Evolution of Cosmological Density Distribution Function from

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

RESCEU-18/02UTAP-429/2002

EvolutionofCosmologicalDensityDistributionFunctionfromthe

LocalCollapseModel

YasuhiroOhta,IsshaKayoandAtsushiTaruya

arXiv:astro-ph/0301567v1 29 Jan 2003DepartmentofPhysicsandResearchCenterfortheEarlyUniverse(RESCEU),SchoolofScience,UniversityofTokyo,Tokyo113,Japan.ohta@utap.phys.s.u-tokyo.ac.jp,kayo@utap.phys.s.u-tokyo.ac.jp,ataruya@utap.phys.s.u-tokyo.ac.jpABSTRACTWepresentageneralframeworktotreattheevolutionofone-pointproba-bilitydistributionfunction(PDF)forcosmicdensityδandvelocity-divergence eldsθ.Inparticular,wederiveanevolutionequationfortheone-pointPDFsandconsiderthestochasticnatureassociatedwiththesequantities.Underthelocalapproximationthattheevolutionofcosmic uid eldscanbecharacter-izedbytheLagrangianlocaldynamicswith nitedegreesoffreedom,evolutionequationforPDFsbecomesaclosedformandconsistentformalsolutionsareconstructed.Adoptingthislocalapproximation,weexplicitlyevaluatetheone-pointPDFsP(δ)andP(θ)fromthesphericalandtheellipsoidalcollapsemodelsastherepresentativeLagrangianlocaldynamics.InaGaussianinitialcondition,whilethelocaldensityPDFfromtheellipsoidalmodelalmostcoincideswiththethatofthesphericalmodel,di erencesbetweensphericalandellipsoidalcollapse

modelarefoundinthevelocity-divergencePDF.Thesebehaviorshavealsobeencon rmedfromtheperturbativeanalysisofhigherordermoments.Importantly,thejointPDFoflocaldensity,P(δ,t;δ′,t′),evaluatedatthesameLagrangianpositionbutatthedi erenttimestandt′fromtheellipsoidalcollapsemodelexhibitsalargeamountofscatter.Themeanrelationbetweenδandδ′doesfailtomatchtheone-to-onemappingobtainedfromsphericalcollapsemodel.Moreover,thejointPDFP(δ;θ)fromtheellipsoidalcollapsemodelshowsasimilarstochasticfeature,bothofwhichareindeedconsistentwiththerecentresultfromN-bodysimulations.Hence,thelocalapproximationwithellipsoidalcollapsemodelprovidesasimplebutamorephysicalmodelthanthesphericalcollapsemodelofcosmologicalPDFs,consistentwiththeleading-orderresultsofexactperturbationtheory.