Evolution of Cosmological Density Distribution Function from(10)

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

whichcoincideswiththeevolutionequation(11).Asforthevelocity-divergencePDFPL(θ;t),weconsistentlyrecovertheevolutionequation(12)withahelpofequation(20):

1 δDθ+ tdt 1dfdδDθ+=dpiPI(p)H(1+f) θdti = PL(θ;t).(24)dtθ

TheapproximatesolutionoftheEulerianone-pointPDFsarealsoobtainedsimilarly,butthefactor1/(1+δ)mustbeconvolvedwiththeLagrangianPDFduetothepresenceofinertialterm(r.h.sofeqs.[17][18]):

PE(δ;t)=1

δDθ+1

dt 1+f(p,t).(26)

Note,however,thatthesePDFsdonotsatisfythefollowingconditions:normalizationcon-dition 1 E=1andzeromeans δ E=0and θ E=0.Thisfactsimplyre ectsthatthe

conservationofEulerianvolumecannotbealwaysguaranteed,incontrasttotheconserva-tionofLagrangianvolumeensuredbythemassconservation.AspointedoutbyFosalba&Gazta naga(1998a)(seealsoProtogeros&Scherrer1997),were-scaletherelationbetweenδandf(p,t)asfollows:

PI(p)δ=g(p,t)≡NE{1+f(p,t)} 1;NE(t)≡dpi

i

dt

dt =δ1PE(δ;t)1 idpiPI(p)dhdg=

θ1+g

dg

H(1+g)