Evolution of Cosmological Density Distribution Function from(14)

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

OnemightfurtherconsidertheevolutionofEulerianjointPDFPE(δ,t;δ′,t′),which

hasbeenindeedshowninN-bodysimulationsbyKayoetal.(2001).Thederivationofevolutionequationitselfisaneasytask,but,theformalsolutioninthelocalapproximationsu ersfromdi cultiesduetothepresenceofadvectiveterm,whichmightberelatedtoanimportante ectonthenon-localnatureof uid ows.SinceeventheLagrangianjointPDFshowsseveralimportantfeatures,onecanexpectthatthequalitativelysimilarbehaviortotheN-bodyresultscanbeseenfromtheLagrangianjointPDF.Hence,wewillpostponetoanalyzetheEulerianjointPDFPE(δ,t;δ′,t′)andinsteadfocusontheLagrangianjointPDF

PL(δ,t′;δ′,t′).

3.DEMONSTRATIONANDRESULTS

Nowweareinapositiontogiveanexplicitevaluationoftheone-pointPDFbasedonthelocalapproximationdiscussedintheprevioussection.Foranillustrativepurpose,weadoptthesphericalandtheellipsoidalcollapsemodelsassimpleandintuitiveLagrangianlocaldynamics.Afterbrie ydescribingthebasicequationsofthesemodelsinsection3.1,wecomputetheEulerianone-pointPDFsPE(δ)andPE(θ)anddiscussthequalitativedi er-

encesarisingfromthechoiceofLagrangianlocaldynamicsinsection3.2.Inparticularthestochasticityinthemulti-variatefunctionoflocaldensityorvelocity-divergenceisexaminedindetailbyevaluatingthejointPDFs,PL(δ,t;δ′,t′)andPE(δ;θ)fromtheellipsoidalcollapse

model.

3.1.ModelsforLagrangianlocaldynamics

FirstconsiderthesimplestcaseoftheLagrangianlocaldynamics,inwhichtheevolutionoflocalquantitiesisdeterminedbythemassinsideasphereofradiusRcollapsingviaself-gravity:d2R4π3;M=ρR=const,(39)R2

whereMisthemassinsidetheradiusand

ρ/ρm 1=(a/R)3 1asfollows: 2dδ4dδ dt1+δ2H2 m(1+δ)δ,(40)

withthequantity mbeingthedensityparameter, m≡8πGρm/(3H2).AsFosalba&Gazta naga(1998a)stated,thisequationisregardedasashear-lessandirrotationalapprox-