Evolution of Cosmological Density Distribution Function from the Local Collapse Model(11)

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

Further,theconditionalmean[θ]δcanbeexpressedas

[θ]δ= 1

dt (31)

δ

fromtheequationofcontinuity(1).Then,thesolutionsofequations(17)and(18)becomes

PE(δ;t)=1

1+gδD(θ h(p,t)).(33)

Onecaneasilyshowthatequations(32)and(33)satisfytheevolutionequations(17)and

(18),withthecorrectnormalizationandthezeromean.

TheabovesolutionsforEulerianPDFcanberegardedasageneralizationofthepreviousstudybasedontheZel’dovichapproximation(Kofmanetal.1994)and/orthesphericalcollapsemodel(Scherrer&Gazta naga2001,seealsoProtogeros&Scherrer1997).Note,however,thatthegeneralexpressionofvelocity-divergencePDFPE(θ;t)di ersfromthe

oneobtainedpreviously.Whilethefactor1/(1+δ)isconvolvedintheintegralinequation

(33),theresultantexpressionbyKofmanetal.(1994)obviouslyomittedthisfactor.Inourprescription,thePDFPE(θ;t)isconstructedfromtheevolutionequation,whichbasically

reliesontheequationofcontinuity.Thismeansthat,evenforthevelocity-divergencePDF,thefactor1/(1+δ)isnecessarytoensurethemassconservation.Infact,theexpression(33)canalsobeobtainedfromthejointPDFPE(δ,θ;t)integratingoverthelocaldensityδ(see

eq.[36]).Althoughthevelocity-divergencePDFinthepreviousstudyhasbeenobtainedinaratherphenomenologicalway,ourpresentapproachusingtheevolutionequationsmightbehelpfulinconstructingtheconsistentPDFs.

Nevertheless,evenatthispoint,thesolutionsofevolutionequationsshouldberegardedasformalexpressions.InordertoevaluatethePDFsexplicitly,weneedtospecifytheLagrangianlocaldynamics.Inotherwords,thequantitativepredictionforPDFsusingthelocalapproximationcruciallydependsonthechoiceofthelocaldynamics.Wewillseeinthesection3thattheexplicitevaluationofPE(δ;t)andPE(θ;t)adoptingthesphericaland

theellipsoidalcollapsemodelsshowsseveralnoticeabledi erences.

2.5.JointPDF

Sofar,wehaverestrictedourattentiontotheone-pointPDFforthesinglelocalvariable,δorθ.Inourtreatmentofthelocalapproximation,theexpressionsforPDFsaregeneral

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