Evolution of Cosmological Density Distribution Function from(2)

We present a general framework to treat the evolution of one-point probability distribution function (PDF) for cosmic density $\delta$ and velocity-divergence fields $\theta$. In particular, we derive an evolution equation for the one-point PDFs and consid

Subjectheadings:cosmology:theory-galaxies:clustering-galaxies:darkmatter-large-scalestructureofuniverse-methods:analytical

1.INTRODUCTION

Thelarge-scalestructureoftheuniverseisthoughttobedevelopedbythegravitationalinstabilityfromthesmallGaussiandensity uctuations.Inauniversedominatedbydarkmatter,theevolutionofmassdistributionisentirelygovernedbygravitationaldynamics.Whileluminousobjectssuchasthegalaxiesandtheclustersaresubsequentlyformedbycomplicatedprocessesincludinggasdynamicsandradiativeprocess,thedarkmatterdistri-butionisthemostfundamentalproductinthehierarchicalclusteringofthecosmicstructureformation.Inparticular,theclusteringfeatureofdarkmatterdistributionisdirectlyob-servedviaweakgravitationallensinge ect(Bartelmann&Schneider2001forreviewandreferencestherein).Thus,thestatisticalstudyofthelarge-scalemassdistributionprovidesausefulcosmologicaltoolinprobingtheformationmechanismofdarkmatterhalos,aswellastheobservedluminousdistribution.

Inprinciple,allthestatisticalinformationofdarkmatterdistributionischaracterizedbytheprobabilitydistributionfunctions(PDFs)formassdensity uctuationandvelocity elds,δandv.Amongthese,theone-pointPDFfordensity eld,P(δ),isoneofthemostfundamentalstatisticalquantitiesandbecauseofitssimplicity,therehasbeennu-meroustheoreticalstudiesonitsevolutionfromaGaussianinitialcondition.Fromtheanalyticalstudyofone-pointPDFs,Kofmanetal.(1994) rstcalculatedthePDFusingtheZel’dovichapproximation.Foraperturbativeconstructionofone-pointPDF,Bernardeau&Kofman(1995)andJuszkiewiczetal.(1995)introducedtheEdgeworthexpansiontoderivetheanalyticformulaforPDFs.Ontheotherhand,basedonthetree-levelapproximation,Bernardeau(1992b)constructedthePDFfromthegeneratingfunctionofthevertexfunc-tion.Interestingly,thevertexfunctioninthetree-levelapproximationisobtainedasanexactsolutionofthesphericalcollapsemodel.Thee ectofsmoothinghasbeenlaterincor-poratedintohiscalculationandtheanalyticPDFwascomparedwithN-bodysimulations(Bernardeau1994b).Followingtheseresults,Fosalba&Gazta naga(1998a)proposedtousethesphericalcollapsemodelforthepredictionofhigher-ordermomentsbeyondthetree-levelapproximation.Themostremarkablepointintheirapproximationisthattheleading-orderpredictionexactlymatchestheoneobtainedfromtherigorousperturbationtheoryandthecorrectionfornext-to-leadingorderiseasilycomputedbysolvingthesimplesphericalcol-lapsedynamics.Further,thesphericalcollapseapproximationisrecentlyextendedtothepredictionofone-pointPDF(Scherrer&Gazta naga2001,seealsoProtogeros&Scherrer