Evolution of Cosmological Density Distribution Function from(12)

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

formsirrespectiveoftheLagrangianlocaldynamics.However,itshouldbeemphasizedthatifthelocaldynamicsischaracterizedbymorethanthetwoinitialparameters,qualitativebe-haviorcouldbesigni cantlychangedfromthelocaldynamicswithsingledegreeoffreedom.Thepointisthattherelationbetweeninitialparametersandtheevolvedresultδorθcannotbedescribedbyaone-to-onemapping.Accordingly,therelationbetweenδandθbecomesnolongerdeterministic.Moreover,thefailureofdeterministicpropertyalsoappearsinthetimeevolutionofsuchlocalvariables.Itisthereforeimportanttodiscussthestochasticnatureofδandθarisingfromthedynamicalevolution.Tocharacterizethis,weconsiderthejointPDF.Withinthelocalapproximation,onecanconstructaconsistentsolutionofEulerianjointPDFbetweenδandθevaluatedatthesametime,PE(δ,θ;t).Further,the

LagrangianjointPDFforthedensity eldevaluatedatthesameLagrangianpositionbutatthedi erenttimes,PL(δ,t;δ′,t′)canalsobeobtained.

TheevolutionequationofPE(δ,θ;t)canbederivedthroughtheexpectationvalueof

anarbitraryfunctiong(δ,θ).Repeatingthesameprocedureasdescribedinsection2.3,weobtain

δdδ θ

δ

dθ tdθPE(δ,θ;t),

θ

dδ tPE(δ,θ;t) Hθ θdθ

dt =

δ,θ11+δ dpiPI(p)dh