Evolution of Cosmological Density Distribution Function from(19)

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

inEulerianexpectationvaluewith1/(1+θ).Ontheotherhand,inourcalculation,moments θn Earerigorouslycomputedaccordingtothede nition(B6),withahelpofthevelocity-

divergencePDF(33)withequation(30).Hence,thepresentcalculationisatleastconsistentwiththelocalapproximationinsection2.4andwebelievethatnoseriouserrorhasappearedinpresentresult.

Apartfromthisdiscrepancy,one ndsthattheleading-order(tree-level)calculationofskewnessS3,0andkurtosisS4,0inbothmodelsexactlycoincideswitheachother.Whilethedi erencesinthehigherordercorrectionforlocaldensityarebasicallysmall,consistentwithFigure2,theresultsinvelocity-divergenceexhibitalargedi erence,especiallyinthe

2varianceσθ.Figure3summarizesthedeparturefromtheleading-orderperturbationsforthevariance(top),theskewness(middle)andthekurtosis(bottom),eachofwhichisnormalized

2bytheleadingterm.Clearly,thehigherordercorrectionsforvarianceσθshowthesigni cant

di erencebetweenthesphericalandtheellipsoidalcollapsemodels,althoughthemodelde-pendenceoftheexternaltideinellipsoidalcollapseisrathersmall.Remarkably,theone-loopcorrectionsθ2,4isnegativeinthesphericalcollapsemodelandtherebythequantityσθdoesnotmonotonicallyincrease.Thisbehaviorindeedmatcheswiththenon-monotonicbehaviorofvelocity-divergencePDFseeninFigure2.Inthissense,theperturbationresultssuccess-fullyexplainthenumericalresultsofPDF.ThisfactfurtherindicatesthatinaGaussianinitialcondition,thein uenceofnon-sphericityore ectofshearcouldbenegligibleintheone-pointstatisticsoflocaldensity,whileitalterstheshapeofthePDFPE(θ),whichmight

beanaturaloutcomeofthemultivariatelocalapproximation.

3.2.2.JointPDFs

NextwefocusonthejointPDFs.LeftpanelofFigure4showstheLagrangianjointPDFPL(δ(z=0);δ(z=9))fromtheellipsoidalcollapsemodel,evaluatedatthepresent

timez=0withvariouslinearvariancesσl.Ontheotherhand,rightpanelofFigure4representstheresults xingthelinear uctuationvaluetoσl=2atpresent,butatdi erentoutputtimes.AlthoughFigure4doesnotrigorouslycorrespondtotheN-bodyresultsobtainedbyKayoetal.(2001)(c.f.Fig.7intheirpaper),thequalitativelysimilarfeaturescanbedrawninseveralpoints.First,thescatterbetweenδ(z=0)andδ(z=9)becomesbroaderasincreasingthetimeandthelinearvariance(toptobottominleftpanel).Second,thenonlinearitybetweentheinitialandtheevolveddensityindicatedfromthecurvatureoftheconditionalmean[δ(z)]δ(z=9)(solid)alsotendstoincreaseastimeelapses(toptobottominrightpanel).Theone-to-onemappingobtainedfromthesphericalcollapsemodel(short-dashed)isverydi erentfromthemeanrelation[δ(z)]δ(z=9),buttheirmeanrelations