Evolution of Cosmological Density Distribution Function from(8)

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

Hence,therelation g/ t E= g(δ) E/ t,nottheequation dg/dt E= g(δ) E/ t,leads

totheevolutionequationfortheEulerianone-pointPDFP(δ,t).Fromequations(13)and

(14),weobtain

dδ

δ

tPE(θ;t)+ dt(t)

Notethatthezero-meanofthevelocitydivergence θ E=0isalwaysguaranteedfrom

equation(18),whichiseasilyshownbyintegratingtheaboveequationoverθdirectly. PE(θ;t)θ =HθPE(θ;t).(18)

2.4.Localapproximation

Theevolutionequationsintheprevioussubsectionarerathergeneralandinderivingthesewedidnotspecifythedynamicsof uidevolution.Inthissense,theyarenotcloseduntilfunctionalformsoftheconditionalmeans[dδ/dt]δ,[θ]δand[dθ/dt]θarespeci ed.Inotherwords,theseequationsrequiretheadditionalequationsgoverningtheevolutionoftheconditionalmeans.However,anattempttoobtaintheclosedsetofevolutionequationssu ersfromseriousmathematicaldi culty,so-calledclosureproblem,whichiswell-knowninthesubjectof uidmechanicsand/orplasmaphysics(e.g.,Chenetal.1989;Goto&Kraichnan1993).SimilartotheBBGKYhierarchy,ifonederivestheevolutionequationsfortheconditionalmeans,thereappearnewunknownconditionalmeans.Thus,wemustfurtherrepeatthederivationofevolutionequationforthenewquantities.Continuingthisoperation,onecould nallyobtainthein nitechainoftheevolutionequations,whichisgenerallyintractable.

Insteadoftheexactanalysistacklingthedi cultproblem,weratherfocusonanapprox-imatetreatmentoftheevolutionofone-pointPDFs,wherethesolutionsfortheevolutionequationsareconsistentlyconstructed.Toimplementthis,weadoptthelocalapproximationthattheevolutionofthelocaldensityδandthevelocity-divergenceθisdescribedbytheLagrangiandynamicswith nitedegreesoffreedom,whoseinitialconditionsarecharacter-izedbytheinitialparameters,p=(p1,p2,···pn),givenatthesameLagrangiancoordinate.