Evolution of Cosmological Density Distribution Function from(13)

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

andthecorrespondingsolutionofequation(34)becomes

PE(δ,θ;t)=1

g(δ(q,t),δ(q,t′)) =dδdδ′g(δ,δ′)dt g(δ(q,t),δ(q,t′))=dt = dδdδ′g(δ,δ′)

TheevolutionequationofLagrangianjointPDFis

dδ

δ

dt dt PL(δ,t;δ′,t′).δ,δ′ dt=

δ,δ′1dtδD(δ f(p,t))δD(δ′ f(p,t′)).

RecallingthatthejointPDFsatisfyingtheevolutionequation(37)shouldbeinvariantunderthetransformation,(δ,t) (δ′,t′),thesolutionconsistentwiththeboundaryconditionPL(δ,t′;δ′,t′)=PL(δ;t′)δD(δ δ′)becomes

PL(δ,t;δ′,t′)=dpiPI(p)δD(δ f(p,t))δD(δ′ f(p,t′)).(38)

i

NoticethatifthelocalLagrangiandynamicsisdescribedbyasingleparameter,theintegralovertheinitialparameterp1inequation(38)canbeformallyperformed.TheresultantexpressionincludesDirac’sdeltafunction,leadingtotheone-to-onemappingbetweenδandδ′.Ontheotherhand,incaseswiththemultivariateinitialparameters,onecannotgenerallyperformtheaboveintegralandtheDirac’sdeltafunctionisnotfactoredout,leadingtothestochasticnatureoflocaldensity elds.