Evolution of Cosmological Density Distribution Function from the Local Collapse Model

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

RESCEU-18/02UTAP-429/2002

EvolutionofCosmologicalDensityDistributionFunctionfromthe

LocalCollapseModel

YasuhiroOhta,IsshaKayoandAtsushiTaruya

arXiv:astro-ph/0301567v1 29 Jan 2003DepartmentofPhysicsandResearchCenterfortheEarlyUniverse(RESCEU),SchoolofScience,UniversityofTokyo,Tokyo113,Japan.ohta@utap.phys.s.u-tokyo.ac.jp,kayo@utap.phys.s.u-tokyo.ac.jp,ataruya@utap.phys.s.u-tokyo.ac.jpABSTRACTWepresentageneralframeworktotreattheevolutionofone-pointproba-bilitydistributionfunction(PDF)forcosmicdensityδandvelocity-divergence eldsθ.Inparticular,wederiveanevolutionequationfortheone-pointPDFsandconsiderthestochasticnatureassociatedwiththesequantities.Underthelocalapproximationthattheevolutionofcosmic uid eldscanbecharacter-izedbytheLagrangianlocaldynamicswith nitedegreesoffreedom,evolutionequationforPDFsbecomesaclosedformandconsistentformalsolutionsareconstructed.Adoptingthislocalapproximation,weexplicitlyevaluatetheone-pointPDFsP(δ)andP(θ)fromthesphericalandtheellipsoidalcollapsemodelsastherepresentativeLagrangianlocaldynamics.InaGaussianinitialcondition,whilethelocaldensityPDFfromtheellipsoidalmodelalmostcoincideswiththethatofthesphericalmodel,di erencesbetweensphericalandellipsoidalcollapse

modelarefoundinthevelocity-divergencePDF.Thesebehaviorshavealsobeencon rmedfromtheperturbativeanalysisofhigherordermoments.Importantly,thejointPDFoflocaldensity,P(δ,t;δ′,t′),evaluatedatthesameLagrangianpositionbutatthedi erenttimestandt′fromtheellipsoidalcollapsemodelexhibitsalargeamountofscatter.Themeanrelationbetweenδandδ′doesfailtomatchtheone-to-onemappingobtainedfromsphericalcollapsemodel.Moreover,thejointPDFP(δ;θ)fromtheellipsoidalcollapsemodelshowsasimilarstochasticfeature,bothofwhichareindeedconsistentwiththerecentresultfromN-bodysimulations.Hence,thelocalapproximationwithellipsoidalcollapsemodelprovidesasimplebutamorephysicalmodelthanthesphericalcollapsemodelofcosmologicalPDFs,consistentwiththeleading-orderresultsofexactperturbationtheory.

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

Subjectheadings:cosmology:theory-galaxies:clustering-galaxies:darkmatter-large-scalestructureofuniverse-methods:analytical

1.INTRODUCTION

Thelarge-scalestructureoftheuniverseisthoughttobedevelopedbythegravitationalinstabilityfromthesmallGaussiandensity uctuations.Inauniversedominatedbydarkmatter,theevolutionofmassdistributionisentirelygovernedbygravitationaldynamics.Whileluminousobjectssuchasthegalaxiesandtheclustersaresubsequentlyformedbycomplicatedprocessesincludinggasdynamicsandradiativeprocess,thedarkmatterdistri-butionisthemostfundamentalproductinthehierarchicalclusteringofthecosmicstructureformation.Inparticular,theclusteringfeatureofdarkmatterdistributionisdirectlyob-servedviaweakgravitationallensinge ect(Bartelmann&Schneider2001forreviewandreferencestherein).Thus,thestatisticalstudyofthelarge-scalemassdistributionprovidesausefulcosmologicaltoolinprobingtheformationmechanismofdarkmatterhalos,aswellastheobservedluminousdistribution.

Inprinciple,allthestatisticalinformationofdarkmatterdistributionischaracterizedbytheprobabilitydistributionfunctions(PDFs)formassdensity uctuationandvelocity elds,δandv.Amongthese,theone-pointPDFfordensity eld,P(δ),isoneofthemostfundamentalstatisticalquantitiesandbecauseofitssimplicity,therehasbeennu-meroustheoreticalstudiesonitsevolutionfromaGaussianinitialcondition.Fromtheanalyticalstudyofone-pointPDFs,Kofmanetal.(1994) rstcalculatedthePDFusingtheZel’dovichapproximation.Foraperturbativeconstructionofone-pointPDF,Bernardeau&Kofman(1995)andJuszkiewiczetal.(1995)introducedtheEdgeworthexpansiontoderivetheanalyticformulaforPDFs.Ontheotherhand,basedonthetree-levelapproximation,Bernardeau(1992b)constructedthePDFfromthegeneratingfunctionofthevertexfunc-tion.Interestingly,thevertexfunctioninthetree-levelapproximationisobtainedasanexactsolutionofthesphericalcollapsemodel.Thee ectofsmoothinghasbeenlaterincor-poratedintohiscalculationandtheanalyticPDFwascomparedwithN-bodysimulations(Bernardeau1994b).Followingtheseresults,Fosalba&Gazta naga(1998a)proposedtousethesphericalcollapsemodelforthepredictionofhigher-ordermomentsbeyondthetree-levelapproximation.Themostremarkablepointintheirapproximationisthattheleading-orderpredictionexactlymatchestheoneobtainedfromtherigorousperturbationtheoryandthecorrectionfornext-to-leadingorderiseasilycomputedbysolvingthesimplesphericalcol-lapsedynamics.Further,thesphericalcollapseapproximationisrecentlyextendedtothepredictionofone-pointPDF(Scherrer&Gazta naga2001,seealsoProtogeros&Scherrer

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

1997).TheapproximationhasbeentestedagainstN-bodysimulationsandagoodagreementwasfoundeveninanon-linearregimeofthedensityperturbation.

Ontheotherhand,fromthenumericalstudy,Kayo,Taruya&Suto(2001)recentlyshowedthatthelog-normalPDFquantitativelyapproximatestheone-pointPDFP(δ)intheN-bodysimulationsemergingfromtheGaussianinitialcondition,irrespectiveoftheshapeofinitialpowerspectra.Thelog-normalPDFhasbeenlongknownasanempiricalmodelcharacterizingtheN-bodysimulations(e.g.,Kofmanetal.1994;Bernardeau&Kofman1995;Taylor&Watts2000)ortheobservedgalaxydistribution(e.g,Hamilton1985;Bouchetetal.1993;Kofmanetal.1994).Recently,agoodagreementwiththelog-normalmodelhasbeenreportedinthenumericalstudyofweaklensingstatistics(Taruyaetal.2002a),andanattempttoclarifytheoriginofthelog-normalbehaviorhasalsomade(Taruyaetal.2002b).Mathematically,thelog-normalPDFisobtainedfromaone-to-onelocalmappingbetweentheGaussianandthenon-lineardensity elds.Thus,thesuccessful ttotheN-bodysimulationmightbeinterpretedtoimplythattheevolutionoflocaldensitycanbewell-approximatedbytheone-to-onelocalmapping.Indeed,theanalyticPDFforthesphericalcollapseapproximationcanalsobeexpressedasaone-to-onelocalmappingviathesphericalcollapsemodel.

TheabovenaivepicturehasbeencriticallyexaminedbyKayo,Taruya&Suto(2001)evaluatingthejointPDFP(δ1,t1;δ2,t2),i.e.,jointprobabilityofthelocaldensity eldsδ1andδ2atthesamecomovingpositionbutatthedi erenttimest1andt2.Ithasbeenfoundthatalargeamountofscatterintherelationbetweenδ1andδ2showsupandtheirmeanrelationsigni cantlydeviatesfromthepredictionfromtheone-to-onelog-normalmapping.Althoughthismightnotbesurprising,thegoodagreementbetweenthelog-normalandthesimulatedPDFsbecomesmorecontrivedanddi culttobeexplainedinastraightforwardmanner.

De nitely,thefailureoftheone-to-onelocalmappingsomehowre ectsthenon-localityofthegravitationaldynamics.Thatis,theevolutionoflocaldensitycannotbedeterminedbytheinitialvalueofthelocaldensityonly.Rather,thelocaldensitycanbeexpressedasmulti-variatefunctionsoflocaldensityandotherlocalquantitiessuchasvelocity,gradientoflocaldensity,velocity-divergenceandsoon.Furthermore,initialvaluesoftheselocalquantitiesarerandomlydistributedoverthethree-dimensionalspace.Asaconsequence,evenifthedynamicsisdeterministic,therelationbetweentheevolvedandtheinitiallocaldensityinevitablybecomesstochastic.Inasense,thesituationisquitesimilartothenon-linearstochasticbiasingofgalaxies,i.e.,thestatisticaluncertaintybetweengalaxiesanddarkmatterarisingfromthehiddenvariables(e.g.,Dekel&Lahav1999;Taruya,Koyama&Soda1999;Taruya&Suto2000).Then,takingaccountofthisstochasticnature,the

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

crucialissueistoconstructasimplebutphysicallyplausiblemodelofone-pointPDF,atleast,consistentwiththeN-bodysimulationsinaqualitativemanner.Further,thein uenceofstochasticityontheevolutionoflocalquantitiesshouldbeinvestigated.

Thepurposeofthispaperistoaddresstheseissuesstartingfromageneraltheoryofevolutionofone-pointPDF.Inparticular,wederiveanevolutionequationforthedensityandthevelocity-divergencePDFsandconsiderhowthestochasticnatureofthelocaldensity eldemerges.Duetotheincompletenessoftheequations,anytheoreticalapproachusingtheevolutionequationsforPDFsgenerallybecomesintractable.However,underthelocalap-proximationthattheevolutionofdensity eld(orvelocity-divergence)isentirelydeterminedbythelocaldynamicswith nitedegreesoffreedom,weexplicitlyshowthattheanalyticalsolutionfortheevolutionequationsisconsistentlyconstructed.Basedonthisapproxima-tion,theone-pointPDFsforthedensityandthevelocity-divergencearecomputedfromtheellipsoidalcollapsemodel,asanaturalextensionoftheone-to-onemappingofthespher-icalcollapseapproximation.Further,thestochasticnaturearisingfromthemulti-variatefunctionoflocalquantitiesisexplicitlyshownevaluatingthejointPDFsofthelocaldensityand/orthevelocity-divergence.Thein uenceofthise ectisdiscussedindetailcomparingwiththesphericalcollapseapproximation.

Thispaperisorganizedasfollows.Insection2,weconsiderageneralframeworktotreattheevolutionofone-pointPDFsandderivetheevolutionequationsfortheEulerianandtheLagrangianPDFs(Sec.2.2and2.3).Then,adoptingthelocalapproximation,consistentsolutionsfortheseequationsareobtained(Sec.2.4).Further,thestochasticnatureoftheevolutionofPDFsisquanti edevaluatingjointPDFs(Sec.2.5).Asanapplicationofthesegeneralconsiderations,insection3,wegiveanexplicitevaluationofone-pointPDFsadoptingthesphericalandtheellipsoidalcollapsemodelsasrepresentativeLagrangianlocaldynamics.ThequalitativefeaturesoftheresultsarecomparedwiththeperturbativeanalysispresentedinappendixBorthepreviousN-bodystudy.Finally,section4isdevotedtotheconclusionandthediscussion.

2.EVOLUTIONEQUATIONFORPROBABILITYDISTRIBUTION

FUNCTION

2.1.Preliminaries

Throughoutthepaper,wetreatdarkmatterasapressure-lessandnon-relativistic uid.Assumingthehomogeneousandisotropicbackgrounduniverse,thedensity eldδ(x,t),thepeculiarvelocity eldv(x,t)andthegravitationalpotentialφforthe uidfollowtheequation

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.

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

Inthefollowingsubsection,we rstconsidertheLagrangianPDFandderivetheevolu-tionequation.ThenwederivetheevolutionequationforEulerianPDFinsection2.3.Theevolutionequationsderivedherearenotyetclosedbecauseoftheunknownfunctions.Insec-tion2.4wediscussanapproximatetreatmentusingthelocaldynamicsmodel,whichenablesustoobtainaclosedformofevolutionequationandtoconstructaconsistentsolution.

grangianone-pointPDF

ToderivetheevolutionequationfortheLagrangianone-pointPDF,weintroduceanarbitraryfunctionoflocaldensity,g(δ),andconsidertheevolutionofitsexpectationvalue, g(δ(q,t)) LevaluatedinaLagrangianframe.Thedi erentiationofthisexpectationvalue

withrespecttotimebecomes

dFig(δ)PL(F;t)=dδg(δ) ti

dtg(δ(q,t)) =

L dgdt =L idFidg(δ)dtPL(F;t).(8)

Theright-hand-sideoftheaboveequationcanbeexpressedbyintegratingbypartas

dg(δ) PL(F;t)= PL(F;t)dFidFig(δ)dtdtii PL(δ;t).= dδg(δ)dtδ

Here,thequantity[A]BdenotestheconditionalmeanofAforagivenvalueofB:

[A]B≡dFiAP(F|B)

Fi=B(9)(10)

withthefunctionP(F|B)beingtheconditionalPDFforagivenB,i.e.,P(F|B)=P(F)/P(B).

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

Now,recallingthefactthatg(δ)isanarbitraryfunctionoflocaldensityδ,equation

(7)mustbeequivalenttoequation(8)intheLagrangeframe.Thecomparisonbetweenequation(7)andequation(9)thenleadstothefollowingevolutionequation: dδ

δ

tPL(θ;t)+ dtPL(θ;t)

θ=0.(12)

2.3.Eulerianone-pointPDF

TheevolutionequationfortheEulerianone-pointPDFscanalsobederivedbyrepeatingthesameprocedureasabove,buttheresultantexpressionsareslightlydi erentfromthoseoftheLagrangianPDFs.Thetimederivativeoftheexpectationvalue g(δ) Ebecomes

tPE(δ;t).(13)

Asfortheexpectationvalueof g/ t,withahelpoftheLagrangiantimederivative,weobtain δdδ1

dδdδdδ

1

dδ

=1av· g(δ(x,t)) E

tg(δ(x,t)) =

E dδg(δ) dt(t) PE(δ;t)+H[θ(t)]δPE(δ;t).(16)δ

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

Hence,therelation g/ t E= g(δ) E/ t,nottheequation dg/dt E= g(δ) E/ t,leads

totheevolutionequationfortheEulerianone-pointPDFP(δ,t).Fromequations(13)and

(14),weobtain

dδ

δ

tPE(θ;t)+ dt(t)

Notethatthezero-meanofthevelocitydivergence θ E=0isalwaysguaranteedfrom

equation(18),whichiseasilyshownbyintegratingtheaboveequationoverθdirectly. PE(θ;t)θ =HθPE(θ;t).(18)

2.4.Localapproximation

Theevolutionequationsintheprevioussubsectionarerathergeneralandinderivingthesewedidnotspecifythedynamicsof uidevolution.Inthissense,theyarenotcloseduntilfunctionalformsoftheconditionalmeans[dδ/dt]δ,[θ]δand[dθ/dt]θarespeci ed.Inotherwords,theseequationsrequiretheadditionalequationsgoverningtheevolutionoftheconditionalmeans.However,anattempttoobtaintheclosedsetofevolutionequationssu ersfromseriousmathematicaldi culty,so-calledclosureproblem,whichiswell-knowninthesubjectof uidmechanicsand/orplasmaphysics(e.g.,Chenetal.1989;Goto&Kraichnan1993).SimilartotheBBGKYhierarchy,ifonederivestheevolutionequationsfortheconditionalmeans,thereappearnewunknownconditionalmeans.Thus,wemustfurtherrepeatthederivationofevolutionequationforthenewquantities.Continuingthisoperation,onecould nallyobtainthein nitechainoftheevolutionequations,whichisgenerallyintractable.

Insteadoftheexactanalysistacklingthedi cultproblem,weratherfocusonanapprox-imatetreatmentoftheevolutionofone-pointPDFs,wherethesolutionsfortheevolutionequationsareconsistentlyconstructed.Toimplementthis,weadoptthelocalapproximationthattheevolutionofthelocaldensityδandthevelocity-divergenceθisdescribedbytheLagrangiandynamicswith nitedegreesoffreedom,whoseinitialconditionsarecharacter-izedbytheinitialparameters,p=(p1,p2,···pn),givenatthesameLagrangiancoordinate.

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

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)

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

Further,theconditionalmean[θ]δcanbeexpressedas

[θ]δ= 1

dt (31)

δ

fromtheequationofcontinuity(1).Then,thesolutionsofequations(17)and(18)becomes

PE(δ;t)=1

1+gδD(θ h(p,t)).(33)

Onecaneasilyshowthatequations(32)and(33)satisfytheevolutionequations(17)and

(18),withthecorrectnormalizationandthezeromean.

TheabovesolutionsforEulerianPDFcanberegardedasageneralizationofthepreviousstudybasedontheZel’dovichapproximation(Kofmanetal.1994)and/orthesphericalcollapsemodel(Scherrer&Gazta naga2001,seealsoProtogeros&Scherrer1997).Note,however,thatthegeneralexpressionofvelocity-divergencePDFPE(θ;t)di ersfromthe

oneobtainedpreviously.Whilethefactor1/(1+δ)isconvolvedintheintegralinequation

(33),theresultantexpressionbyKofmanetal.(1994)obviouslyomittedthisfactor.Inourprescription,thePDFPE(θ;t)isconstructedfromtheevolutionequation,whichbasically

reliesontheequationofcontinuity.Thismeansthat,evenforthevelocity-divergencePDF,thefactor1/(1+δ)isnecessarytoensurethemassconservation.Infact,theexpression(33)canalsobeobtainedfromthejointPDFPE(δ,θ;t)integratingoverthelocaldensityδ(see

eq.[36]).Althoughthevelocity-divergencePDFinthepreviousstudyhasbeenobtainedinaratherphenomenologicalway,ourpresentapproachusingtheevolutionequationsmightbehelpfulinconstructingtheconsistentPDFs.

Nevertheless,evenatthispoint,thesolutionsofevolutionequationsshouldberegardedasformalexpressions.InordertoevaluatethePDFsexplicitly,weneedtospecifytheLagrangianlocaldynamics.Inotherwords,thequantitativepredictionforPDFsusingthelocalapproximationcruciallydependsonthechoiceofthelocaldynamics.Wewillseeinthesection3thattheexplicitevaluationofPE(δ;t)andPE(θ;t)adoptingthesphericaland

theellipsoidalcollapsemodelsshowsseveralnoticeabledi erences.

2.5.JointPDF

Sofar,wehaverestrictedourattentiontotheone-pointPDFforthesinglelocalvariable,δorθ.Inourtreatmentofthelocalapproximation,theexpressionsforPDFsaregeneral

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

formsirrespectiveoftheLagrangianlocaldynamics.However,itshouldbeemphasizedthatifthelocaldynamicsischaracterizedbymorethanthetwoinitialparameters,qualitativebe-haviorcouldbesigni cantlychangedfromthelocaldynamicswithsingledegreeoffreedom.Thepointisthattherelationbetweeninitialparametersandtheevolvedresultδorθcannotbedescribedbyaone-to-onemapping.Accordingly,therelationbetweenδandθbecomesnolongerdeterministic.Moreover,thefailureofdeterministicpropertyalsoappearsinthetimeevolutionofsuchlocalvariables.Itisthereforeimportanttodiscussthestochasticnatureofδandθarisingfromthedynamicalevolution.Tocharacterizethis,weconsiderthejointPDF.Withinthelocalapproximation,onecanconstructaconsistentsolutionofEulerianjointPDFbetweenδandθevaluatedatthesametime,PE(δ,θ;t).Further,the

LagrangianjointPDFforthedensity eldevaluatedatthesameLagrangianpositionbutatthedi erenttimes,PL(δ,t;δ′,t′)canalsobeobtained.

TheevolutionequationofPE(δ,θ;t)canbederivedthroughtheexpectationvalueof

anarbitraryfunctiong(δ,θ).Repeatingthesameprocedureasdescribedinsection2.3,weobtain

δdδ θ

δ

dθ tdθPE(δ,θ;t),

θ

dδ tPE(δ,θ;t) Hθ θdθ

dt =

δ,θ11+δ dpiPI(p)dh

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

andthecorrespondingsolutionofequation(34)becomes

PE(δ,θ;t)=1

g(δ(q,t),δ(q,t′)) =dδdδ′g(δ,δ′)dt g(δ(q,t),δ(q,t′))=dt = dδdδ′g(δ,δ′)

TheevolutionequationofLagrangianjointPDFis

dδ

δ

dt dt PL(δ,t;δ′,t′).δ,δ′ dt=

δ,δ′1dtδD(δ f(p,t))δD(δ′ f(p,t′)).

RecallingthatthejointPDFsatisfyingtheevolutionequation(37)shouldbeinvariantunderthetransformation,(δ,t) (δ′,t′),thesolutionconsistentwiththeboundaryconditionPL(δ,t′;δ′,t′)=PL(δ;t′)δD(δ δ′)becomes

PL(δ,t;δ′,t′)=dpiPI(p)δD(δ f(p,t))δD(δ′ f(p,t′)).(38)

i

NoticethatifthelocalLagrangiandynamicsisdescribedbyasingleparameter,theintegralovertheinitialparameterp1inequation(38)canbeformallyperformed.TheresultantexpressionincludesDirac’sdeltafunction,leadingtotheone-to-onemappingbetweenδandδ′.Ontheotherhand,incaseswiththemultivariateinitialparameters,onecannotgenerallyperformtheaboveintegralandtheDirac’sdeltafunctionisnotfactoredout,leadingtothestochasticnatureoflocaldensity elds.

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

OnemightfurtherconsidertheevolutionofEulerianjointPDFPE(δ,t;δ′,t′),which

hasbeenindeedshowninN-bodysimulationsbyKayoetal.(2001).Thederivationofevolutionequationitselfisaneasytask,but,theformalsolutioninthelocalapproximationsu ersfromdi cultiesduetothepresenceofadvectiveterm,whichmightberelatedtoanimportante ectonthenon-localnatureof uid ows.SinceeventheLagrangianjointPDFshowsseveralimportantfeatures,onecanexpectthatthequalitativelysimilarbehaviortotheN-bodyresultscanbeseenfromtheLagrangianjointPDF.Hence,wewillpostponetoanalyzetheEulerianjointPDFPE(δ,t;δ′,t′)andinsteadfocusontheLagrangianjointPDF

PL(δ,t′;δ′,t′).

3.DEMONSTRATIONANDRESULTS

Nowweareinapositiontogiveanexplicitevaluationoftheone-pointPDFbasedonthelocalapproximationdiscussedintheprevioussection.Foranillustrativepurpose,weadoptthesphericalandtheellipsoidalcollapsemodelsassimpleandintuitiveLagrangianlocaldynamics.Afterbrie ydescribingthebasicequationsofthesemodelsinsection3.1,wecomputetheEulerianone-pointPDFsPE(δ)andPE(θ)anddiscussthequalitativedi er-

encesarisingfromthechoiceofLagrangianlocaldynamicsinsection3.2.Inparticularthestochasticityinthemulti-variatefunctionoflocaldensityorvelocity-divergenceisexaminedindetailbyevaluatingthejointPDFs,PL(δ,t;δ′,t′)andPE(δ;θ)fromtheellipsoidalcollapse

model.

3.1.ModelsforLagrangianlocaldynamics

FirstconsiderthesimplestcaseoftheLagrangianlocaldynamics,inwhichtheevolutionoflocalquantitiesisdeterminedbythemassinsideasphereofradiusRcollapsingviaself-gravity:d2R4π3;M=ρR=const,(39)R2

whereMisthemassinsidetheradiusand

ρ/ρm 1=(a/R)3 1asfollows: 2dδ4dδ dt1+δ2H2 m(1+δ)δ,(40)

withthequantity mbeingthedensityparameter, m≡8πGρm/(3H2).AsFosalba&Gazta naga(1998a)stated,thisequationisregardedasashear-lessandirrotationalapprox-

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

imationof uidequations,sinceonecanderivethefollowingequationfromequations(1)to

(3): d2δ4dδ mδ+σijσij ωijωij,(41)dt1+δ2

withahelpoftheLagrangiantimederivative.Herethequantitiesσijandωijrespectivelydenotetheshearandtherotationgivenby

σ1

ij=j

x+ v

j3θδij,

ω1 v

ij=j

xj

√

dt2αi= 4πGρmαi

∞ 1+δ2δ+λext,idτ ,bi=α1α2α3

03,

andtherelationbetweenδandαibecomes

3

δ=a

35 ;linearexternaltide,(42)(45)(46)

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

whereD(t)isthelineargrowthrate.Thevariablesλirepresenttheinitialparametersofprincipalaxes,andintermsofthese,theinitialconditionsreduceto

dαi(t0)=a(t0){1 D(t0)λi},(49)

dt2+2Hdδ

31dt α˙i3 2=H2(1+δ) α˙2 3δij.3Hα1α3

Notethatsimilartothesphericalcollapseapproximation,thethreeinitialparametersλiareregardedastherandomvariables.Whentheinitialconditionofdensity eldisassumedtobeaGaussianrandomdistribution,theexpressionfortheinitialparameterdistributionPI(λi)isanalyticallyobtainedasfollows(e.g.,Doroshkevich1970;Bardeenetal.1986):

23375I1PI(λi)=(λ1 λ2)(λ2 λ3)(λ1 λ3),(52)exp 3262σ5πσll

wherethequantitiesI1andI2denoteI1≡λ1+λ2+λ3andI2≡λ1λ2+λ2λ3+λ3λ1,respectively.

BasedontheseLagrangianlocalmodels,wenumericallycalculatethePDFsassumingtheEinstein-deSitteruniverse( m=1, Λ=0),inwhichthelineargrowthrateDissimplyproportionaltothescalefactora.Forabetterunderstandingofthelateranalysis,inFigure1,weplottheevolutionoflocaldensityδfromtheellipsoidalcollapsemodelforsomeinitialconditions(e,p)givenbye=(λ1 λ3)/(2δl)andp=(λ1+λ3 2λ2)/(2δl).Theresultsarethendepictedasafunctionoflinearlyextrapolateddensityδl=λ1+λ2+λ3andarecomparedwiththeonefromtheone-to-onemappingofsphericalcollapsemodel(solid).Figure1showsthatthelocaldensityoftheellipsoidalcollapsemodelgenerallytakesalargervaluethanthatofthesphericalcollapsemodel.Further,thevarietyofevolveddensityfor xedδlsuggeststhatalargeamountofscatterwillappearinthejointPDFPL(δ,t;δ′,t′)andtheresultantone-pointPDFsPE(δ)andPE(θ)cannot,ingeneral,coincidewiththose

obtainedfromthesphericalcollapsemodel.

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

3.2.Results

IncomputingthePDFsfromtheabovelocalcollapsemodels,onemaypracticallyen-counterthecasewhenthelocaldensityin nitelydivergesat niteelapsedtimeforsomeregionsintheinitialparameterspace,whichhasnotbeentreatedinprevioussection.Toavoidtheunphysicaldivergences,wemustrestricttheintegralinthePDFstotheinitialpa-rameterspaceV(t),inwhichthelocaldensityδdoesnotdiverge.Indeed,thismodi cationslightlya ectsthenormalizationconditionforboththeLagrangianandtheEulerianPDFs,i.e., 1 L,E=1.AlthoughthisdoesnotalteranyqualitativefeaturesofPDFs,weconsider

somemodi cationstokeepthecorrectnormalizationandadoptingthisprocedureinap-pendixA,andtheresultsforone-pointandjointPDFsarepresentedbelow.Note,however,thattheperturbationcalculationdiscussedin3.2.1isfreefromtheseriousdivergencesandwithintheperturbativetreatment,onecanrigorouslydevelopthelocalapproximationforPDFs.

3.2.1.One-pointPDFs

Letusshowtheresultsoftheone-pointPDF.Figure2plotstheone-pointEulerianPDFsofthelocaldensity(top)andthevelocity-divergence(bottom)evaluatedatvariouslinear uctuationvaluesσl.Inbothpanels,thethicklinesrepresenttheresultsobtainedfromtheellipsoidalcollapsemodelwithlinearexternaltide,whilethethinlinesdenotethePDFsfromthesphericalcollapsemodels.IncomputingthePDFs,theLagrangianlocaldynamicsarenumericallysolvedwiththevariousinitialconditionspintheinitialparameterspaceV.Then,weightingbythePDFoftheinitialparameterPI(p),thePDFsareconstructed

bybinningtheevolvedresultsofthedensityδandthevelocity-divergenceθ,togetherwithappropriateconvolutionfactors(seeAppendixA).

Asexpected,theoverallbehaviorsofbothPDFsinFigure2arequalitativelysimilar,irrespectiveoftheLagrangianlocalmodels.Asincreasingthelinear uctuationvalueσl,whilethedensityPDFPE(δ)extendsoverthehigh-densityregionδ 1,thevelocity-

divergencePDFPE(θ)isnegativelyskewedanditextendsoverthenegativeregionθ 1.

Inlookingatthedi erencesineachlocalmodel,wereadilyobserveseveralremarkablefeatures.First,thedensityPDFscomputedfromboththesphericalandtheellipsoidalcollapsemodelsalmostagreewitheachother.At rstglance,thisseemstocontradictwithanaiveexpectationfromthelocaldynamicsinFigure1.However,onemightrathersuspectthattheagreementindensityPDFsisaccidental,duetothedistributionofinitialparametersPI(λi),whichis,atleast,consistentwithanaivepicturethatjointPDFPL(δ,t;δ′,t′)fromthe

ellipsoidalcollapsemodelexhibitsalargemountscatterandthemeanrelationbetweenδand

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)

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

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

basicallyre ectthequalitativebehavioroflocaldynamicsinFigure1.Thatis,theevolvedresultsoflocaldensityintheellipsoidalcollapsetendstotakealargervaluethanthatinthesphericalcollapse.Moreover,recallthefactthatboththeinitialandthe nalPDFsoflocaldensityPE(δ)showagoodagreementbetweenthesphericalandtheellipsoidalcollapse

model(seeFig.2).ThisisindeedthesamesituationasintheN-bodysimulation;apartfromthedetaileddi erences,asimplemodelofPDFsprovidesanessentialingredientforthestochasticnatureofN-bodyresults.Inthissense,thelocalapproximationwithellipsoidalcollapsemodelscanberegardedasaconsistentandphysicalmodelofone-pointstatistics,whichsuccessfullyexplainstheN-bodysimulations.

Finally,usingtheellipsoidalcollapsemodelwithlinearexternaltide,weexaminetheEulerianjointPDFoflocaldensityandvelocity-divergenceevaluatedatthesametime,i.e.,PE(δ;θ).InFigure5,contourplotsofjointPDFPE(δ;θ)forvariouslinear uctuationvalues

σlaredepictedasfunctionof θandδ.Thisistheso-calleddensity-velocityrelation,whichmightbeofobservationalinterestinmeasuringthedensityparameter mfromthevelocity-densitycomparisonthroughthePOTENTanalysis(e.g.,Bertschinger&Dekel1989).Alongthelineofthis,theoreticalworksbasedontheEulerianperturbationtheoryhaveattractedmuchattention,aswellastheN-bodystudy(e.g.,Bernardeau1992a;Chodorowski&Lokas1997;Bernardeauetal.1999).Basedonthesolutionofthelocalapproximation(36),onecaneasilycalculatetheperturbationseriesofvelocity-densityrelation[θ]δasfunctionoflocaldensityanddensity-velocityrelation[δ]θasfunctionofvelocity-divergence,theleading-orderresultsofwhichareexpectedtocoincidewithpreviousearlyworksinthenon-smoothingcase,exactly.Beyondtheperturbationanalysis,Figure5revealsthegeneraltrendofthestochasticnatureinthevelocity-densityrelation.Asincreasingσl,thescatterbecomesmuchbroaderandtheconditionalmeans[δ]θ(dot-dashed)and[θ]δ(solid)doesnotcoincidewitheachother.Ofcourse,theone-to-onemappingobtainedfromthesphericalcollapsemodel(short-dashed)failstomatchthebothconditionalmeans.ThesequalitativebehaviorisinfactconsistentwiththeN-bodyresultsbyBernardeauetal.(1999)andthepresentmodelprovidesasimplewaytoderivethenon-linearandstochasticvelocity-densityrelation.

4.CONCLUSIONANDDISCUSSION

Inthispaper,startingfromageneraltheoryofevolutionofone-pointPDFs,wederivedtheevolutionequationsforPDFandwithinalocalapproximation,consistentformalsolu-tionsofPDFareconstructedinboththeLagrangianandtheEulerianframes(seeeqs.[21][22]forLagrangianPDFsandeqs.[32][33]witheqs.[27][30]forEulerianPDFs).InordertorevealthestochasticnaturearisingfromthemultivariateLagrangiandynamics,wefurthercon-

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

sidertheEulerianjointPDFPE(δ,θ;t)(eq.[36])andtheLagrangianjointPDFPL(δ,t;δ′,t′)

(eq.[38]).Then,adoptingthesphericalandtheellipsoidalcollapsemodelsasrepresentativeLagrangianlocaldynamics,weexplicitlyevaluatetheEulerianPDFs,PE(δ)andPE(θ),as

wellasthejointPDFs.Theresultsfromtheellipsoidalcollapsemodelshowseveraldis-tinctproperties.WhilethePDFPE(δ)almostcoincideswiththeone-to-onemappingof

thesphericalcollapsemodel,thetailsofvelocity-divergencePDFPE(θ)largelydeviatefrom

thoseobtainedfromthesphericalmodel.Thesebehaviorshavealsobeencon rmedfromtheperturbativeanalysisofhigherordermoments.Ontheotherhand,evaluatingtheLa-grangianjointPDFoflocaldensity,PL(δ,t;δ′,t′),alargescatterintherelationbetweenthe

initialandtheevolveddensity eldswasfoundandtheirmeanrelationfailstomatchtheone-to-onemappingofsphericalcollapsemodel.Thisremarkablyreproducesthesamesitu-ationintheN-bodysimulation.Therefore,thelocalapproximationwithellipsoidalcollapsemodelprovidesasimpleandphysicallyreasonablemodelofone-pointstatistics,consistentwiththeleading-orderresultsofexactperturbationtheory.

Sincethepresentformalismdescribedinsection2isquitegeneral,theapproachdoesnotrestrictitsapplicabilitytothepressure-lesscosmological uid.Rather,onemayapplytothevarious uidsystemsinpresenceorabsenceofgravity.Asmentionedinsection

2.4,however,theapplicabilityorthevalidityoflocalapproximationofPDFs,inprinciple,sensitivelydependsonthechoiceofLagrangianlocaldynamics.Inthelastsection,simpleandintuitiveexampleswereexaminedfortheillustrativepurposes.TheresultsindicatethatthemultivariateLagrangiandynamicsratherthanthelocalmodelwithasinglevariablecandescribevariousstatisticalfeaturesof uidevolutionincludingthestochasticnature.

Perhaps,astraightforwardextensionofthepresenttreatmentistoincludethee ectofredshift-spacedistortionorprojectione ect,whichispracticallyimportantforpropercomparisonwithobservation.Beforeaddressingthisissue,however,rememberthemostprimarilyimportanceofthesmoothinge ect.Whiletheappropriateprescriptionfortop-hatsmoothing lterdoesexistinthelocalapproximationwiththesphericalcollapsemodel(e.g.,Bernardeau1994b;Protogeros&Scherrer1997;Fosalba&Gazta naga1998a),thesmoothinge ectontheapproximationusingellipsoidalcollapsemodelstillneedstobeinvestigated.Thisstepisinparticularacrucialtaskinordertoconstructamorephysicalprescriptionforone-pointstatisticsofcosmic eldsandtheanalysisisnowinprogress.Theresultswillbepresentedelsewhere(Ohta,Kayo&Taruya,inpreparation).

WearegratefultoY.Sutoforreadingthemanuscriptandcomments.I.KacknowledgesthesupportbyTakenaka-IkueikaiFellowship.Thisworkissupportedinpartbythegrand-in-aidforScienti cResearchofJapanSocietyforPromotionofScience(No.1470157).

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