Analysis of Large Scale Structure using Percolation, Genus and Shape Statistics

arXiv:astro-ph/98031v1 17 Mar 1998VARUNSAHNI

Inter-UniversityCentreforAstronomy&Astrophysics,PostBag4,Pune411007,India

Abstract.

WeprobegravitationalclusteringinN-bodysimulationsusinggeomet-ricaldescriptorssensitiveto‘connectedness’:thegenuscurve,percolationandshapestatistics.Asgravitationalclusteringadvances,thedensityfieldinN-bodysimulationsshowsanincreasinglypronounceddeparturefromGaussianityreflectedinthechangingshapeofthepercolationcurveandthechangingamplitudeandshapeofthegenuscurve.WefeelthatbothgenusandpercolationcurvesprovidecomplementaryprobesoflargescalestructuretopologyandcouldbeusedtodiscriminatebetweenmodelsofstructureformationandtheanalysisofobservationaldatasuchasgalaxycatalogsandMBRmaps.Thefillingfactorinclusters&superclustersatpercolationissmallindicatingthatmatterismorelikelytolieinfilamentsandpancakes.Ananalysisof‘shapes’inN-bodysimulationshasshownthatfilamentsaremorepronouncedthanpancakes.Toprobeshapesofclustersandsuperclustersmorerigorouslyweproposeanewshapestatisticwhichdoesnotfitisodensitysurfacesbyellipsoids(asdoneearlier).InsteadourshapestatisticisderivedfromfundamentalpropertiesofacompactbodysuchasitsvolumeV,surfaceareaS,integratedmeancurvatureC,andconnectivity(characterizedbytheGenus).Thenewshapestatisticgivessensibleresultsfortopologicallysimplesurfacessuchastheellipsoid,andformorecomplicatedsurfacessuchasthetorus.

[Invitedtalk,toappearin:ProceedingsoftheIAUSymposiumNo.183:“CosmologicalparametersandevolutionoftheUniverse”,Kyoto,JapanAug.1997,ed.K.Sato(KluwerAcademicPubl.)]

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1.Introduction

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TheUniverseasweperceiveitseemsabundantlyrichinvisualform.Itslargescalestructureconsistingofclustersandsuperclustersofgalaxieshasbeenvariouslyperceivedtobe‘acosmicweb’,‘networkofsurfaces’,‘sponge-like’,bubble-like’etc.Attemptstodescribeitslargescalefea-turesquantitativelyhavebeenmadeusinganumberofstatisticalindi-catorssensitivetothe‘connectedness’oflargescalestructureincludingthegenuscurveandpercolationstatistics(Zeldovich1982;Shandarin1983;Gott,Melott,&Dickinson1986);minimalspanningtrees(Barrowetal.1995);Minkowskifunctionals;(Meckeetal.1994)andstatisticssensitivetoshape(seeSahni&Coles(1995)andreferencestherein).

InthistalkweassesstherelativemeritsofgenusandpercolationcurvesbyapplyingthemtothesameN-bodysimulationsinanΩ=1Uni-versewithscale-invariantinitialconditionsP(k)≡󰀌|δk|2󰀍∝kn,n=−2,−1,0,+1,(forsimplicityweshowresultsonlyforn=−2whichmaybeconsideredthelowerlimitoftheslopeoftheinitialspectrumongalaxyscales).Resultsareshownatseveralepochseachcharacterizedbythescaleofnonlinearity,kNL,atthatepochmeasuredinunitsofthefundamentalmode2π/L,whereListhelengthofthesimulationbox.N-bodysimula-tionswereperformedona1283gridusingaparticle-meshalgorithm(Melott&Shandarin1993).Areducedgridofsize3wasusedtoconstructden-sityfieldsfromparticlepositionsandtheanalysisofpercolationandgenuscurveswasthenperformedonthesefields.

WealsointroduceanewstatisticsensitivetoshapebasedonMinkowskifunctionals.

2.Growthofnon-GaussianityduringCosmologicalgravitationalclustering.

Conventionalmodelsofgravitationalclusteringusuallyassumethatprimor-dialdensityperturbationshadascale-invariantHarrison-Zeldovichspec-trumandweredistributedinthemannerofaGaussianrandomfield.Ar-gumentswhichsupportthishypothesisstemfromthecentrallimittheoremandtheInflationaryparadigm.Thenon-GaussianitywhichweobserveintheUniversetoday(clusters,superclusters,voids)isattributedtonon-linearevolutionandtheresultingphasecorrelationbetweenmodes.Tworobustandwidelyusedstatisticalindicatorsofclusteringaretheproba-bilitydensityfunction(PDF)andthetwopointcorrelationfunctionξ(r).Howeverneithercharacterizesthenonlinear󰀁distributionofmatteruniquely.Thetwopointcorrelationfunctionξ(x)=d3󰀩kexp(i󰀩k·󰀩x)P(k)beingsen-sitiveonlytothepowerspectrumP(k)≡󰀌|δk|2󰀍andnottothephasesφkofindividualmodes[δ(󰀩k)=|δk|exp(iφk)]missesfeaturesarisingbecause

PERCOLATION,GENUSANDSHAPESTATISTICS.3

ofphasecorrelationsinthenonlinearregime.Ontheotherhand,thePDFdoesnotcharacterizeadistributionuniquelyinthenonlinearregime:distri-butionswithidenticalPDF’scanhaveverydifferenttopologicalpropertiesand,conversely,distributionsdifferingintheirPDF’smayhaveidenticalgeometricalproperties(thishappensforinstanceinthecaseofdistributionsrelatedbyamappingδNL=F(δLIN),suchasthelog-normal).3.PercolationandGenusCurves

Itisclearthattraditionalindicatorsofclustering:thetwopointcorrelationfunctionandtheprobabilitydensityfunction,mustbecomplementedinthenon-linearregimeifwearetogetabetterunderstandingoftheissueofnon-Gaussianity.Onewayofachievingthisistousegeometricalmeasureswhicharesensitivetotheconnectednessofadistribution.Twosuchindicators–percolationandthegenuscurve,willbestudiedinthissection,athirdshapestatistics,willbediscussedinthenext.

Oneoftheaimsofpercolationtheoryistostudytheconnectednessofstructureasafunctionofthedensitythreshold.Varyingthedensitythresholdfromahightolowvalue,leadstoa‘percolationtransition’asthevolumefractioninthelargestclusterchangesrapidlyfromalmostzerotounitywhenthedensitythresholdcrossesacriticalvalue.Itisconvenienttocharacterizepercolationintermsoffillingfactor–henceforthFF–thetotalvolumeinallclusters/voidsabove/belowthedensitycontrastthresholddividedbythesimulationvolume1.GaussianrandomfieldspercolateatthecriticalfillingfactorFFC≃16%regardlessofthespectrum.DensityfieldsevolvingundergravitationalinstabilitytypicallypercolateatlowerlevelsofFFCdependingupontheinitialspectrumandtheextentofnon-linearevolution(Yess&Shandarin1996).SimilarconclusionsarealsoreachedinthecaseofpointlikedistributionsalthoughthenaturalreferenceinthiscaseisthePoissondistribution(Klypin&Shandarin1993).

EarlierworkongravitationalclusteringfocussedonFFCasadiagnosticmeasure(Shandarin1983;Dominik&Shandarin1992;Klypin&Shandarin1993).However,althoughusefulinprobingtheextentofnonlinearevolutionFFCdoeshavecertaindrawbacks,forinstanceitissensitivetoresolution,numberofparticlesandsamplegeometry(Dekel&West1985).Apow-erfulnewstatisticwithouttheabovelimitationsisthepercolationcurve(PC).Considerthevolumefractionvmaxdefinedastheratioofthevolumeinthelargestcluster/voidtothetotalvolumeinallclusters/voidslyingabove/belowadensitycontrastthreshold.Thepercolationcurvedescribesthevolumefractionvmaxasafunctionofthedensitycontrastthreshold(orfillingfactor).

1

FFisthecumulativeprobabilitydistributionfunction:FF=P(δ>δT).

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ThepercolationcurveisplottedinFig.1forevolveddensityfieldsfromN-bodysimulationswiththepower-lawinitialspectrumn=−2.Perco-lationcurvesforclusters(thicksolidlines)andvoids(thickdashedlines)areplottedseparately.Verticalthinsolidandthindashedlinesshowthepercolationdensitythresholdbelowwhichclustersandabovewhichvoidspercolate.FromFig.1wefindthatatveryhigh(low)thresholdsthenumberofclusters(voids)isverysmallandvmax<<1.Asthethresholdisgradu-allydecreased(increased)thevolumefractioninthelargestcluster(void)increasesasclusters(voids)begintomergeuntilthepercolationtransitionwhenthelargest‘supercluster’(supervoid)spanstheentiresimulationbox.(Decreasing(increasing)thedensitythresholdcorrespondstoincreasingthefillingfactorforclusters(voids).)FromFig.1weseethatasthesimulationevolvesδCincreasesmonotonicallyaspowerinlongerwavelengthscausesstructurestoformandalignonincreasinglylargerscales.Forspectrawithlesserlongrangepowersuchasn=0,δCinitiallyincreasesbutlaterbeginstodecreasesignalingtheformationofsmall,isolatedclumps(notshown)(Sahni,Sathyaprakash&Shandarin1997a).

Ananalysissimilartopercolationcanalsobeperformedusingthegenuscurve(GC)whichcanbeformallyexpressedasanintegralovertheGaus-siancurvatureKoftheiso-densitysurfacesSνlyingabove/below󰀁adensitythresholdν=δ/σδbytheGauss-Bonnettheorem:4πG(ν)=−SνKdA.ForGaussianRandomfieldsthegenuscurvehasa‘bellshaped’form:G(ν)=A(1−ν2)exp(−ν2/2)(Hamiltonetal.1986;Gottetal.1987;Gottetal.19).(Ananalyticalexpressionforthegenusintheweaklynon-linearregimehasbeenobtainedin(Matsubara1994).)Multiplycon-nectedsurfaceshaveG≥0whilesimplyconnectedhaveG<0.TheupperrighthandpanelsofFig.1showthegenuscurveplottedasafunctionofthedensitycontrast.Itisinterestingtonotethatzero-crossingsofthegenuscurvearequiteclosetothepercolationthresholdforbothclustersandvoids.Thisreflectsthefactthatthestructuretransformsfromsimplyconnectedtomultiplyconnectedatthezero-crossingofGwhichallowsittopercolate.Onecandiscernastrongincreaseinnon-Gaussianityasthesimulationevolves,reflectedbyanevolutioninshapeofbothpercolationandgenuscurves.

InsteadofplottingGCandPCagainstthedensitycontrast(whichisnotanormalizedquantity),itmaybemoreappropriatetoplotthemagainstthefillingfactor.Thishelpstodistinguishbetweendistributionsrelatedbyamappingδ→f(δ)(suchasthelog-normal)whichhaveidenticaltopologi-calpropertiesbutcanhavequitedifferentPDF’s.Thelowerpanelsinfig.1showPC,GCforclusters(solid)andvoids(dashed)plottedagainstFF.Thethreeverticallinesshowthefillingfactoratpercolationforclusters,voidsandGaussianrandomfieldswithidenticalspectra.Bothpercolation

PERCOLATION,GENUSANDSHAPESTATISTICS.5

andgenuscurvesnowresemble‘hysteresis’curves,theareabetweenvoidandclustercurvesindicatingthedegreeofnon-Gaussianityinthedistri-bution.ForPCwenoticeamarkedincreaseinnon-Gaussianityreflectedintheincreasingdifferencebetweenpercolationthresholdsforclustersandvoidsmeasuredby:FFC(voids)−FFC(clusters).Thegenuscurvedoesnotappeartoevolvemuch,whichissurprising.HowevertheamplitudeofGCdoesdecreasewithepoch,aneffectwhichismorepronouncedforspectrawithgreatersmallscalepower,andwhichweattributetotherapidbuildupofphasecorrelationsduetononlinearmodecouplingduringadvancedgravitationalclustering(Sahni,Sathyaprakash&Shandarin1997a).

ComparingthegeometricalpropertiesofadistributiontoafeaturelessGaussian,onecanmakestatementsregardingits‘connectednessortopol-ogy’.InFig.1wehaveindicatedthepercolationthresholdofGaussianran-domfieldsbyadottedverticalline.Comparingthesepercolationthresholdswiththoseofclustersandvoidsweconcludethatpercolationis‘easier’forclustersandmore‘difficult’forvoids.ClusterspercolatingatlowerFFthanGaussianaresaidtopossessa‘network-like’topology.VoidsontheotherhandpercolateathigherFFthanGaussianandsohavea‘bubble-like’topology.Thisappearstobeagenericfeatureofmostmodelsofgravita-tionalclusteringwithareasonableamountoflong-wavelengthpowerintheinitialspectrum(i.en≤0)(Sahni,Sathyaprakash&Shandarin1997a).4.Shape-statistics

Asdiscussedinthelastsection,gravitatingsystemsclusteringfromGaus-sianinitialconditionspercolateatlowvaluesofthefillingfactor.ForCDM&CHDMmodelsthefillingfactorcanbeassmallas2%−7%,muchsmallerthanthe16%expectedforarandomGaussianfield(Klypin&Shandarin1993).Thisimmediatelysuggeststhatthepercolatingphaseismorelikelytobesheetorfilament-likesincesuchdistributionsarelikelytooccupyalargerlineardimension(foranequalamountofmass)andwillthereforepercolatemoreeasily.SomeindicationthatthisisindeedthecasealsocomesfromtheZeldovichapproximation(Shandarin&Zeldovich19).Adetailedstudyof‘shapes’inscaleinvariantmodelsofgravitationalclusteringrevealedthatonedimensional‘filaments’aremoreabundantthantwo-dimensional‘sheets’.Thefilamentarityandpancakenessofstructuresgrowswithtime,leadingtothedevelopmentofalongcoherencelengthscaleinsimulations(Sathyaprakash,Sahni,&Shandarin1996).Exploringthe‘connectedness’oflargescalestructuresemi-analytically,Bondandcol-laboratorsrecentlyconcludedthatclustersandsuperclustersappeartobeinterwovenina‘cosmicweb’,withsuperclustersactingascluster-cluster‘bridges’.Morepronouncedbridgesarelikelytoformbetweenclustersof

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galaxieswhicharealignedandclosetogether(Bond,Kofman&Pogosyan1996).

Thesupercluster-voidmorphologyislikelytovaryfordifferentsce-nario’sofstructureformation,itisunlikelythatstructureformationmodelsbasedongravitationalinstabilitywillhaveidenticaltopologicalpropertiesasthosebasedonstring/texturemodelsorexplosions.Wethereforefeelthatastudyofshapesofsuperclusters/voidscouldhelpdistinguishbetweendif-ferentalternativesoncegalaxycataloguesbecomefullythreedimensionalandadetailedcomparisonbetweentheoryandobservationsbecomespos-sible.ForthcomingredshiftsurveyssuchasSDSSand2dFpromisetoshedmorelightonissuessuchaswhetherthe‘greatwalls’appearinginNorthernandSouthernskysurveysareplanarobjectsoraremorelikefilamentsor‘ribbons’.

Mostshapestatisticsproposedsofarstudytheshapeofacollectionofpointsbymeasuringitsmomentofinertiatensor,aprocedurewhichisquitesimilartofittingbyanellipsoid.Althoughthismethodhasyieldedsomeinterestingresultsitisfairtosaythatnoneofthestatisticsappliedtoshapesisentirelysatisfactory(Sathyaprakash,Sahni,Shandarin&Fisher1997).Toillustratethisconsidertwoexamples:(1)theshapeofanemptycupasdeterminedfromitsmomentofinertiatensorisapproximatelyel-lipsoidalwhereasthecupisreallyacurvedtwo-dimensionalobject.(TheZeldovichapproximationinfactsuggeststhatthecausticsurfacesofthefirstpancake-likesingularitiesaremorelikelytobecurvedthanflat.)(2)Atorushasbothplanarandfilamentaryproperties;fittingwithanellipsoidwouldsuggestanoblateshape,whereasa‘thin’torusisclearlymorelikeacurvedfilament.

ResultsofN-bodysimulationsclearlydemonstratethatshapesofiso-densitysurfacesvarywidelywhenviewedatdifferentdensitythresholds.Athighthresholdsdensitypeaksaremostlyspheroidal,whereasatclosertopercolationthresholds,surfacesgetrather‘spongy’withacomplicatedtopology.

Toassesstheshapesofobjectswhichmaybetopologicallynon-trivial,wehaverecentlyproposedashapestatisticbasedonthefourMinkowskifunctionalsofacompactsurface:(i)itsVolumeV,(ii)surfaceareaS,(iii)integratedmeancurvature:C=1

H2+H1,

K2=

H3−H2

PERCOLATION,GENUSANDSHAPESTATISTICS.7

(1)pancakewithvanishingthickness:H3≃H2>>H1andK≃(1,0),(2)filamentwithinfinitesimaldiameter:H3>>H2≃H1andK≃(0,1),

(3)sphere:H3≃H2≃H1andK≃(0,0),(4)ribbon:H3>>H2>>H1andK≃(1,1).

Realisticsurfaceswillberepresentedaspointsona‘shapeplane’(K1,K2),withidealpancakes,filaments,ribbonsandspheresdefiningitsfourver-tices:(1,0),(0,1),(1,1),(0,0).

Todemonstratetheeffectivenessoftheshapestatisticweapplyittotwosurfaces–anellipsoidandatorus.Thesurfaceofthetriaxialellipsoidhastheparametricform

r=a(sinθcosφ)ˆx+b(sinθsinφ)ˆy+c(cosθ)ˆz

where0≤φ≤2π,0≤θ≤π.

Intable1weshowresultsfordeformationsofthisellipsoid.

(1)

TABLE1.Deformationsofatriaxialellipsoidwithaxisa,b,c.TheshapefunctionsV/S,S/C,Chavedimensionsoflength,(K1,K2)aredimensionless.

(100,100,(100,80,(100,50,(100,20,(100,10,(100,3,

100)80)50)20)10)3)

sphere→filament

100.0085.4558.5125.0412.673.82

100.0086.1261.9229.2215.324.70

100.0086.9769.01.6851.5050.19

(0,0)

(3.9×10,4.9×10−3)(2.8×10−2,5.4×10−2)(7.7×10−2,0.30)(9.5×10−2,0.)(0.10,0.83)

−3

(100,100,3)(100,70,3)(100,30,3)(100,10,3)(100,3,3)

pancake→filament

5.985.975.905.473.8263.8852.2727.8110.794.7078.5767.32.8850.8850.19(0.83,(0.80,(0.65,(0.33,(0.10,0.10)0.13)0.33)0.65)0.83)

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wherea,cTABLE2.Shapefunctionsforanellipticaltoruswithaxisb,a,c,(b>a,c).(100,99,3)(100,3,99)(100,3,3)(150,20,2)(150,2,20)(20,19,19)

Pancake1Pancake2FilamentRibbon1Ribbon2Sphere-with-hole

(0.90,2.9×10−2)(0.88,0.20)(0.14,0.93)(0.70,0.80)(0.70,0.80)(0.14,−0.09)

7.057.0..4.28.5

136.114.666.025.8825.8738.0

144.94173.03157.08235.56235.6531.42

PERCOLATION,GENUSANDSHAPESTATISTICS.9

galaxycatalogsandMBRmaps.Thesmallnessofthefillingfactorinclus-ters&superclustersatpercolationindicatesthatabulkofthematterislikelytolieinfilamentsandpancakes.Ananalysisof‘shapes’inN-bodysimulationsshowsthatfilamentsgrowmorepronouncedasthesimulationevolvesandaremoreprominentforspectrawithgreaterlargescalepower.Toprobe‘shapes’morerigorouslyweintroduceanewshape-statisticwhichstudiesshapesofcompactsurfaces(iso-densitysurfacesingalaxysurveysorN-bodysimulations)withoutfittingthemtoellipsoidalconfigurationsasdoneearlier.Thenewshape-indicatorsarisefromsimple,geometricalconsiderationsandarederivedfromfundamentalpropertiesofacompactbodysuchasitsvolumeV,surfaceareaS,integratedmeancurvatureC,andGenus.Thenewshapestatisticscanbeappliedtotopologicallysimpleandcomplicatedsurfacesandappearstobequiterobust.

Acknowledgments:TheworkreportedherewasdoneincollaborationwithB.S.SathyaprakashandSergeiShandarintowhomIexpressmygrat-itudeandthanksformanyyearsoffruitfulinteraction.IalsoacknowledgestimulatingdiscussionswithSanjeevDhurandharandSomakRaychaud-hury.References

Barrow,J.D.,Bhavsar,S.P.&Sonoda,D.H.1985MNRAS,21617.Bond,J.R.,Kofman,L.&Pogosyan,D.1996,Nature,380,603.deLapparent,V.,Geller,M.J.&Huchra,J.P.1991,ApJ,369,273DekelA.&WestM.J.1985,ApJ,288,411.

DominikK.&ShandarinS.F.1992,ApJ,393,450

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Hamilton,A.J.S.,Gott,J.R.&Weinberg,D.H.,1986,ApJ,309,1.KlypinA.A.&ShandarinS.F.1993,ApJ,413,48Matsubara,T.1994,ApJ,434,L43

Melott,A.L.&Shandarin,S.F.1993,ApJ,410,469

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SahniV.,SathyaprakashB.S.&ShandarinS.F.1997a,ApJ476,L1-L5.SahniV.,SathyaprakashB.S.&Shandarin,S.F.1998,ApJ,495,L5.SathyaprakashB.S.,SahniV.&ShandarinS.F.1996,ApJ,462,L5-L8.

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ZeldovichYa.B.1982,SovietAstron.Lett.,8,102

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0.15

0.80.60.40.200.80.60.40.200.80.60.40.20-1

0

1

2

0.10.05

0-0.050.150.10.050-0.050.150.10.05

0-0.05

-1

0

1

2

0.15

0.80.60.40.200.80.60.40.200.80.60.40.200

0.1

0.2

0.3

0.4

0.5

0.10.05

0-0.050.150.10.050-0.050.150.10.05

0-0.05

0

0.1

0.2

0.3

0.4

0.5

Figure1.

Percolation(leftpanels)andgenus(rightpanels)curvesareshownas

functionsofthedensitycontrastδ(above)andfillingfactor(below)forascalefreeinitialspectrumn=−2.InplotsshowingPC(leftpanels),solid(dashed)curvescorrespondtothevolumefractioninthelargestcluster(void)–vmax.Verticalsolid(dashed)linesshowthethresholddescribingpercolationbetweenoppositefacesofthecubeforclusters(voids).ThethindottedlineinthelowerlefthandpanelshowsthefillingfactoratpercolationforaGaussianrandomfieldwiththesamepowerspectrumasofevolveddensityfields(fordetailsseeSahni,Sathyaprakash&Shandarin1997a).

PERCOLATION,GENUSANDSHAPESTATISTICS.11

Pancake1

Pancake2

Ribbon1

Ribbon2

Sphere-with-hole

Filament

Figure2.Deformationsofanellipticaltorus.