Evolution of Cosmological Density Distribution Function from the Local Collapse Model(18)

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

δ′signi cantlydeviatesfromone-to-onemappingofsphericalcollapsemodel(seeSec.3.2.2andFig.4).Ontheotherhand,thevelocity-divergencePDFsfromtheellipsoidalcollapsemodelexhibitlongernon-Gaussiantails,comparedwiththoseobtainedfromthesphericalcollapsemodel.ThedeviationbetweenbothmodelsinPDFPE(θ)becomessigni cantas

increasingthevalueσl.Interestingly,inthenon-linearregimeσl≥1,tailsofPDFPE(θ)fromthesphericalcollapsemodelshowtheoppositetendency,i.e.,theamplitudedecreasesasincreasingσl.

Inordertocharacterizethequalitativebehaviorsmoreexplicitly,weperturbativelysolvetheevolutionequationsforboththesphericalandtheellipsoidalcollapsemodels.Thedi erencesarethenquanti edevaluatingthehigherordermomentsofone-pointstatisticsforthelocaldensityandvelocity-divergence.InappendixB,basedontheformalsolutionofPDFsinsection2.4,perturbativecalculationsoflocalcollapsemodelsarebrie ysummarizedandthesolutionsuptothe fthorderarepresented.Theresultantexpressionsforthehigherordermomentsofdensityandvelocity-divergenceareobtainedasaseriesexpansionoflinearvarianceσl2,uptothetwo-looporderforthevarianceandtheone-looporderfortheskewnessandthekurtosis:

σ2≡ δ2 E=σl2+s2,4σl4+s2,6σl6+···,

δ3 ES3≡

σ6

forthelocaldensityand

24θ6σθ≡ θ2 E=σl2+sθ2,4σl+s2,6σl+···,

θ3 ET3≡

6σθ(53)=S4,0+S4,2σl2+···(55)(56)=T4,0+T4,2σl2+···(58)

forthevelocity-divergence.Then,allthecoe cientsintheaboveexpansionsyieldtherigorousfractionalnumberandTable1displaysasummaryoftheresults.Thecalculationinsphericalcollapsemodelessentiallyreproducesthenon-smoothingresultsobtainedbyFosalba&Gazta naga(1998a,b).Note,however,thatthediscrepancyhasappearedinthehigherordercorrectionofvelocity-divergence(c.f.,eq.[12]withγ=0ofFosalba&Gazta naga1998b).Perhaps,incomputingthevelocity-divergencemoments,Fosalba&Gazta naga(1998b)incorrectlyusedthecumulantexpansionformulaforδlistedinFosalba&Gazta naga(1998a).Further,wesuspectthattheyerroneouslyreplacedtheconvolutionfactor1/(1+δ)

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