Evolution of Cosmological Density Distribution Function from(6)

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

Inthefollowingsubsection,we rstconsidertheLagrangianPDFandderivetheevolu-tionequation.ThenwederivetheevolutionequationforEulerianPDFinsection2.3.Theevolutionequationsderivedherearenotyetclosedbecauseoftheunknownfunctions.Insec-tion2.4wediscussanapproximatetreatmentusingthelocaldynamicsmodel,whichenablesustoobtainaclosedformofevolutionequationandtoconstructaconsistentsolution.

grangianone-pointPDF

ToderivetheevolutionequationfortheLagrangianone-pointPDF,weintroduceanarbitraryfunctionoflocaldensity,g(δ),andconsidertheevolutionofitsexpectationvalue, g(δ(q,t)) LevaluatedinaLagrangianframe.Thedi erentiationofthisexpectationvalue

withrespecttotimebecomes

dFig(δ)PL(F;t)=dδg(δ) ti

dtg(δ(q,t)) =

L dgdt =L idFidg(δ)dtPL(F;t).(8)

Theright-hand-sideoftheaboveequationcanbeexpressedbyintegratingbypartas

dg(δ) PL(F;t)= PL(F;t)dFidFig(δ)dtdtii PL(δ;t).= dδg(δ)dtδ

Here,thequantity[A]BdenotestheconditionalmeanofAforagivenvalueofB:

[A]B≡dFiAP(F|B)

Fi=B(9)(10)

withthefunctionP(F|B)beingtheconditionalPDFforagivenB,i.e.,P(F|B)=P(F)/P(B).