Evolution of Cosmological Density Distribution Function from(20)

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

basicallyre ectthequalitativebehavioroflocaldynamicsinFigure1.Thatis,theevolvedresultsoflocaldensityintheellipsoidalcollapsetendstotakealargervaluethanthatinthesphericalcollapse.Moreover,recallthefactthatboththeinitialandthe nalPDFsoflocaldensityPE(δ)showagoodagreementbetweenthesphericalandtheellipsoidalcollapse

model(seeFig.2).ThisisindeedthesamesituationasintheN-bodysimulation;apartfromthedetaileddi erences,asimplemodelofPDFsprovidesanessentialingredientforthestochasticnatureofN-bodyresults.Inthissense,thelocalapproximationwithellipsoidalcollapsemodelscanberegardedasaconsistentandphysicalmodelofone-pointstatistics,whichsuccessfullyexplainstheN-bodysimulations.

Finally,usingtheellipsoidalcollapsemodelwithlinearexternaltide,weexaminetheEulerianjointPDFoflocaldensityandvelocity-divergenceevaluatedatthesametime,i.e.,PE(δ;θ).InFigure5,contourplotsofjointPDFPE(δ;θ)forvariouslinear uctuationvalues

σlaredepictedasfunctionof θandδ.Thisistheso-calleddensity-velocityrelation,whichmightbeofobservationalinterestinmeasuringthedensityparameter mfromthevelocity-densitycomparisonthroughthePOTENTanalysis(e.g.,Bertschinger&Dekel1989).Alongthelineofthis,theoreticalworksbasedontheEulerianperturbationtheoryhaveattractedmuchattention,aswellastheN-bodystudy(e.g.,Bernardeau1992a;Chodorowski&Lokas1997;Bernardeauetal.1999).Basedonthesolutionofthelocalapproximation(36),onecaneasilycalculatetheperturbationseriesofvelocity-densityrelation[θ]δasfunctionoflocaldensityanddensity-velocityrelation[δ]θasfunctionofvelocity-divergence,theleading-orderresultsofwhichareexpectedtocoincidewithpreviousearlyworksinthenon-smoothingcase,exactly.Beyondtheperturbationanalysis,Figure5revealsthegeneraltrendofthestochasticnatureinthevelocity-densityrelation.Asincreasingσl,thescatterbecomesmuchbroaderandtheconditionalmeans[δ]θ(dot-dashed)and[θ]δ(solid)doesnotcoincidewitheachother.Ofcourse,theone-to-onemappingobtainedfromthesphericalcollapsemodel(short-dashed)failstomatchthebothconditionalmeans.ThesequalitativebehaviorisinfactconsistentwiththeN-bodyresultsbyBernardeauetal.(1999)andthepresentmodelprovidesasimplewaytoderivethenon-linearandstochasticvelocity-densityrelation.

4.CONCLUSIONANDDISCUSSION

Inthispaper,startingfromageneraltheoryofevolutionofone-pointPDFs,wederivedtheevolutionequationsforPDFandwithinalocalapproximation,consistentformalsolu-tionsofPDFareconstructedinboththeLagrangianandtheEulerianframes(seeeqs.[21][22]forLagrangianPDFsandeqs.[32][33]witheqs.[27][30]forEulerianPDFs).InordertorevealthestochasticnaturearisingfromthemultivariateLagrangiandynamics,wefurthercon-