Evolution of Cosmological Density Distribution Function from(16)

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

whereD(t)isthelineargrowthrate.Thevariablesλirepresenttheinitialparametersofprincipalaxes,andintermsofthese,theinitialconditionsreduceto

dαi(t0)=a(t0){1 D(t0)λi},(49)

dt2+2Hdδ

31dt α˙i3 2=H2(1+δ) α˙2 3δij.3Hα1α3

Notethatsimilartothesphericalcollapseapproximation,thethreeinitialparametersλiareregardedastherandomvariables.Whentheinitialconditionofdensity eldisassumedtobeaGaussianrandomdistribution,theexpressionfortheinitialparameterdistributionPI(λi)isanalyticallyobtainedasfollows(e.g.,Doroshkevich1970;Bardeenetal.1986):

23375I1PI(λi)=(λ1 λ2)(λ2 λ3)(λ1 λ3),(52)exp 3262σ5πσll

wherethequantitiesI1andI2denoteI1≡λ1+λ2+λ3andI2≡λ1λ2+λ2λ3+λ3λ1,respectively.

BasedontheseLagrangianlocalmodels,wenumericallycalculatethePDFsassumingtheEinstein-deSitteruniverse( m=1, Λ=0),inwhichthelineargrowthrateDissimplyproportionaltothescalefactora.Forabetterunderstandingofthelateranalysis,inFigure1,weplottheevolutionoflocaldensityδfromtheellipsoidalcollapsemodelforsomeinitialconditions(e,p)givenbye=(λ1 λ3)/(2δl)andp=(λ1+λ3 2λ2)/(2δl).Theresultsarethendepictedasafunctionoflinearlyextrapolateddensityδl=λ1+λ2+λ3andarecomparedwiththeonefromtheone-to-onemappingofsphericalcollapsemodel(solid).Figure1showsthatthelocaldensityoftheellipsoidalcollapsemodelgenerallytakesalargervaluethanthatofthesphericalcollapsemodel.Further,thevarietyofevolveddensityfor xedδlsuggeststhatalargeamountofscatterwillappearinthejointPDFPL(δ,t;δ′,t′)andtheresultantone-pointPDFsPE(δ)andPE(θ)cannot,ingeneral,coincidewiththose

obtainedfromthesphericalcollapsemodel.