Evolution of Cosmological Density Distribution Function from(9)

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

Aswillbeexplicitlyshowninthenextsection,forinstance,ifthesphericalcollapsemodelisadoptedasLagrangianlocaldynamics,theevolutionoflocaldensityischaracterizedbythesinglevariable,whichcanbesetasthelinearlyextrapolateddensity uctuation,δl.Ifadoptingtheellipsoidalcollapsemodel,thedynamicaldegreesoffreedomreducetothree,representingtheprincipalaxesofinitialhomogeneousellipsoid,λ1,λ2andλ3.Thus,inthisapproximation,thedensity uctuationscanbeexpressedasδ=f(p,t),andusingthisexpression,thevelocity-divergenceisgivenbyθ= (df/dt)/H(1+f)fromtheequationofcontinuity(1).Withinthelocalapproximation,providedtheinitialdistributionfunctionPI(p),theformoftheconditionalmeanscanbecompletelyspeci edanditcanbeexpressed

intermsofthefunctionsPI(p)andf(p,t).

Letus rstconsidertheLagrangianPDF.Inthiscase,theformalexpressionsfortheconditionalmeans[dδ/dt]δand[dθ/dt]θaregivenby

dδdf(p,t)dpiPI(p)PL(δ;t)i 111= δDθ+.(20)dtθdtdtdt

Withtheseexpressions,theevolutionequations(11)and(12)becomeaclosedformandtheconsistentsolutionscanbeconstructedasfollows: PL(δ;t)=dpiPI(p)δD(δ f(p,t)),(21)

PL(θ;t)=

ii dpiPI(p)δDθ+1dt .(22)

Theproofthattheaboveequationsindeedsatisfytheevolutionequations(11)and(12)canbeeasilyshownbydi erentiatingequations(21)and(22)withrespecttotime.ForthePDFofthelocaldensity,onehas

δD(δ f(p,t)) t dfδD(δ f(p,t))=dpiPI(p) δi

fromthepropertyoftheDirac’sdeltafunction.Intheaboveequation,theoperator / δinthelastlinecanbefactoredoutandonecanusetheexpressionofconditionalmean(19).Then,thetimederivativeoftheone-pointPDFPL(δ;t)isrewrittenas dδ

δ