Evolution of Cosmological Density Distribution Function from the Local Collapse Model(5)

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

ofcontinuity,theEulerequationandthePoissonequationasfollows:

δ

a

v

a ·{(1+δ)v}=0,1(1)(v· )v+Hv=

xj (x,t),···,(4)

andde nethejointPDF,P(F;t),whichgivesaprobabilitydensitythatthequantityFtakessomevaluesof(δ,v, δ,···)atthetimet.Intermsofthis,theone-pointPDFofthedensity uctuationsP(δ;t)isgivenas

dFiP(F;t),(5)P(δ;t)=

Fi=δ

andsimilarlytheone-pointPDFofthedimensionlessvelocitydivergence,P(θ;t),is

P(θ;t)=dFiP(F;t),

Fi=θ(6)

whereθ≡ ·v/(aH).

Ingeneral,astatisticalcharacterizationofthelarge-scalestructureiscoordinate-dependent.Physically,thereareatleasttwomeaningfulcoordinates,i.e.,theLagrangianandtheEule-riancoordinates.WhiletheEuleriancoordinateis xedonacomovingframe,theLagrangiancoordinateis xedon uidparticlesandfollowsthemotionofthe uid ow.Hence,astimegoeson,highdensityregionsintheLagrangianspaceoccupylargervolumethanthoseintheEulerianspace.WethusconsiderboththeLagrangianPDFPLandtheEulerianPDF

PE,de nedintheLagrangianandtheEuleriancoordinates,qandx,respectively.The

correspondingexpectationvalues, ··· Land ··· Earealsointroduced.

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