In the proof of Lemma 1 we have shown in \\cref{eq: lemma1} that for any $\\omega \\in \\Theta_{\\xi(k)}^*$\n\\begin{align}\n\\| \\theta_{k+1} -\\omega \\|^2 &\\leq \\| \\theta_{k} - \\omega \\|^2- \\shaocong{2} \\eta[ \\ell_{\\xi(k)}(\\theta_{k+1} ) - \\ell_{\\xi(k)}^*  ]  - \\big(1 - \\shaocong{\\eta L} \\big)\\|\\theta_{k+1}  - \\theta_k \\|^2. \\nonumber\n\\end{align}   \nWe can choose any $\\omega \\in \\Theta^*$  and sum the above bound over the $B$-th epoch to obtain that\n\\begin{align*}\n\\| \\theta_{n(B+1)} -\\omega \\|^2 \\leq \\| \\theta_{nB} - \\omega \\|^2- \\shaocong{2}\\eta \\sum_{k=nB}^{n(B+1)-1} \\big(\\ell_{\\xi(k)}(\\theta_{k+1} ) - \\ell_{\\xi(k)}^* \\big) - \\sum_{k=nB}^{n(B+1)-1}\\big(1 - \\shaocong{\\eta L} \\big)\\|\\theta_{k+1}  - \\theta_k \\|^2. \n\\end{align*}  \nRearranging the above inequality yields that\n\\begin{align*}\n\\sum_{k=nB}^{n(B+1)-1} \\Big[\\big( \\ell_{\\xi(k)}(\\theta_{k+1} ) - \\ell_{\\xi(k)}^\\ast \\big) + \\big(\\frac{1}{\\shaocong{2} \\eta} - \\frac{L}{2}\\big)\\|\\theta_{k+1}  - \\theta_k \\|^2 \\Big] \\leq \\frac{1}{\\shaocong{2} \\eta} \\big(\\| \\theta_{nB} -\\omega \\|^2 -\\| \\theta_{n(B+1)} -\\omega \\|^2\\big).\n\\end{align*}  \n\nFurther summing the above bound over the epochs $K=0,...,B-1$ yields that\n\\begin{align}\n\\sum_{K=0}^{B-1}\\sum_{k=nK}^{n(K+1)-1} \\Big[\\big( \\ell_{\\xi(k)}(\\theta_{k+1} ) - \\ell_{\\xi(k)}^\\ast \\big) + \\big(\\frac{1}{\\shaocong{2} \\eta} - \\frac{L}{2}\\big)\\|\\theta_{k+1}  - \\theta_k \\|^2 \\Big]\\leq \\frac{1}{\\shaocong{2} \\eta}\\| \\theta_{0} -\\omega \\|^2 .\n\\end{align}\nNote that $ \\ell_{\\xi_k} (\\theta_{k+1}) - \\ell_{\\xi_k} ^\\ast$ is non-negative, and $\\big(\\frac{1}{\\shaocong{2} \\eta} - \\frac{L}{2}\\big)\\|\\theta_{k+1}  - \\theta_k \\|^2$ is also non-negative if we choose \\shaocong{$\\eta \\leq \\frac{1}{L}$}. Also, the left hand side of the above inequality is  bounded above for all $B$. Therefore, it implies that $ \\ell_{\\xi_k}(\\theta_{k+1}) -  \\ell_{\\xi_k} ^\\ast \\overset{k}{\\to} 0$, $\\|\\theta_{k+1}  - \\theta_k\\|\\overset{k}{\\to} 0$. In particular, for all subsequences $\\{i(T) \\}_T, i=1,...,n$, we have $ \\ell_{i}(\\theta_{i(T)+1}) -  \\ell_{i} ^\\ast \\overset{T}{\\to} 0$. Therefore, by continuity of the sample losses, we conclude that all the limit points of $\\{\\theta_{i(T)+1}\\}_T$ belong to the set $\\Theta_i^*$ for all $i$. Since $\\|\\theta_{k+1}  - \\theta_k\\|\\overset{k}{\\to} 0$, we conclude that all the limit points $\\mathfrak{X}_i$ of $\\{\\theta_{i(T)}\\}_T$ belong to the set $\\Theta_i^*$ for all $i$, and item 1 is proved.\n\n\nNext, we prove item 2. It suffices to show that $\\mathfrak{X}_i = \\mathfrak{X}_j$ for all $i\\ne j$. Consider any $\\omega \\in \\mathfrak{X}_i$ with a corresponding subsequence $\\theta_{i(T_k)} \\overset{k}{\\to} \\omega$. By the random reshuffle sampling, we have $|i(T_k) - j(T_k)| \\le n$ for all $i,j,k$. Also, note that $\\|\\theta_{k+1}  - \\theta_k\\|\\overset{k}{\\to} 0$. We obtain that\n\\begin{align}\n\\|\\theta_{j(T_k)} - \\omega \\| \\le \\|\\theta_{j(T_k)} - \\theta_{i(T_k)} \\| + \\|\\theta_{i(T_k)}  - \\omega\\| \\overset{k}{\\to}  0.\n\\end{align}\nTherefore, we showed that every $\\omega \\in \\mathfrak{X}_i$ is also in any other $\\mathfrak{X}_j$. In summary, $\\mathfrak{X}_i = \\mathfrak{X}_j = \\mathfrak{X}$.\nMoreover, since item 1 shows that $\\mathfrak{X}_i \\subset \\Theta_i^*$, we further obtain that $\\mathfrak{X} \\subset \\bigcap_{i=1}^n \\Theta_i^*$.\n\n\\section{Proof of  \\Cref{thm: conv}}\n\t\nWe prove it by contradiction. Assume there exists $\\omega_1, \\omega_2 \\in \\mathfrak{X}$ such that $\\omega_1 \\ne \\omega_2$. Let $\\theta_{q(k)} \\to \\omega_1$ and $\\theta_{p(k)} \\to \\omega_2$ be two converged subsequences. Without loss of generality, we can always assume that $p(k) > q(k)$ (if not, simply take a subsequence of $\\{p(k)\\}_k$ such that this property is satisfied). \n\nApply the inequality in Lemma 1 with any $\\omega \\in \\mathfrak{X}\\subset \\Theta^*$ and note that $p(k)>q(k)$, we obtain that\n\\begin{align}\n\\| \\theta_{p(k)} - \\omega \\|  \\leq \\| \\theta_{q(k)} - \\omega\\|.\n\\end{align}\nIn particular, set $\\omega = \\omega_1$,\tthe right hand side of the above inequality converges to $0$ because $\\omega_1$ is the unique limit point of $\\theta_{q(k)}$ by our choice. Therefore, we conclude that $\\omega_1$ is also a limit point of $\\{ \\theta_{p(k)} \\}_{k}$, and hence $\\omega_1 = \\omega_2$, contradiction.\n\n\n\n\n\\section{Proof of \\Cref{thm: rate_coherent-SC}}\nConsider the $k$-th iteration with sample $\\xi(k)$. By smoothness of $\\ell_{\\xi(k)}$, we obtain that\n\\begin{align*}\n\\ell_{\\xi(k)}(\\theta_{k+1} )\\leq \\ell_{\\xi(k)} (\\theta_k) + \\langle \\theta_{k+1}-\\theta_k , \\nabla \\ell_{\\xi(k)}(\\theta_k) \\rangle + \\frac{L}{2}\\|\\theta_{k+1}  - \\theta_k \\|^2.\n\\end{align*}\nOn the other hand, by restricted strong convexity of $\\ell_{\\xi(k)}$, we have: for all $\\omega \\in \\Theta_{\\xi(k)}^*$,\n\\begin{align}\n\\ell_{\\xi(k)}(\\omega) \\geq \\ell_{\\xi(k)}(\\theta_k) + \\langle \\omega - \\theta_k, \\nabla \\ell_{\\xi(k)}(\\theta_k) \\rangle + \\frac{\\mu_{\\xi(k)}}{2}\\|\\theta_{k} - \\omega\\|^2.\n\\end{align}\nCombining both inequalities above, we obtain that: for all $\\omega \\in \\Theta^*$,\n\\begin{align}\n\\ell_{\\xi(k)}(\\theta_{k+1} ) &\\leq  \\ell_{\\xi(k)}(\\omega)  + \\langle  \\theta_{k+1} -\\omega, \\nabla \\ell_{\\xi(k)}(\\theta_k) \\rangle   + \\frac{L}{2}\\|\\theta_{k+1}  - \\theta_k \\|^2 - \\frac{\\mu_{\\xi(k)}}{2}\\|\\theta_{k} - \\omega\\|^2 \\nonumber\\\\\n&=  \\ell_{\\xi(k)}(\\omega)  + \\langle  \\theta_{k+1} -\\omega, \\frac{1}{\\eta} (\\theta_{k} - \\theta_{k+1}) \\rangle   + \\frac{L}{2}\\|\\theta_{k+1}  - \\theta_k \\|^2 - \\frac{\\mu_{\\xi(k)}}{2}\\|\\theta_{k} - \\omega\\|^2 \\nonumber\\\\\n&= \\ell_{\\xi(k)}(\\omega)  + \\frac{1}{\\shaocong{2}\\eta}(\\|\\theta_{k} -\\omega  \\|^2 - \\| \\theta_{k+1} -\\omega\\|^2)   - \\big(\\frac{1}{\\shaocong{2}\\eta}-\\frac{L}{2} \\big)\\|\\theta_{k+1}  - \\theta_k \\|^2  - \\frac{\\mu_{\\xi(k)}}{2}\\|\\theta_{k} - \\omega\\|^2.\\nonumber \n\\end{align} \n\nNow let \\shaocong{$\\eta= \\frac{1}{L}$}. We further obtain that: for all $\\omega \\in \\Theta^*$,\n\\begin{align}\n\\|\\theta_{k+1} - \\omega\\|^2 &\\leq \\shaocong{\\big(1 -  {\\mu_{\\xi(k)}\\eta}  \\big)}\\|\\theta_{k} -\\omega \\|^2 - \\shaocong{2}\\eta \\Big(   \n\\ell_{\\xi(k)}  (\\theta_{k+1}) - \\ell_{\\xi(k)}^* \\Big) \\nonumber \\\\\n&\\leq \\big(1 - \\frac{\\mu_{\\xi(k)}}{L} \\big)\\|\\theta_{k} -\\omega \\|^2. \\label{eq: 3}\n\\end{align} \nTelescoping the above inequality over the $B$-th epoch and by sampling with random reshuffle, we conclude that: for all $\\omega \\in \\Theta^*$,\n\\begin{align*}\n\\| \\theta_{n(B+1)} - \\omega\\|^2 &\\leq \\prod_{i=1}^n \\big( 1- \\frac{\\mu_{i}}{L} \\big)   \\|\\theta_{nB} - \\omega \\|^2 \\nonumber\\\\\n&= \\alpha \\|\\theta_{nB} - \\omega \\|^2. \\nonumber\n\\end{align*}\nIn particular, choose $\\omega = \\argmin_{u\\in \\Theta^*} \\|\\theta_{nB} - u\\|$, the above inequality further implies that\n\\begin{align}\n\\mathrm{dist}_{\\Theta^*}^2(\\theta_{n(B+1)}) \\le \\| \\theta_{n(B+1)} - \\omega\\|^2 \\le \\alpha \\|\\theta_{nB} - \\omega \\|^2 = \\alpha\t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{nB}). \\nonumber\n\\end{align}\nThe desired result follows by telescoping the above inequality over the epoch index $B$.\n\n\\section{Proof of \\Cref{prop: rate_coherent_SGD}}\nOne can check that \\cref{eq: 3} still holds for SGD with random sampling, i.e., \n$$\\|\\theta_{k+1} - \\omega\\|^2  \\leq \\big(1 - \\frac{\\mu_{\\xi(k)}}{L} \\big)\\|\\theta_{k} -\\omega \\|^2.$$\nTaking expectation on both sides of the above inequality yields that\n$$\\mathbb{E} \\|\\theta_{k+1} - \\omega\\|^2  \\leq \\big(1 - \\frac{ \\bar{\\mu} }{L} \\big)\\mathbb{E} \\|\\theta_{k} -\\omega \\|^2,$$\nwhere $\\bar{\\mu} := \\frac{1}{n}\\sum_{i=1}^n \\mu_i$. \nTelescoping the above inequality over the $B$ epochs yields that\n$$\\mathbb{E} \\|\\theta_{nB} - \\omega\\|^2  \\leq \\big(1 - \\frac{ \\bar{\\mu} }{L} \\big)^{nB}\\mathbb{E} \\|\\theta_{0} -\\omega \\|^2.$$\n \n\n\n\n\n\\section{Proof of \\Cref{lemma: 2}}\nConsider the $k$-th iteration with sample $\\xi(k)$. By smoothness of $\\ell_{\\xi(k)}$, we obtain that\n\\begin{align*}\n\\ell_{\\xi(k)}(\\theta_{k+1} )\\leq \\ell_{\\xi(k)} (\\theta_k) + \\langle \\theta_{k+1}-\\theta_k , \\nabla \\ell_{\\xi(k)}(\\theta_k) \\rangle + \\frac{L}{2}\\|\\theta_{k+1}  - \\theta_k \\|^2.\n\\end{align*}\nOn the other hand, by restricted strong convexity of $\\ell_{\\xi(k)}$, we have for  $\\omega = \\proj{\\Theta^*}(\\theta_k)$,\n\\begin{align}\n\\ell_{\\xi(k)}(\\omega) \\geq \\ell_{\\xi(k)}(\\theta_k) + \\langle \\omega - \\theta_k, \\nabla \\ell_{\\xi(k)}(\\theta_k) \\rangle + \\frac{\\mu_{\\xi(k)}}{2}\\|\\theta_{k} - \\omega\\|^2.\n\\end{align}\nCombining both of the above inequalities, we obtain that\n\\begin{align}\n\\ell_{\\xi(k)}(\\theta_{k+1} ) &\\leq  \\ell_{\\xi(k)}(\\omega)  + \\langle  \\theta_{k+1} -\\omega, \\nabla \\ell_{\\xi(k)}(\\theta_k) \\rangle   + \\frac{L}{2}\\|\\theta_{k+1}  - \\theta_k \\|^2 - \\frac{\\mu_{\\xi(k)}}{2}\\|\\theta_{k} - \\omega\\|^2 \\nonumber\\\\\n&=  \\ell_{\\xi(k)}(\\omega)  + \\langle  \\theta_{k+1} -\\omega, \\frac{1}{\\eta} (\\theta_{k} - \\theta_{k+1}) \\rangle   + \\frac{L}{2}\\|\\theta_{k+1}  - \\theta_k \\|^2 - \\frac{\\mu_{\\xi(k)}}{2}\\|\\theta_{k} - \\omega\\|^2 \\nonumber\\\\\n&= \\ell_{\\xi(k)}(\\omega)  + \\frac{1}{\\shaocong{2}\\eta}(\\|\\theta_{k} -\\omega  \\|^2 - \\| \\theta_{k+1} -\\omega\\|^2)   - \\big(\\frac{1}{\\shaocong{2}\\eta}-\\frac{L}{2} \\big)\\|\\theta_{k+1}  - \\theta_k \\|^2  - \\frac{\\mu_{\\xi(k)}}{2}\\|\\theta_{k} - \\omega\\|^2.\\nonumber \n\\end{align} \nChoose \\shaocong{$\\eta= \\frac{1}{L}$} and rearrange the above inequality, we obtain that\n\\begin{align}\n\\|\\theta_{k+1} - \\omega\\|^2 &\\leq \\shaocong{\\big(1 - {\\mu_{\\xi(k)}} \\eta \\big)}\\|\\theta_{k} -\\omega \\|^2 - \\shaocong{2}\\eta \\Big(   \n\\ell_{\\xi(k)}  (\\theta_{k+1}) - \\ell_{\\xi(k)} (\\omega) \\Big)  \\nonumber\\\\\n&\\leq \\big(1 - \\frac{\\mu_{\\xi(k)}}{L} \\big)\\|\\theta_{k} -\\omega \\|^2 - \\shaocong{2}\\eta \\Big(\\ell_{\\xi(k)}  (\\theta_{k+1}) - \\ell_{\\xi(k)}^* + \\ell_{\\xi(k)}^*-\\ell_{\\xi(k)} (\\omega) \\Big) \\nonumber\\\\\n&\\leq \\big(1 - \\frac{\\mu_{\\xi(k)}}{L} \\big)\\|\\theta_{k} -\\omega \\|^2 - \\shaocong{2}\\eta \\Big(\\ell_{\\xi(k)}  (\\theta_{k+1}) - \\ell_{\\xi(k)}^* + \\ell_{\\xi(k)}^*-\\sup_{\\omega\\in \\Theta^*} \\ell_{\\xi(k)} (\\omega) \\Big) \\nonumber\\\\\n&\\leq \\big(1 - \\frac{\\mu_{\\xi(k)}}{L} \\big)\\|\\theta_{k} -\\omega \\|^2 + \\shaocong{2}\\eta \\epsilon, \\label{eq: 4}\n\\end{align} \nwhere the last inequality uses the definition of minimizer incoherence, which is bounded by $\\epsilon$. \n Telescoping the above inequality over the iterations of the $B$-th epoch, we obtain that\n\\begin{align}\n\\| \\theta_{n(B+1)} - \\omega\\|^2 \\leq \\prod_{i=1}^n \\Big( 1- \\frac{\\mu_i}{L}\\Big)  \\|\\theta_{nB} - \\omega \\|^2 +  \\shaocong{2}\\eta\\epsilon \\sum_{k = nB}^{n(B+1) - 1} \\prod_{s=k+1}^{n(B+1)-1}  \\Big(1 - \\frac{\\mu_{\\xi(s)}}{L} \\Big)  , \\label{eq: tele}\n\\end{align}\nwhere we define $\\prod_{s=n(B+1)}^{n(B+1)-1}  \\Big(1 - \\frac{\\mu_{\\xi(s)}}{L} \\Big) = 1$ by default. Note that the above inequality is an epochwise contraction with a bounded error term $\\eta\\epsilon \\sum_{k = nB}^{n(B+1) - 1} \\prod_{s=k+1}^{n(B+1)-1}  \\Big(1 - \\frac{\\mu_{\\xi(s)}}{L} \\Big)$, we conclude that $\\| \\theta_{n(B+1)} - \\omega\\|^2$ is bounded for all $B$ and hence $\\{\\theta_k \\}_k$ is bounded.\n\n\n\n\\section{Proof of \\Cref{thm: incohe_sc}}\nNote that \\cref{eq: tele} further implies that\n\\begin{align}\n\\mathrm{dist}_{\\Theta^*}^2(\\theta_{n(B+1)}) &\\le \\| \\theta_{n(B+1)} - \\omega\\|^2 \\nonumber\\\\\n&\\le  \\prod_{i=1}^n \\Big( 1- \\frac{\\mu_i}{L}\\Big)  \\|\\theta_{nB} - \\omega \\|^2 + \\shaocong{2} \\eta   \\epsilon\\sum_{k = nB}^{n(B+1) - 1} \\prod_{s=k+1}^{n(B+1)-1}  \\Big(1 - \\frac{\\mu_{\\xi(s)}}{L} \\Big) \\nonumber\\\\\n&= \\prod_{i=1}^n \\Big( 1- \\frac{\\mu_i}{L}\\Big) \\mathrm{dist}_{\\Theta^*}^2(\\theta_{nB}) + \\shaocong{2} \\eta   \\epsilon \\sum_{k = nB}^{n(B+1) - 1} \\prod_{s=k+1}^{n(B+1)-1}  \\Big(1 - \\frac{\\mu_{\\xi(s)}}{L} \\Big). \\label{eq: 1}\n\\end{align}\nNext, denote $\\sigma_B$ as the random shuffle permutation performed in epoch $B$ and define the quantity\n\\begin{align}\nM(\\sigma_B) := \\sum_{k = n(B-1)}^{nB - 1} \\prod_{s=k+1}^{nB-1}  \\Big(1 - \\frac{\\mu_{\\xi(s)}}{L} \\Big)  . \\nonumber\n\\end{align}\nIt is clear that $M(\\sigma_B)$ is a random variable that depends on the permutation $\\sigma_B$. We define its expectation as $\\mathbb{E}_\\sigma M(\\sigma_B) := \\overline{M}$, which is a fixed constant for every epoch $B$. Then, taking expectation on both sides of \\cref{eq: tele} yields that\n\\begin{align}\n\\mathbb{E} \t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{n(B+1)}) \\leq \\alpha \\mathbb{E} \t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{nB}) + \\shaocong{2} \\eta \\epsilon\\overline{M}. \\nonumber\n\\end{align}\nRearranging the above inequality further yields that\n\\begin{align}\n\\mathbb{E} \t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{n(B+1)}) - \\frac{\\shaocong{2}\\eta\\epsilon\\overline{M}}{1-\\alpha} \\le \\alpha \\Big(\\mathbb{E} \t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{nB}) - \\frac{\\shaocong{2}\\eta\\epsilon\\overline{M}}{1-\\alpha} \\Big), \\nonumber\n\\end{align}\nwhich, after telescoping over $B$, further gives that: for all $B$,\n\\begin{align}\n\\mathbb{E} \t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{nB}) \\le \\alpha^B \\Big(\t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{0}) - \\frac{\\shaocong{2}\\eta\\epsilon\\overline{M}}{1-\\alpha} \\Big) + \\frac{\\shaocong{2}\\eta\\epsilon\\overline{M}}{1-\\alpha}. \\nonumber\n\\end{align}\nLastly, note that we choose \\shaocong{$\\eta = \\frac{1}{L}$}.\n\n\n\\section{Proof of \\Cref{coro: 2}}\nThe proof is similar to that of \\Cref{thm: incohe_sc}. The only difference is that the sampling order of the index $\\{\\sigma(k) \\}_k$ is now deterministic. \n\nOne can check that \\cref{eq: 1} is valid for SGD with incremental sampling by replacing $\\xi(s)$ with $\\sigma(s)$, and we have \n\\begin{align}\n\\mathrm{dist}_{\\Theta^*}^2(\\theta_{n(B+1)}) &\\le \\prod_{i=1}^n \\Big( 1- \\frac{\\mu_i}{L}\\Big) \\mathrm{dist}_{\\Theta^*}^2(\\theta_{nB}) +  \\shaocong{2}\\eta   \\epsilon \\sum_{k = nB}^{n(B+1) - 1} \\prod_{s=k+1}^{n(B+1)-1}  \\Big(1 - \\frac{\\mu_{\\sigma(s)}}{L} \\Big) \\nonumber\\\\\n&\\overset{\\text{def}}{:=} \\prod_{i=1}^n \\Big( 1- \\frac{\\mu_i}{L}\\Big) \\mathrm{dist}_{\\Theta^*}^2(\\theta_{nB}) +  \\shaocong{2}\\eta   \\epsilon \\widetilde{M}. \\nonumber\n\\end{align}\nThen, the desired result follows from a standard telescoping over $B$ and \\shaocong{$\\eta = \\frac{1}{L}$}.\n\n\n\\section{Proof of \\Cref{prop: error_bound}}\nOne can check that \\cref{eq: 4} still holds for SGD with random sampling and step size \\shaocong{$\\eta=\\frac{1}{L}$}. Taking expectations on both sides of the inequality and simplifying yields that\n\\begin{align}\n\\mathbb{E} \\mathrm{dist}_{\\Theta^*}^2(\\theta_{k+1}) &\\leq \\big(1 - \\frac{\\overline{\\mu}}{L} \\big) \\mathbb{E}\\mathrm{dist}_{\\Theta^*}^2(\\theta_{k}) + \\shaocong{2}\\eta {\\epsilon}, \n\\end{align} \nRearranging and simplifying  the above inequality yields that\n\\begin{align}\n\\mathbb{E} \t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{k+1}) - \\frac{\\shaocong{2}\\eta {\\epsilon}}{1-(1-\\overline{\\mu}\/L)} \\le  \\big(1 - \\frac{\\overline{\\mu}}{L} \\big) \\Big(\\mathbb{E} \t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{k}) - \\frac{\\shaocong{2}\\eta {\\epsilon}}{1-(1-\\overline{\\mu}\/L)} \\Big), \\nonumber\n\\end{align}\nwhich, after telescoping over $k$, further gives that: for all $k=nB$,\n\\begin{align}\n\\mathbb{E} \t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{nB}) \\le \\big(1 - \\frac{\\overline{\\mu}}{L} \\big)^{nB} \\Big(\t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{0}) - \\frac{\\shaocong{2} \\eta {\\epsilon}L}{\\overline{\\mu}} \\Big) + \\frac{\\shaocong{2} \\eta {\\epsilon}L}{\\overline{\\mu}}. \\nonumber\n\\end{align}\n\n\n\n\n\n\n\n\n\\section{Analysis under Full Minimizer Coherence}\nIn this section, we study the convergence properties of SGD with random reshuffle under full minimizer coherence, i.e., $\\epsilon_i=0$ for all $i=1,...,n$ (see \\Cref{def: incoherence}) and hence all sample losses share a set of global minimizers $\\Theta^*$. \n\n\\subsection{Convergence of SGD Trajectory}\nWe note that all the results in this subsection only require the sample losses to be restricted convex.\nWe first characterize the boundedness of the optimization trajectory of SGD with random reshuffle.\n\\begin{lemma}[Bounded trajectory]\\label{lemma: 3}\n\tLet Assumptions \\ref{assum: exist} and \\ref{assum: f} hold and assume that the problem (P) has full minimizer coherence. Apply SGD with random reshuffle with step size \\shaocong{$\\eta\\leq \\frac{1}{L}$} to solve the problem. Then, the SGD trajectory $\\{\\theta_k\\}_k$ is bounded.\n\\end{lemma}\n\nAs the trajectory of SGD with random reshuffle $\\{\\theta_k \\}_k$ is bounded, it has a compact set of limit points and we denote it as $\\mathfrak{X}$. Also, note that the iteration index sequence  $\\{k\\}_{k\\in \\mathbb{N}}$ can be decomposed into $n$ subsequences $\\{i(T)\\}_T, i=1,...,n$, each of which tracks the SGD iterations that sample the $i$-th data point in the epochs $T=1,2,...$. In particular, we denote $\\mathfrak{X}_i$ as the set of limit points of \\shaocong{$\\{\\theta_{i(T)}\\}_T$} and it holds that $\\mathfrak{X}=\\bigcup_{i=1}^n \\mathfrak{X}_i$. Moreover, we obtain the following properties regarding the limit point sets of the trajectory of SGD with random reshuffle.\n\n\\begin{proposition}[Limit points]\\label{prop: 2}\n\tUnder the same conditions as those of \\Cref{lemma: 3}, the trajectory of SGD with random reshuffle satisfies the following properties.\n\t\\begin{enumerate}[topsep=0pt, noitemsep, leftmargin=*]\n\t\t\\item $\\mathfrak{X}_i \\subset \\Theta_i^*$ for all $i=1,...,n$;\n\t\t\\item $\\mathfrak{X}_i = \\mathfrak{X} \\subset \\Theta^*$ for all $i=1,...,n$.\n\t\\end{enumerate}\n\\end{proposition}\n\nTo elaborate, item 1 shows that each sub-trajectory $\\{\\theta_{i(T)}\\}_T$ generated by SGD with random reshuffle is a minimizing sequence for the corresponding sample loss $\\ell_i$. Item 2 further strengthens item 1 by showing that  all the sub-trajectories $\\{i(T)\\}_T, i=1,...,n$ share the same set of limit points, which is a subset of the global minimizer set of the total loss. Intuitively, this is due to the fact that all the sample losses share a set of global minimizers under full minimizer coherence, which guarantees the sub-trajectories of SGD with random reshuffle to have consistent asymptotic properties. \n\nThe proof of \\Cref{prop: 2} consists of two major steps. We first exploit full minimizer coherence to prove item 1 and the stationary condition $\\|\\theta_{k+1} - \\theta_k\\| \\overset{k}{\\to} 0$. Then, the stationary condition further guarantees that all sub-trajectories share the same set of limit points and hence implies item 2.\n\nOur main result below further strengthens the convergence properties of the SGD trajectory.\n\\begin{thm}[Trajectory convergence]\\label{thm: conv}\n\tUnder the same conditions as those of \\Cref{lemma: 3}, every trajectory $\\{\\theta_k\\}_k$ generated by SGD with random reshuffle converges to a certain global minimizer in $\\Theta^*$, i.e., it has a single limit point.\n\\end{thm}\n\nThe above result shows that the entire trajectory of SGD with random reshuffle converges to a certain global minimizer in the case of full minimizer coherence. This implies that full minimizer coherence helps suppress the randomness of the random reshuffle and leads to \\shaocong{the point-wise convergence}. Such a deterministic convergence result of SGD with random reshuffle is stronger than other {\\em in-expectation} convergence results of SGD with random sampling that are established under various loss conditions (e.g., strong growth condition, interpolation) that imply full minimizer coherence.   \n\n\n\\subsection{Convergence Rate Analysis}\nIn this subsection, we further study the convergence rate of SGD with random reshuffle under full minimizer coherence. For any point $\\theta\\in \\mathbb{R}^d$, we denote its distance to an arbitrary set $A\\subset \\mathbb{R}^d$ as $\\mathrm{dist}_{A}(\\theta):=\\inf_{u\\in A} \\|\\theta-u\\|$.\n\nWe obtain the following convergence rate result. \n\\begin{thm}[Random reshuffle]\\label{thm: rate_coherent-SC}\n\tLet Assumptions \\ref{assum: exist} and \\ref{assum: f} hold and assume that the problem (P) has full minimizer coherence. Apply SGD with random reshuffle with step size \\shaocong{$\\eta = \\frac{1}{L}$} to solve the problem. Then, for all epochs $B=1,2,...$, it holds that\n\t\\begin{align}\\label{eq: rate1}\n\t\t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{nB}) \\le \\prod_{i=1}^n \\Big(1-\\frac{\\mu_i}{L}\\Big)^B \\mathrm{dist}_{\\Theta^*}^2(\\theta_{0}).\n\t\\end{align}\n\n\\end{thm}\n\nThe above theorem establishes the linear convergence rate of SGD with random reshuffle under full minimizer coherence and the constant step size \\shaocong{$\\eta = \\frac{1}{L}$}. In particular, the convergence rate depends on the curvature incoherence parameter $\\alpha = \\prod_{i=1}^n (1-\\frac{\\mu_i}{L})$ (see \\Cref{def: cur_incoherence}) that characterizes the quality of the condition numbers of all the sample losses. We also note that in the special case that all the sample losses have the same condition number $\\frac{\\mu}{L}$, the above convergence rate of SGD with random reshuffle is of order $\\mathcal{O}(1-\\frac{\\mu}{L})^{nB}$, which meets the convergence rate of full gradient descent under strong convexity. \n\n\n\\subsection{Comparison to Other Sampling Schemes} \nWe further analyze the convergence rates of SGD with incremental sampling (i.e., cyclic sampling without random reshuffle) and random sampling under full minimizer coherence and compare them with that of SGD with random reshuffle. \n\nIn fact, under full minimizer coherence, our proof of \\Cref{thm: rate_coherent-SC} only rely on the fact that the random reshuffle scheme samples every data point once in each epoch, which is also satisfied by the incremental sampling scheme. Therefore, the convergence rate result in \\Cref{thm: rate_coherent-SC} also applies to SGD with incremental sampling and we obtain the following corollary.\n\n\\begin{coro}[Incremental sampling]\\label{coro: 1}\n\tUnder the same settings as those of \\Cref{thm: rate_coherent-SC} and apply SGD with incremental sampling and step size \\shaocong{$\\eta = \\frac{1}{L}$} to solve the problem (P). Then, the convergence rate is also characterized by \\cref{eq: rate1}.\n\\end{coro}\n\nOn the other hand, we obtain the following result for SGD with random sampling.\n\n\n\\begin{proposition}[Random sampling]\\label{prop: rate_coherent_SGD}\n\tUnder the same settings as those of \\Cref{thm: rate_coherent-SC} and apply SGD with random sampling and step size \\shaocong{$\\eta = \\frac{1}{L}$} to solve the problem (P). Then, for all epochs $B=1,2,...$, it holds that\n\t\\begin{align}\n\t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{nB}) \\leq \\Big(1 - \\frac{\\bar{\\mu}}{L}\\Big)^{nB} \\mathrm{dist}_{\\Theta^*}^2(\\theta_{0}),\n\t\\end{align} \n\twhere $\\bar{\\mu} := \\frac{1}{n}\\sum_{i=1}^n \\mu_i$. \n\\end{proposition}\nThe above result establishes a linear convergence rate for SGD with random sampling. Note that the convergence rate depends on the average of the condition numbers of the sample losses. This is different from the convergence rate of SGD with random reshuffle, which depends on the product of the condition numbers of all the sample losses. In particular, by the arithmetic mean-geometric mean (AM-GM) inequality, it holds that\n\\begin{align}\n\t\\prod_{i = 1}^n \\Big( 1 - \\frac{\\mu_i}{L} \\Big) \\leq \\Big( 1 - \\frac{\\bar{\\mu}}{L} \\Big)^n. \\label{eq: am-gm}\n\\end{align}\nTherefore, under full minimizer coherence, SGD achieves a faster convergence rate under random reshuffle than that under random sampling. Such a result provides a theoretical justification for the superior performance of SGD with random reshuffle in training over-parameterized models. \n\nWe note that \\cite{haochen2018random} also obtains a similar comparison of convergence rate between SGD with random sampling and SGD with random reshuffle. However, their analysis requires the loss to be uniformly strongly convex, whereas our result applies to the broader class of restricted strongly convex functions. Moreover, we established trajectory convergence of SGD with random reshuffle under the existence of multiple global minimizers, whereas their result establishes convergence in expectation under the existence of a unique global minimizer.\n\n\\subsection{Empirical Verification}\nIn this subsection, we verify our theoretical results via experiments. \nWe first study the impact of curvature incoherence $\\alpha$ on the convergence of SGD with random reshuffle. In specific, we train a Resnet 18 network using SGD with random reshuffle on a mini MNIST dataset that consists of 1000 images of digit ``1'' and 1000 images of digit ``8''. To model different distributions of curvature incoherence, we divide the data samples evenly into 50 fixed mini-batches and consider two different settings: 1) each mini-batch contains 50\\% images of digit ``1'' and 50\\%  images of digit ``8''; and 2) each mini-batch contains either images of digit ``1'' or images of digit ``8''. In both settings, the average condition numbers of the sample losses are different. \n\\Cref{fig: 4} (Left) shows the training loss curves of SGD with random reshuffle in these two settings starting from the same initialization point. It can be observed that SGD with random reshuffle converges faster in the second setting. This implies that the average condition number of the sample losses in the first setting is better than that in the second setting.\n\n\nNext, we further compare the empirical convergence of SGD under random reshuffle with that under incremental sampling and random sampling. We train a Resnet18 on $4096$ images sampled from CIFAR10 using SGD with the three sampling schemes. We use learn rate $\\eta=0.03$, batch-size $128$ and a fixed initialization model that is trained by SGD with random reshuffle for one epoch (with learning rate $0.0025$) using a pre-trained ImageNet model. \\Cref{fig: 4} (Right) shows the training loss curves of the three algorithms. It can be seen that SGD with random reshuffle and incremental sampling have a comparable convergence speed, both of which are faster than that of SGD with random sampling. This observation fully supports our theoretical comparison in \\cref{eq: am-gm}.\n\\begin{figure}[bth]\\centering  \n\t\\includegraphics[width=0.49\\linewidth]{FIGs\/NN3.png} \n\t\\includegraphics[width=0.49\\linewidth]{FIGs\/NN.png}\n\t\\caption{Left: Comparison of convergence of SGD with random reshuffle under different curvature incoherence distributions. Right: Comparison of convergence of SGD under random reshuffle, incremental sampling and random sampling.}\\label{fig: 4}\n\\end{figure}\n\n\n   \n\n\n\n\n\n\\subsection{Discussion on Sampling Scheme}\n\n\n\\subsection{Optimal Batch Size}\n\\begin{definition}[batch-model incoherence] Assume the dataset with $n$ observations is divided into $m$ mini-batches ($b_1, \\dots, b_m$).Then $\\frac{n}{m}$ is the batch size and the batch-model incoherence of the $i$-th mini-batch is defined as \n\t$$\\epsilon_{b_i} = \\sup_{\\theta \\in \\Theta} \\frac{m}{n}\\sum_{j \\in b_i} l_j(\\theta) - \\frac{m}{n}( \\sum_{j \\in b_i} l_j )^\\ast. $$\n\\end{definition}\n\\begin{remark}\n\tFor gradient decent, it is easy to see $\\epsilon = 0$.\n\\end{remark}\n\n\\begin{thm}\\label{batchsize1}\n\tLet $b$ be a given mini-batch and $\\epsilon_b$ be the batch-model incoherence on the batch $b$. Then \n\t$$\\epsilon_b \\leq (\\sum_{j\\in b } \\epsilon_i) \\cdot \\frac{m}{n}.$$\n\\end{thm}\n\n\\begin{example}\n\tUnder strongly convex assumption  \n\t$$\\epsilon_b \\leq (\\sum_{j \\in b} \\epsilon_j- \\sum_{j \\in b}\\alpha_j \\| \\theta_b - \\theta_j^\\ast \\|^2)\\cdot \\frac{m}{n}.$$\n\\end{example}\n\n\\begin{lemma}[Batch $\\alpha$-diverse]\\label{lemma: batch_alpha}\n\tFor any $\\theta \\notin \\bigcup_{i=1}^n \\Theta_i^*$, there exists $\\alpha \\le 1$ such that\n\t\\begin{align*}\n\tf(\\theta) - \\ell_{b(j)}(\\theta)  \\leq &\\alpha \\left( f(\\theta) - f^\\ast \\right) + \\frac{1}{m}\\sum_{i=1}^m (\\ell_{b(i)}^\\ast+\\epsilon_{b(i)}) - \\ell_{b(j)}(\\theta). \n\t\\end{align*}\n\\end{lemma}\n$K=mB$.\n\\begin{thm} \n\tLet \\Cref{assum: exist} and \\ref{assum: f} hold for the problem (P). Apply SGD with cyclic-types of sampling schemes and learning rate $\\eta<\\frac{2}{L}$ to solve the problem (P). Then, the average of the total loss along the optimization path $\\{\\theta_k\\}_k$ satisfies\n\t\\begin{align*}\n\t\\frac{1}{K}\\sum_{k=0}^{K-1}  f( \\theta_{k}) - f^\\ast  \\leq \\frac{L\\mathrm{dist}_\\Theta^2(\\theta_{0})}{2K(1-\\alpha)} + \\frac{\\sum_{i=1}^m \\epsilon_i}{m(1-\\alpha)}.\n\t\\end{align*}\n\\end{thm}\n\nIf all loss are $\\mu_i$ r.s.c\n\n\\subsection{Effect of Sampling Scheme}\nSampling scheme may influence the batch model incoherence.  \n\\begin{definition} Let $\\sigma \\in \\{1,\\dots, n\\}^n$. Assume $n$ observations is divided into $m$ mini-batches ($b_1, \\dots, b_m$). The batch-model incoherence of the $i$-th mini-batch corresponding to the permutation $\\sigma$ is defined as \n\t$$\\epsilon_{b_i}(\\sigma) = \\sup_{\\theta \\in \\Theta} \\frac{m}{n}\\sum_{j \\in b_i} l_{\\sigma(j)}(\\theta) - \\frac{m}{n}( \\sum_{j \\in b_i} l_{\\sigma(j)} )^\\ast. $$\n\\end{definition}\nGiven $m$ mini-batches $\\{ b_1, \\dots, b_m \\}$, define $\\mathcal{L}: S_n \\to \\mathbb{R}_+$ as\n\\begin{align*}\n\\mathcal{L}: \\sigma \\mapsto \\frac{1}{m}\\sum_{i=1}^m \\epsilon_{b_i}(\\sigma).\n\\end{align*} \nLet $\\sigma^\\ast := \\argmin_{\\sigma \\in S_n} \\mathcal{L}$. \\textbf{Question:} Is it the best choice?\n\n\n\n\\section{Introduction to Model Incoherence}\nIn this section, we introduce two notions of model incoherence.\nRecall the finite-sum optimization problem\n\\begin{align}\n\\min_{x\\in\\mathbb{R}^d} f(\\theta) := \\frac{1}{n}\\sum_{i=1}^n \\ell_i(\\theta). \\tag{P}\n\\end{align} \nWe make the following standard assumption on the existence of solution set of the problem (P). \n\\begin{assum}[Existence of solution set]\\label{assum: exist}\n\tEach sample loss $\\ell_i, i=1,...,n$ has a solution set $\\Theta_i^*\\subset \\mathbb{R}^d$, on which its global minimum $\\ell_i^* >-\\infty$ is attained. The total loss $f$ has a solution set $\\Theta^*\\subset \\mathbb{R}^d$, on which its global minimum $f^*>-\\infty$ is attained.      \n\\end{assum}\n\nIn general, the solution sets $\\{\\Theta_i^*\\}_{i=1}^n$ of the sample losses can be different from the solution set $\\Theta^*$  of the total loss.\n\nWe also make the following standard assumptions on the sample losses, where we denote $\\proj{A}(x)$ as the Euclidean projection of $x$ onto set $A$. \n\\begin{assum}\\label{assum: f}\n\tThe problem (P) satisfies:\n\t\\begin{enumerate}[topsep=0pt, noitemsep, leftmargin=*]\n\n\t\t\\item The sample losses are $L$-smooth, i.e., $\\forall i$ and $x,y\\in\\mathbb{R}^d$,\n\t\t\\begin{align}\n\t\t\t\\ell_{i}(x) \\le \\ell_i(y) + \\inner{x-y}{\\nabla \\ell_i(y)} + \\frac{L}{2}\\|x-y\\|^2; \\nonumber\n\t\t\\end{align}\n\t\t\\item Every sample loss $\\ell_i$ \\shaocong{is $\\mu_i$-weakly strong convex on $\\Theta_i^* \\cup \\Theta^*$}, i.e., \n\t\t\\begin{align}\n\t\t\t\\ell_i(\\omega) \\ge \\ell_i(x) + \\inner{\\omega - x}{\\nabla \\ell_i(x)} + \\frac{\\mu_i}{2}\\|x-\\omega\\|^2 \\nonumber\n\t\t\\end{align} \n\t\tholds for all $x\\in \\mathbb{R}^d, \\omega \\in \\proj{\\Theta_i^*}(x),  \\proj{\\Theta^*}(x) $.\n\t\\end{enumerate}\n\\end{assum}\n\\shaocong{We note that the weak strong convexity is a weaker condition than the usual strong convexity and covers a wide range of non-convex problems including phase retrieval \\cite{Zhou2016b,zhang_2018}, neural networks \\cite{zhong_2017,Zhou2017}, low-rank matrix factorization \\cite{Tu_2016}, blind deconvolution \\cite{Li_2018}, etc. Also, the weak strong convexity implies the restricted strong convexity under an additional convexity condition. We refer to \\cite{karimi2016linear} for further discussions.}\n\n\n\\subsection{Minimizer Incoherence}\nIn this subsection, we introduce minimizer incoherence to measure the discrepancy between the sample loss solution sets $\\{\\Theta_i^*\\}_{i=1}^n$  and the total loss solution set $\\Theta^*$. In \\Cref{subsec: compare_condition}, we provide a discussion that outlines the connections between the minimizer incoherence and other loss conditions that have been studied in the existing literature.\n\n\\begin{definition}[Minimizer incoherence]\\label{def: incoherence}\n\tThe minimizer incoherence $\\epsilon_i>0$ of every sample loss $\\ell_i$ is defined as\n\t\\begin{align*}\n\t\\epsilon_i :=\\sup_{\\theta \\in \\Theta^*}   \\ell_i(\\theta)  -  \\ell_i^\\ast, \\quad i=1,...,n.\n\t\\end{align*} \n\\end{definition}\n\nTo elaborate, the minimizer incoherence corresponds to the gap between the highest sample loss that is achievable on the total loss solution set and the global minimum of the sample loss. Intuitively, it measures the incoherence between the sample loss solution set $\\Theta_i^*$ and the total loss solution set $\\Theta^*$. In particular, when minimizer incoherence vanishes, the following inclusion properties of the solution sets hold.\n\\begin{proposition}[Minimizer coherence]\\label{prop: vanish_incoherence}\n\tThe definition of minimizer incoherence implies that\n\t\\begin{enumerate}[topsep=0pt, noitemsep, leftmargin=*]\n\t\t\\item If $\\epsilon_i=0$ for some $i$, then $\\Theta^* \\subset \\Theta_i^*$;\n\t\t\\item If $\\epsilon_i=0$ for all $i$, then $\\Theta^* = \\cap_{i=1}^n \\Theta_i^*$.\n\t\\end{enumerate}\n\\end{proposition}\nIn particular, the second item corresponds to the case where we have full minimizer coherence, i.e., all the sample losses share the set of global minimizers $\\Theta^*$. This is common in deep learning applications where the models are over-parameterized to overfit all the data samples (hence have full minimizer coherence) and have multiple global minimizers. \nMoreover, our minimizer incoherence generalizes the interpolation condition proposed in \\cite{ma2017power}, which requires all the sample losses to share a unique global minimizer under strong convexity. \n\n\\begin{figure}[bth]\n\t\\centering\n\t\\includegraphics[width=0.4\\linewidth]{FIGs\/min_incohere.pdf}\n\t\\includegraphics[width=0.4\\linewidth]{FIGs\/min_cohere.pdf}\n\t\\caption{Left: Illustration of minimizer incoherence. Right: Illustration of full minimizer coherence.}\\label{fig: 1} \n\\end{figure}\n\n\\Cref{fig: 1} illustrates the cases of both minimizer incoherence and full minimizer coherence via quadratic sample losses. In fact, many nonconvex machine learning problems have been shown to have either vanishing or small minimizer incoherence, and we provide two illustrative examples below.\n\\begin{itemize}[topsep=0pt, noitemsep, leftmargin=*]\n\t\\item Phase retrieval \\cite{zhang_2018}: In this problem, we take linear measurements of an underlying complex signal $x_0\\in \\mathbb{C}^d$ with multiple Gaussian vectors $\\{a_i \\}_{i=1}^n$ and make phaseless observations $y_i = |a_i^\\intercal x_0|$. The goal is to recover the complex signal up to a global phase shift by solving the problem\n\t\\begin{align}\n\t\t\\min_{x\\in\\mathbb{C}^d} f(x) := \\frac{1}{2n}\\sum_{i=1}^{n} \\big(y_i - |a_i^\\intercal x|\\big)^2. \\nonumber\n\t\\end{align}\n\tIt is clear that all the sample losses share the set of minimizers $\\{x_0e^{j\\phi}~|~\\phi\\in (0, 2\\pi] \\}$ and hence have full minimizer coherence.\n\t\\vspace{2pt}\n\t\\item Over-parameterized neural networks: In deep learning, the neural network model $\\theta$ is typically over-parameterized so that the predictor $h_\\theta$ can be trained to overfit all the training samples, i.e., $h_\\theta(x_i) \\approx y_i$ for all $i=1,...,n$. Such overfitting usually achieves a small total loss as well as small sample losses. To justify this, we train a Resnet18 network to overfit the MNIST dataset \\shaocong{with the cross-entropy loss. We do not apply any regularization}. \\Cref{fig: Resnet18} shows the distribution of the sample losses after $50$ training epochs. One can see that most of the sample losses are below $3\\times 10^{-3}$, implying that deep models have very small minimizer incoherence. \n\n\t\\begin{figure}[bth]\\centering\n\t\t\\includegraphics[width=0.8\\linewidth]{FIGs\/nnzero.png} \n\t\t\\caption{Distribution of sample losses after training ResNet18 for $50$ epochs on MNIST dataset.} \\label{fig: Resnet18}\n\t\\end{figure}\n\\end{itemize}\n\n\\subsection{Curvature Incoherence}\n\n\nIn this subsection, we introduce the curvature incoherence. Recall that the condition number of each sample loss $\\ell_i$ is $\\frac{L}{\\mu_i}$. Then, we define the following curvature incoherence. \n\n\n\\begin{definition}[Curvature incoherence]\\label{def: cur_incoherence}\n\tThe curvature incoherence $\\alpha$ of the sample losses is defined as \n\t\\begin{align*}\n\t\\alpha:= \\prod_{i=1}^n\\Big(1 - \\frac{\\mu_i}{L} \\Big).\n\t\\end{align*}\nIn particular, $\\alpha$ belongs to the range $[0,1)$.\n\\end{definition}\nAs an intuitive understanding, if all the sample losses have a good condition number (i.e., $\\frac{L}{\\mu_i}\\to 1$), then $\\alpha$ vanishes and the curvatures of all sample losses are highly coherent. \n\n\n\n\n\n\n\\section{Analysis under Minimizer Incoherence}\nIn this section, we study the convergence of SGD with random reshuffle under minimizer incoherence where $\\epsilon_i > 0$ for some $i\\in\\{1,...,n\\}$. In such a case, the minimizer sets of the sample losses are different from that of the total loss. For simplicity, we assume the minimizer incoherences of all the sample losses are bounded by $\\epsilon:= \\max_{i\\in \\{1,...,n\\}} \\epsilon_i$. \n\n\n\\subsection{Convergence Rate Analysis}\n\n\nWe first show the boundedness of the trajectory of SGD with random reshuffle under minimizer incoherence and a constant step size.\n\\begin{lemma}[Bounded trajectory]\\label{lemma: 2}\n\tLet Assumptions \\ref{assum: exist} and \\ref{assum: f} hold and assume that the problem (P) has bounded minimizer incoherence $\\epsilon$. Apply SGD with random reshuffle with step size \\shaocong{$\\eta = \\frac{1}{L}$} to solve the problem. Then, the SGD trajectory $\\{\\theta_k\\}_k$ is bounded.\n\\end{lemma}\nThe above result generalizes the bounded trajectory result in \\Cref{lemma: 3}, which is proved under full minimizer coherence. \nNext, we obtain the following result regarding the convergence rate of SGD with random reshuffle under minimizer incoherence and a constant step size.\n\n\\begin{thm}[Random reshuffle]\\label{thm: incohe_sc}\n\tLet Assumptions \\ref{assum: exist} and \\ref{assum: f} hold. Apply SGD with random reshuffle with step size $\\eta=\\frac{1}{L}$ to solve the problem (P). Then, for all epochs $B=1,2,...$,\n\t\t\\begin{align}\n\t\t&\\mathbb{E} \t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{nB}) \\le \\alpha^B \\Big(\t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{0}) \\!-\\! \\frac{2\\epsilon \\overline{M}}{L(1\\!-\\!\\alpha)} \\Big) \\!+\\! \\frac{2\\epsilon \\overline{M}}{L(1\\!-\\!\\alpha)}, \\nonumber \n\t\t\\end{align}\n{where} $\\overline{M} = \\mathbb{E}_{\\xi}  \\bigg[\\sum_{k = 0}^{n - 1} \\prod_{s=k+1}^{n-1}  \\Big(1 - \\frac{\\mu_{\\xi(s)}}{L} \\Big) \\bigg]$ and $\\alpha$ corresponds to the curvature incoherence.\n\\end{thm}\n\nThe above result shows that SGD with random reshuffle converges linearly to a neighborhood of the global minimizer set under minimizer incoherence. Similar to the full minimizer coherence case, the convergence rate coefficient is determined by the curvature incoherence $\\alpha$. Moreover, the size of the neighborhood is characterized by the minimizer incoherence $\\epsilon$ and the condition numbers of the sample losses. This explains why over-parametrized models such as neural networks can be trained to achieve a small loss by SGD with constant step size: they have very small minimizer incoherence, as demonstrated by the experiment in \\Cref{fig: Resnet18}. \nIn general, a higher minimizer incoherence and worse condition numbers lead to a larger convergence error of SGD. \n\n\n\\subsection{Comparison to Other Sampling Schemes}\nWe also obtain the convergence rates of SGD with incremental sampling and random sampling under minimizer incoherence and a constant step size. \n\nIn specific, for SGD with incremental sampling, we denote $\\{\\sigma(0), \\sigma(1),...,\\sigma(n-1)\\}$ as a specific permutation of the data sample indexes. Such a permutation is fixed throughout the entire training process under incremental sampling. We obtain the following result on SGD with incremental sampling.\n\n\\begin{coro}[Incremental sampling]\\label{coro: 2}\n\tLet Assumptions \\ref{assum: exist} and \\ref{assum: f} hold. Apply SGD with incremental sampling with step size $\\eta=\\frac{1}{L}$ to solve the problem (P). Then, for all epochs $B=1,2,...$,\n\t\\begin{align}\n\t&\\mathrm{dist}_{\\Theta^*}^2(\\theta_{nB}) \\le \\alpha^B \\Big(\t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{0}) - \\frac{2\\epsilon \\widetilde{M}}{L(1\\!-\\!\\alpha)} \\Big) \\!+\\! \\frac{2\\epsilon \\widetilde{M}}{L(1\\!-\\!\\alpha)}, \\nonumber\n\t\\end{align}\n\t{where} $\\widetilde{M} = \\sum_{k = 0}^{n - 1} \\prod_{s=k+1}^{n-1}  \\Big(1 - \\frac{\\mu_{\\sigma(s)}}{L} \\Big)$ and $\\alpha$ corresponds to the curvature incoherence.\n\\end{coro}\nTo elaborate, the permutation map $\\sigma$ used by the incremental sampling can be viewed as a particular realization of the random permutation of the random reshuffle scheme. In particular, the convergence error term $\\overline{M}$ in \\Cref{thm: incohe_sc}  corresponds to the average of the convergence errors over all possible random permutations of the data indexes, whereas the convergence error term $\\widetilde{M}$ in \\Cref{coro: 2}  is determined by the specific permutation map $\\sigma$ used. \nTherefore, depending on the quality of the permutation map, the convergence error of SGD under incremental sampling can be either larger or smaller than that of SGD under random reshuffle.\n\nFor SGD with random sampling, we obtain the following convergence rate under minimizer incoherence and a constant step size.\n\\begin{proposition}[Random sampling]\\label{prop: error_bound}\n\tLet Assumptions \\ref{assum: exist} and \\ref{assum: f} hold. Apply SGD with random sampling with learning rate $\\eta=\\frac{1}{L}$ to solve the problem (P). Then, for all $k=nB, n=1,2,...$, \n\t\\begin{align}\n\t\\mathbb{E} \t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{nB}) \\le \\big(1 - \\frac{\\overline{\\mu}}{L} \\big)^{nB} \\Big(\t\\mathrm{dist}_{\\Theta^*}^2(\\theta_{0}) - \\frac{2 {\\epsilon}}{\\overline{\\mu}} \\Big) + \\frac{2 {\\epsilon}}{\\overline{\\mu}}. \\nonumber\n\t\\end{align}\n\twhere $\\bar{\\mu} := \\frac{1}{n}\\sum_{i=1}^n \\mu_i$.\n\\end{proposition}\n\nComparing the above result with that in \\Cref{thm: incohe_sc}, one can see that under minimizer incoherence, SGD with random reshuffle has a better convergence rate coefficient $\\alpha=\\prod_{i=1}^n (1-\\frac{\\mu_i}{L})$  than that $(1 - \\frac{\\overline{\\mu}}{L} )^{n} $ of SGD with random sampling (due to the AM-GM inequality). Moreover, regarding the convergence error, one can show that the convergence error $\\frac{2\\epsilon \\overline{M}}{L(1-\\alpha)}$ of SGD with random reshuffle is smaller than that $\\frac{2{\\epsilon}}{\\overline{\\mu}}$ of SGD with random sampling, and we outline the proof below.\n\\begin{align}\n\t\\frac{2\\epsilon \\overline{M}}{L(1-\\alpha)} &= \\frac{2\\epsilon \\sum_{k = 0}^{n - 1} \\mathbb{E}_{\\xi} \\big[ \\prod_{s=k+1}^{n-1}  \\big(1 - \\frac{\\mu_{\\xi(s)}}{L} \\big) \\big]}{L(1-\\alpha)} \\nonumber\\\\\n\t&\\overset{(i)}{\\le} \\frac{2\\epsilon \\sum_{k = 0}^{n - 1} \\big(1 - \\frac{\\overline{\\mu}}{L} \\big)^{n-k-1}}{L(1-\\alpha)} \\nonumber\\\\\n\t&= \\frac{2\\epsilon L}{\\overline{\\mu}} \\frac{1 -  \\big(1 - \\frac{\\overline{\\mu}}{L} \\big)^{n}}{L(1-\\alpha)} \\nonumber\\\\\n\t&\\overset{(ii)}{\\le}  \\frac{2\\epsilon}{\\overline{\\mu}}. \\nonumber \n\\end{align}\nTo elaborate, consider the quantity $\\mathbb{E}_{\\xi} \\big[ \\prod_{s=k+1}^{n-1}  \\big(1 - \\frac{\\mu_{\\xi(s)}}{L} \\big) \\big]$ in $\\overline{M}$. Note that for each fixed $k$, the samples $\\{\\xi(s)\\}_{s=k+1}^{n-1}$ are drawn from $\\{1,...,n\\}$ uniformly at random {\\em without replacement} due to the {random reshuffle} scheme, and hence the expectation over $\\{\\xi(s)\\}_{s=k+1}^{n-1}$ consists of $\\binom{n}{n-k-1}$ number of different combinations. Therefore, the inequality $(i)$ follows from the Maclaurin's inequality. Moreover, the inequality $(ii)$ follows from the AM-GM inequality. Such a comparison result reveals the statistical advantage of random reshuffle over random sampling: random reshuffle visits all data permutations in expectation and leads to a convergence error in spirit of geometric series (i.e., the $\\sum\\prod$ term in $\\overline{M}$), whereas random sampling samples each data uniformly with replacement and leads to a convergence error in spirit of arithmetic mean (i.e., the $\\overline{\\mu}$ term). \n\n\n\\subsection{Empirical Verification} \nWe verify our theoretical results obtained in this section via experiments on nonconvex phase retrieval. In specific, consider an underlying complex signal $x_0\\in \\mathbb{C}^d$ with a set of Gaussian measurement vectors $\\{a_i \\}_{i=1}^m$. The nonconvex phase retrieval model is written as\n$y_i = |\\langle a_i, x_0\\rangle|+ n,$\nwhere $\\{y_i \\}_{i=1}^m$ are the phaseless observations and $n$ denotes a Gaussian random noise. To retrieve the signal based on the phaseless observations and the Gaussian measurement vectors, we aim to solve the  following nonconvex problem.\n$$\\min_{x \\in \\mathbb{C}^d}  \\frac{1}{2n}\\sum_{i=1}^n \\big( y_i -  |a_i^\\intercal x |  \\big)^2.$$\nDue to noise corruption, the sample losses do not share a minimizer and hence have minimizer incoherence. In particular, the minimizer incoherence increases as the noise level increases. Specifically, we generate $x_0$, $a_i$, and $y_i$ from normal distribution with $d=128, n=512$. We repeat each experiment for $300$ times and use learning rate $\\eta=0.1$.\n\n\\begin{figure}[bth]\\centering\n\t\\includegraphics[width=0.98\\linewidth]{FIGs\/phaseret.png} \n\n\n\t\\caption{Left: Impact of minimizer incoherence on convergence error of SGD. Right: Convergence curves of SGD with different sampling schemes under minimizer incoherence.}\\label{fig: 5}\n\\end{figure}\n\nWe first explore how the level of noise (i.e., level of minimizer incoherence) in phase retrieval affects the convergence error of SGD with different sampling schemes. \\Cref{fig: 5} (Left) presents the box plot of convergence errors of SGD with random sampling and random reshuffle under different levels of Gaussian noise corruptions. For SGD with incremental sampling, we plot the smallest and largest errors achieved in the repeated experiments.\nIt can be seen that as the noise increases (i.e., minimizer incoherence increases), the convergence errors of these SGDs increase accordingly, which matches our theoretical characterizations of the convergence error. In particular, it can be observed that SGD with random reshuffle consistently has smaller convergence error than SGD with random sampling. Moreover, SGD with incremental sampling can sometimes outperforms SGD with random reshuffle when the permutation map happen to be good.\n\\Cref{fig: 5} (Right) shows the training loss curves of these algorithms under noise variance $\\sigma^2=9$. Under minimizer incoherence, it can be seen that SGD with random reshuffle and SGD with incremental sampling have a comparable convergence speed (i.e., a comparable slope of the training curves), and both of them converge faster than SGD with random sampling. These empirical results validate our convergence rate results of SGD under minimizer incoherence obtained in this section.\n\n\n\n\n\n\\section{Introduction}\\label{sec: intro}\nWe study the following finite-sum optimization problem that covers many important machine learning applications.\n\\begin{align}\n\t\\min_{\\theta\\in \\mathbb{R}^d} f(\\theta) := \\frac{1}{n}\\sum_{i=1}^{n} \\ell_i(\\theta), \\tag{P}\n\\end{align}\nwhere $\\theta\\in \\mathbb{R}^d$ corresponds to the model parameters, $f: \\mathbb{R}^d\\to \\mathbb{R}$ denotes the total loss and  each $\\ell_i$ corresponds to the sample loss of the $i$-th data sample. Such a problem formulation covers a variety of machine learning problems including support vector machine, logistic regression, matrix completion and neural network training, etc. \\shaocong{The common approach is minimizing the error of a predictive model over all data samples in a dataset, an i.i.d. assumption on the data typically decomposes the error into a sum of sample errors.}\n \nA standard and widely-applied algorithm that solves the problem (P) is the stochastic gradient descent (SGD) algorithm, which\nhas been well studied in both convex optimization \\cite{bottou2018optimization,robbins_1951,Nemirovski_2009,Lan_2012} and nonconvex optimization \\cite{bottou2018optimization,Ghadimi_2016,Ghadimi_2016b}. In these works, the SGD adopts a random sampling with replacement scheme (referred to as {\\em random sampling}) and its analysis is based on a bounded stochastic variance assumption. \n Although such an SGD framework yields theoretically optimal convergence rate \\cite{Rakhlin2012}, it cannot fully explain the superior practical performance of SGD in modern machine learning applications where the models are typically over-parameterized and SGD usually adopts the incremental sampling with random reshuffle scheme (referred to as {\\em random reshuffle}). Hence, it is desired to develop novel SGD frameworks that provide better understanding of the superior practical performance of SGD with random reshuffle.\n\nToward this goal, some existing works have proposed various novel analysis frameworks that lead to improved convergence rates of SGD with random sampling. In specific, \\cite{tseng1998incremental,solodov1998incremental} introduced a strong growth condition that bounds the maximum sample loss gradient norm in terms of the total loss gradient norm. Under such a condition, \\cite{schmidt2013fast} established a sublinear convergence rate and a linear convergence rate for SGD with random sampling in convex and strongly convex optimization, respectively. More recently, \\cite{vaswani2018fast} proposed a more relaxed weak growth condition and established similar convergence rate results for SGD with random sampling. In another recent work, \\cite{ma2017power} considered an interpolation setting where the model overfits all the data points so that all the sample losses share a unique global minimizer, and they showed that SGD with random sampling achieves a linear convergence rate under strong convexity. On the other hand, another line of works studied SGD with random reshuffle and established sublinear convergence rates under strong convexity, e.g.,  \\cite{haochen2018random,nagaraj2019sgd,shamir2016without}. However, the analysis in these works are based on traditional assumptions (e.g., Lipshcitzness, boundedness) that do not emphasize model characteristics, and a sufficiently small step size (typically $\\mathcal{O}(n^{-1})$) is required to justify the advantage of random reshuffle over random sampling. \nIn particular, these technical settings are not practical in modern machine learning training scenarios where the models are typically {\\em over-parameterized} and a {\\em constant-level} step size is adopted.\nTherefore, it is of great importance and interest to develop a novel theoretical framework for SGD with random reshuffle that characterizes the impact of model characteristics on its convergence under a practical constant step size. In specific, we are interested in studying SGD with random reshuffle in the following aspects.\n \\begin{itemize}[topsep=0pt,noitemsep, leftmargin=*]\n \n \t\\item The convergence results of SGD with random reshuffle studied in the existing works are {\\em in-expectation} with regard to the randomness of reshuffle under a sufficiently small step size. Can we prove {stronger} type of convergence of SGD with random reshuffle under over-parameterized models and a constant step size?\n \t\\vspace{2pt}\n \t\\item It has been observed in many practical scenarios that SGD with random reshuffle converges faster than SGD with random sampling under a constant step size. Therefore, the framework that we develop for analyzing SGD with random reshuffle is expected to provide theoretical justifications for this phenomenon.\n \t\\vspace{2pt}\n \t\\item The existing theoretical frameworks for analyzing SGD either assume the loss is strongly convex or assume the sample losses share a single global minimizer, both of which rule out many practical machine learning problems that are nonconvex and have multiple global minimizers. Therefore, our framework for analyzing SGD with random reshuffle must cover nonconvex scenarios and allow the existence of multiple global minimizers.\n \\end{itemize}\n\n\\subsection{Our Contributions}\nWe analyze the convergence of SGD with random reshuffle under a constant step size by exploiting two notions of model incoherence. In specific, we introduce a minimizer incoherence that measures the discrepancy between the global minimizer of a sample loss and that of the total loss. In particular, full minimizer coherence implies that all the sample losses share a global minimum and hence the model is over-parameterized. We also introduce a curvature incoherence that measures the quality of the condition numbers of the sample losses.\nOur theoretical results are in two-fold: \n \\begin{itemize}[topsep=0pt,noitemsep, leftmargin=*]\n \t\\item We first consider the case of full minimizer coherence where all the sample losses share a set of global minimizers. In such a case, we show that the variable sequence generated by SGD with random reshuffle converges to a certain global minimizer under a constant step size, and therefore the algorithm converges deterministically. Then, under full minimizer coherence and restricted strong convexity, we show that SGD with random reshuffle achieves a linear convergence rate, in which the contraction parameter is determined by the curvature incoherence of the sample losses. Moreover, we establish a linear convergence rate for SGD with random sampling in our framework and prove that SGD achieves a faster linear convergence rate under random reshuffle than that under random sampling, which provides justification to the superior performance of SGD with random reshuffle in training over-parameterized models. We further verify these theoretical results via experiments on over-parameterized neural network training.\n \t\\vspace{2pt}\n \t\\item Then, we analyze SGD with random reshuffle in the case of minimizer incoherence where the sample losses do not share any global minimizer. Under a constant step size and restricted strong convexity, we show that SGD with random reshuffle converges to a neighborhood of the global minimizer set at a linear convergence rate. In specific, the convergence rate depends on the curvature incoherence of the sample losses and the convergence error is determined by the minimizer incoherence of the sample losses. We show that the convergence rate of SGD with random reshuffle is faster than that of SGD with random sampling, and the convergence error of SGD is smaller under random reshuffle than that under random sampling.\n We verify our theoretical results via experiments on nonconvex phase retrieval.\n \\end{itemize}\n\nOur analysis shows that the convergence rate and convergence error of SGD with random reshuffle are in the form of geometric mean, whereas those of SGD with random sampling are in the form of arithmetic mean. Therefore, random reshuffle leads to a better convergence statistics for SGD than random sampling.\n\n\n\n\n\n\n\\subsection{Related Works}\n\n\\paragraph{SGD with random sampling:} Various theoretical frameworks have been developed for analyzing SGD with random sampling. In specific, \\cite{schmidt2013fast} exploited the strong growth condition to show that SGD with random sampling achieves a sublinear convergence rate in the convex case and achieves a linear convergence rate in the strongly convex case. \\cite{ma2017power} introduced an interpolation setting, in which they showed that SGD with random sampling achieves a linear convergence rate in the strongly convex case. \\cite{vaswani2018fast} proposed a relaxed weak growth condition and established a linear convergence rate for SGD with random sampling under strong convexity. \\cite{bottou2018optimization} studied SGD with random sampling under a second moment condition. In \\cite{gower2019}, they introduced an expected smooth condition and established linear convergence of SGD with random sampling to a neighborhood of the global minimum.\n\n\n\n\n\n\n\n\\textbf{SGD with random reshuffle:} It has been noticed that incremental SGD can achieve a faster convergence rate compared to SGD with random sampling in \\cite{bottou2009curiously}. The first theoretical analysis was given in \\cite{gurbuzbalaban2015random}, where incremental SGD  is shown to outperform SGD with random sampling under a diminishing stepsize. Random reshuffle has been shown to further improve the convergence rate of traditional SGD from $\\mathcal{O}(\\frac{1}{k})$ to $\\mathcal{O}(\\frac{1}{k^2})$ in the strongly convex case. Then, in more recent works \\cite{haochen2018random}, \\cite{nagaraj2019sgd}, and \\cite{ying2018stochastic}, it was shown that SGD with random reshuffle outperforms SGD with random sampling after finite epochs under a sufficiently small constant step size and strong convexity. \n\n\\section{Conclusion}\nIn this paper, we propose a model incoherence framework to study the impact of model incoherence on convergence of SGD. When the model has full minimizer coherence, we prove that SGD with random reshuffle converges to a global minimum deterministically and achieves a faster convergence rate than that of SGD with random sampling. When the sample losses have incoherent minimizers, we further show that SGD with random reshuffle has a smaller convergence error than that of SGD with random sampling. Our results reveal the statistical difference between the two random sampling schemes and characterize the impact of model incoherence on the optimization convergence. In the future work, we will further explore the generalization ability of SGD under different sampling schemes and develop a proper analysis framework for it. \n\n\\newpage\n\\section*{Acknowledgement}\nWe greatly thank the anonymous reviewers for providing many valuable feedback that help to substantially improve the quality of the paper. \n\n\n\\section{SGD with Random Reshuffle}\nIn this section, we introduce the SGD with random reshuffle algorithm and provide some preliminary results on it. \n\nThe SGD algorithm starts with an initialization $\\theta_0\\in \\mathbb{R}^d$ and applies the following update rule iteratively.\n\\begin{align}\n\t\\text{(SGD):}~\\theta_{k+1} = \\theta_k - \\eta \\nabla\\ell_{\\xi(k)}(\\theta_k), ~k=0,1,...,\n\\end{align}\nwhere $\\eta>0$ is the step size and $\\xi(k)$ corresponds to the index of data sample drawn from $\\{1,...,n\\}$ randomly in the  $k$-th iteration. In this work, we focus on the widely-used incremental sampling with random reshuffle scheme, which is formally defined as follows and is referred to as {\\em random reshuffle} for simplicity throughout the paper, . \n\n\\textbf{(Random reshuffle):} {\\em In each epoch, we apply a random permutation to the sample indexes, i.e., $\\{1,2,...,n\\}\\overset{\\textrm{permute}}{\\longrightarrow} \\{\\xi(0), \\xi(1),..., \\xi(n-1) \\}$. Then, the sample $\\xi(k)$ is used in the $k$-th iteration of this epoch.}\n\nWe obtain the following preliminary result for SGD with random reshuffle. \n\\begin{lemma}  \\label{lemma: 1}\n\tLet Assumptions \\ref{assum: exist} and \\ref{assum: f} hold. Apply SGD with random reshuffle  to solve the problem (P) with step size \\shaocong{$\\eta\\leq\\frac{1}{L}$}. \n\t Then, the variable sequence $\\{\\theta_k\\}_k$ generated by the algorithm satisfies: for all $k\\in \\mathbb{N}$ and all $\\omega\\in \\Theta_{\\xi(k)}^*$,\n\t\\begin{align*}\n\t\\|\\theta_{k+1} - \\omega\\|^2 \\le \\|\\theta_{k} -\\omega \\|^2  - \\eta \\big(   \n\t\\ell_{\\xi(k)}(\\theta_{k+1}) -\\ell_{\\xi(k)}^*\n\t\\big). \n\t\\end{align*} \n\\end{lemma} \nWe note that the proof of \\Cref{lemma: 1} only requires the sample losses to be restricted convex (i.e., $\\mu_i$ can be zero in \\Cref{assum: f}.3).\nThe above lemma characterizes the per-iteration progress of SGD with random reshuffle towards any global minimizer of the sample loss used in the $k$-th iteration. In particular, it implies that $\\|\\theta_{k+1} - \\omega\\| \\le \\|\\theta_{k} -\\omega \\|$, i.e., SGD makes monotonic progress towards the minimizer of the sample loss used in the $k$-th iteration.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}