Evolution of Cosmological Density Distribution Function from(7)

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

Now,recallingthefactthatg(δ)isanarbitraryfunctionoflocaldensityδ,equation

(7)mustbeequivalenttoequation(8)intheLagrangeframe.Thecomparisonbetweenequation(7)andequation(9)thenleadstothefollowingevolutionequation: dδ

δ

tPL(θ;t)+ dtPL(θ;t)

θ=0.(12)

2.3.Eulerianone-pointPDF

TheevolutionequationfortheEulerianone-pointPDFscanalsobederivedbyrepeatingthesameprocedureasabove,buttheresultantexpressionsareslightlydi erentfromthoseoftheLagrangianPDFs.Thetimederivativeoftheexpectationvalue g(δ) Ebecomes

tPE(δ;t).(13)

Asfortheexpectationvalueof g/ t,withahelpoftheLagrangiantimederivative,weobtain δdδ1

dδdδdδ

1

dδ

=1av· g(δ(x,t)) E

tg(δ(x,t)) =

E dδg(δ) dt(t) PE(δ;t)+H[θ(t)]δPE(δ;t).(16)δ