Evolution of Cosmological Density Distribution Function from the Local Collapse Model(4)

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

crucialissueistoconstructasimplebutphysicallyplausiblemodelofone-pointPDF,atleast,consistentwiththeN-bodysimulationsinaqualitativemanner.Further,thein uenceofstochasticityontheevolutionoflocalquantitiesshouldbeinvestigated.

Thepurposeofthispaperistoaddresstheseissuesstartingfromageneraltheoryofevolutionofone-pointPDF.Inparticular,wederiveanevolutionequationforthedensityandthevelocity-divergencePDFsandconsiderhowthestochasticnatureofthelocaldensity eldemerges.Duetotheincompletenessoftheequations,anytheoreticalapproachusingtheevolutionequationsforPDFsgenerallybecomesintractable.However,underthelocalap-proximationthattheevolutionofdensity eld(orvelocity-divergence)isentirelydeterminedbythelocaldynamicswith nitedegreesoffreedom,weexplicitlyshowthattheanalyticalsolutionfortheevolutionequationsisconsistentlyconstructed.Basedonthisapproxima-tion,theone-pointPDFsforthedensityandthevelocity-divergencearecomputedfromtheellipsoidalcollapsemodel,asanaturalextensionoftheone-to-onemappingofthespher-icalcollapseapproximation.Further,thestochasticnaturearisingfromthemulti-variatefunctionoflocalquantitiesisexplicitlyshownevaluatingthejointPDFsofthelocaldensityand/orthevelocity-divergence.Thein uenceofthise ectisdiscussedindetailcomparingwiththesphericalcollapseapproximation.

Thispaperisorganizedasfollows.Insection2,weconsiderageneralframeworktotreattheevolutionofone-pointPDFsandderivetheevolutionequationsfortheEulerianandtheLagrangianPDFs(Sec.2.2and2.3).Then,adoptingthelocalapproximation,consistentsolutionsfortheseequationsareobtained(Sec.2.4).Further,thestochasticnatureoftheevolutionofPDFsisquanti edevaluatingjointPDFs(Sec.2.5).Asanapplicationofthesegeneralconsiderations,insection3,wegiveanexplicitevaluationofone-pointPDFsadoptingthesphericalandtheellipsoidalcollapsemodelsasrepresentativeLagrangianlocaldynamics.ThequalitativefeaturesoftheresultsarecomparedwiththeperturbativeanalysispresentedinappendixBorthepreviousN-bodystudy.Finally,section4isdevotedtotheconclusionandthediscussion.

2.EVOLUTIONEQUATIONFORPROBABILITYDISTRIBUTION

FUNCTION

2.1.Preliminaries

Throughoutthepaper,wetreatdarkmatterasapressure-lessandnon-relativistic uid.Assumingthehomogeneousandisotropicbackgrounduniverse,thedensity eldδ(x,t),thepeculiarvelocity eldv(x,t)andthegravitationalpotentialφforthe uidfollowtheequation

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