198 8ApJ. . .326. .539B -1 _1 1 1 1- _1 1 (Ling, Frenk,andBarrow 1987;N.Kaiser,private out to30/iMpc;onlarger scales thedataarelesssignificant r >100/zMpc.(Herehis the Hubbleparameterinunitsof clusters ofrichnessR>1,forwhichd&55h~Mpc, is approximated byequation(1)betweenr%7h~Mpc and the givenclass.Forexample,correlationfunctionofAbell where disthemeanseparationbetweenneighboringclusters of (Bahcall andSoneira1983;KlypinKopylov1983) cluster correlationfunctionwhichcanbeapproximated by . Itischaracterized,foreachrichnessclass,byacluster- 100 kms"Mpc.)Thesignal issignificantlypositiveatleast as theneutrinoscenario). scenario whichisbasedonthe“standard”assumptions,such nontrivial difficultyforthisCDMscenario(andanyother larger scales,perhapsupto~100/zMpc,whichposesa But thereisrecentevidenceforsignificantstructureoneven the “voids”ofafewtensmegaparsecs(Whiteetal1987). mation itcanalsoreproducetheclusteringofgalaxieson scales upto~10/z“Mpc(Davisetal1985)andperhapseven hereafter BFPR;Frenketal1985;Blumenthal1986; Dekel andSilk1986).Withappropriate“biased”galaxyfor- initial fluctuations,isverysuccessfulinexplainingthemajor observed propertiesofgalaxies(e.g.,Blumenthaletal1984, tions ofcriticaldensity(Q=1)andGaussian,scale-invariant sort ofcolddarkmatter(CDM),withthe“standard”assump- 1 The AstrophysicalJournal,326:539-550,1988March15 © 1988.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. LickObservatoryBulletin,No.1084. One crucialobservationisthesuperclusteringofclusters of The cosmologywheretheuniverseisdominatedbysome 1 correlation functionandmeanstreamingvelocitytobecompatiblewiththeobservations. Subject headings:—darkmatterelementaryparticlesgalaxies:clustering linear analysisandsimulationsbasedontheZeldovichapproximation,wefindresultantcluster-cluster model whereCDMandbaryonscontributecomparablytothemeanmassdensity.Inthisis trum offluctuationsatthepre-recombinationJeansmass,givingrisetostructureonverylargescales.Using responsible forgalaxyformation,whilethepresenceofbaryonsinducesalarge-scalefeatureinspec- h Mpc.Assumingthattheseobservationsarebeinginterpretedcorrectly,weconsideranopenFriedmann erate scales,failstoaccountfortheobservedclusteringofclustersandstreamingvelocitiesonscales-30-100 fluctuations, whichissosuccessfulinexplainingthemajorpropertiesofgalaxiesandtheirclusteringonmod- © American Astronomical Society • Provided by the NASA Data System VERY LARGESCALESTRUCTUREINANOPENCOSMOLOGYOFCOLDDARKMATTER The ‘standard”cosmologyofcolddarkmatter(CDM)inaflatuniversewithGaussian,scale-invariant 18 Ur)^(2r/dy- ,(1) I. INTRODUCTION Santa CruzInstituteofParticlePhysics,BoardStudiesinUniversityCaliforniaat Lick Obsefvatory,BoardofStudiesinAstronomyandAstrophysics,UniversityCaliforniaatSantaCruz galaxies: formation Racah InstituteofPhysics,TheHebrewUniversity Received 1987May18;acceptedAugust28 George R.Blumenthal 1 AND BARYONS Joel R.Primack Avishai Dekel ABSTRACT 539 AND -1 -1 2 studied inmoredetail,seems tobedominatedbyaspherical and Robertson1986).Thevelocity fieldinthisregion,when background frameofreference. Relatedresultshavealsobeen obtained byotherobservers(Rubin etal1976;Collins,Joseph, velocity ofV=599±104km srelativetothemicrowave and independentlyobtained distances,yieldsastreaming (Burstein etal1986;Dressier1987).Thebestfit of a uniform bulkmotiontothedata,whichconsistsofredshifts spherical region~60/zMpcindiameteraround us the bulkmotioninferredforellipticalgalaxiesandclusters in a tions.) the clusterrichness—alsoindisagreementwithobserva- by lightneutrinos,¿wasfoundnumerically[Barneset al with theobservedresult.(Incaseofauniversedominated Thus, equation(2)impliesanegativeÇforr>,inconflict required biasitwouldgonegativeatanevensmallerscale. r =18h~Mpc(BondandEfstathiou1984),with the “standard” nonbiasedCDM^isexpectedtogonegativeat which becomesasimplelinearproportionfor^(r)1.In (Kaiser 1984;PolitzerandWiseseeJensenSzalay 1985] nottobemuchlargerthanÇandindependent of 1986 andBardeenetalforamoredetaileddiscussion) matter, ^(r),by imately relatedtothetwo-pointcorrelationfunctionof cc c being smoothedontherelevantscale.Then£isapprox- c c threshold vainthelocallyGaussiandensityfluctuationsafter axies andtheclustersareassumedtoarisefrompeaksabovea goes negative.InthestandardbiasedCDMpicturegal- communication), butthereisnoevidencethatthefunction cc The otherpuzzlingfindingofpotentiallygreatimportance is r2 èJ) =exp[(v/0.05, have alsocalledattentiontothispossibilityinearlierpub- ute comparableamountsofthematterinuniverse.(We hybrid modelinwhichcolddarkmatterandbaryonscontrib- this paper,wedirectspecialattentiontothebaryon-photon generate suchlarge-scalepowerwhileretainingthedesirable (eq. [3]),whichisusedfornormalization,muchlessaffected Fluctuations onsmallerscalesareJeansstableandcannot by thepoweronlargescalesthanVintegral(eq.[4]). velocity. Notethat,becauseofthekterm,ÔM/Mintegral coherence length,are“superbiased”intoformingin expected tohavealargeamplitudequalitativelysimilarthe on largescalesproduceshighervaluesforthebulkmotion “ superpancakes.”Andextrapowerinthefluctuationspectrum observed onebecausetheclusters,asaresultoflarge-scale eq bcdm 1980), tot 1986; Primack1986.)Ifbaryonsprovideasignificantfraction b cdm A dangerinconsidering a fluctuationspectrumwith Although itconflictswiththedogmaofcosmicinflationthat In ordertopreservethedesirablefeaturesofCDM It isnoteasytoimaginephysicalmechanismswhichmight 21 Àj^SOiQh)- Mpc.(6) 2 Dj, <0.035h~(7) Vol. 326 1988ApJ...326..539B -2 detail inAppendixA. Bexplainshowwedefine nique maybeofuseinother contexts,wedescribeitinsome technique nottousethefastFourier transform.Sincethetech- nificant weightoveralarge range ofk.Itisessentialinthis here, sincewearesimulating afluctuationspectrumwithsig- logarithmic intervalofwave numberk.Thiswasimportant using atechniquethatallowsonetoputequalweightineach difficulties withtheC+Bmodel,andconsideralternatives. sidered alargerclassofmodels. and Efstathiou(1987)whousedadifferenttechniquecon- have recentlybeenreachedbyBond(1987)andBardeen,Bond, account forthelarge-scaleobservations.Similarconclusions large-scale power,suchasariseintheC+Bmodel,canindeed support thehypothesisthatfluctuationspectrawithenhanced amplitude tocorrespondtheobserved¿.Theseresults both therightshape(~r)andapproximately rithm. Remarkably,thecluster-clustercorrelationfunction has are identifiedinourrealizationsusingaclusterfindingalgo- tatively similartothoseofthelinearcalculation.Richclusters chosen pointsintherealizationvolume.Theresultsarequali- lated withinspheresofvarioussizessurroundingrandomly C +Bfluctuationspectrum.Thestreamingvelocityiscalcu- applying theZeldovichapproximationtorealizationsof and theresultsoflinearcalculationbulkmotion. indeed accountfortheobservationsofenhancedclustering Section HIcontainsourmainresults,whichareobtainedby the physicaloriginoffluctuationspectruminmoredetail, rich clustersandoflarge-scalebulkmotion.In§II,wediscuss which modelssuchasC+Bwithextralargescalepowercan Ocdm =0.2andQ0.4. consider thelarge-scalestructureinaG+BmodelwithQ= on Q.Inordertoexploreparameterspacebetter,wealso synthesis calculationsthatleadtotherestrictionequation(7) the assumptionsunderlyingstandardbigbangnucleo- straints. Therehavebeenseveralrecentsuggestionstomodify approach todecreasethepotentialconflictwithAT/Tcon- Krauss 1984).Finally,increasingQisanotheralternative consistent withinflation(Peebles1984;Turner,Steigman,and curvature termandthusmakethemodelflattherefore would betoinvokeanonzerocosmologicalconstant(e.g.,Vit- c torio andSilk1985),whichcouldalsoremovethecosmological model duetotheearlyformationofsubgalacticobjectsfrom the CDMcomponentoffluctuations.Anotherwayout (1984) issomewhatoverstated.Thereareseveralotherpos- out byreionization,whichcouldnaturallyoccurinthishybrid sibilities. Thesmall-angleAT/Tfluctuationscanbewashed culty. Perhaps,asKaiserandLasenby(1987)havesuggested, the small-angleAT/TupperlimitofUsonandWilkinson proceed toconsidertheC+Bmodeldespitethispotentialdiffi- tot attitude, forthepurposesofpresentpaper,willbeto culty onsmallangles,discussedinmoredetail§IV.Our b No. 2,1988 Bond, privatecommunication).Butthereispossiblyadiffi- dicted anisotropiesat6°areclosetobutnotinexcessofthe ft angular scales(seealsoSilkandVittorio1987),thepre- model withtheisotropyofmicrowavebackgroundonlarge is anontrivialtask.NodifficultyexpectedinourC+B Melchiorri etal(1981)95%confidencelevelbound(J.R. nario thatcansatisfysimultaneouslytheoppositeconstraints constrain theamplitudeofspectrumfromaboveonvarious scales (seethereviewbyKaiserandSilk1987).Findingasce- The Zeldovichrealizationsdiscussedinthispaperweredone In §IVwediscussourresults.Wealsoattempttoassessthe Our mainobjectiveinthispaperistoexploretheextent © American Astronomical Society • Provided by the NASA Astrophysics Data System VERY LARGESCALESTRUCTUREINOPENCOSMOLOGY -1 14 _ 1 2 just discussed.ItisalsoenhancedovertheD=0.4C+Bspec- equation (4)inthelinearapproximation. Thisisplottedin to eachofthesefluctuation spectra canbecalculatedfrom results forexampleinarelatively latetimeofgalaxyformation. Figure 2asafunctionoftheradius Rofthesphericalwindow. However, acorollaryisthereduced small-scalepower,which scale powerintheQ=0.2C+Bmodelisconsiderable. reduced byabiasfactor,theeffectiveenhancementoflarge- malization ofanQ=1CDMspectrumwilltypicallybefurther scales inmoreopenuniversesdiscussed§I.Sincethe nor- trum andtheQ=1Cspectrumbyshiftofpowertolarger over theQ=0.2colddarkmatterspectrum(C)byeffect large-scale poweroftheQ=0.2C+Bspectrumisenhanced ÀM/M versusR,assumingh=0.5.Oneobservesthat the spectrum. result istoboosttheamplitudeonlargescalesofC +B dard procedureandnormalizebothfluctuationspectrasothat ôM/M =1ontherelativelysmallscaleof8/zMpc, net tion amplitudeissmaller.However,sincewefollowthestan- contribute nogrowthintheC+Bmodel,sototalfluctua- essentially thesameamount.Butonsmallerscales,baryons primordial amplitude.Onscaleslargerthan2j,fluctuations that startatearlytimeswiththesameamplitudegrowby to apureCDMmodelwiththesameQandinitial enhanced large-scalepower,itisperhapsclearesttocompare scales beforerecombination. radiation-baryon fluidoscillateratherthangrowonthese catches upto<5.Inthismodel,thesefluctuationsare spectrum belowreflectthefactthatfluctuationsin responsible forforminggalaxiesandclusters.Wigglesinthe recombination, theamplitudeofbaryonicfluctuationsô than ~10MarecompletelyerasedbySilkdamping.After smaller than2j,andbaryonicfluctuationsonscales inhibited byComptondraguntilrecombinationonscales portional toaafterzanduntilthemeandensityofuni- crossing andmatterdominationatz,withgrowthpro- density. Growthoffluctuationsinthebaryoniccomponentis verse deviatessignificantlyfromtheinstantaneouscritical in theusualcolddarkmattermodel(see,e.g.,BFPR). a functionofscalefactor=(1+z)occursbetweenhorizon with smallerpeakssuperposedonthesmoothspectrumat Logarithmic growthinthecolddarkmatteramplitudeôas smaller scales.Thesmoothspectrumonscalesarisesas large peakatthescaleofbaryon-photonJeansmass2j, ate dampingtime. erasing theradiationplusbaryonfluctuationsatappropri- matter, andthreespeciesofmasslessneutrinosaretreatedas treated tosufficientaccuracybyexpandinginmoments. separate components.Freestreamingoftheneutrinosis cold darkmatterplusbaryon(C+B)modelwascalculated tot Photon diffusionorSilkdampingisputin“byhand,”by using acodeinwhichtheradiationandbaryons,colddark batic Gaussianfluctuations.Thefluctuationspectrumforthe Harrison-Zeldovich scalefree(\S\&k)spectrumofadia- clusters forthepurposeofcalculatingtheircorrelationfunc- cdm tion. b 0 eq eq cdm k The root-mean-squarebulkmotion velocitycorresponding This isillustratedinFigure1,whichplotsthenormalized In ordertounderstandhowtheC+Bspectrumgets The characteristicfeatureofthisfluctuationspectrumisa We startfromtheusualCDMassumptionsofaprimordial II. LINEARCALCULATIONS 541 198 8ApJ. . .326. .539B 3 -1 3-1 _1 zations usingtheproceduredescribedinAppendixA.We use 542 (the Nyquistfrequency). large rangeofwavenumbersfrom1.2x10“to0.3hMpc This procedureenablesustorepresentthespectrumover a N =32tc/6particlesinasphereofradiusR160/zMpc. had verylittleeffect.Thusthe peaksandtheunderlyingmass the “galaxies”asdefinedin Appendix B,withoutapplyinga culation.) Thepositionandpeculiar velocityofeach“” distribution haveessentiallythe samebulkmotioninthiscal- to thewayweselect“galaxies”; imposingdifferent<5values <5 densitythreshold.(Theresults werefoundtobeinsensitive in galaxyandclusterhalos,thusthevisiblegalaxiesmay motion whichismuchsmallerthanthatclaimedintherecent biasing factorofatleast2(Davisetal1985;seeareviewby tion spectrumandhencethevelocityshouldbeloweredbya unreasonable fortheQ=1CDMmodel,whichfluctua- trace themassfairlywell;i.e.,thereisnobiasing.Itprobably as inFigure1.ThisisreasonablefortheQ=0.2C+Bmodel, In eachcase,thespectrumhasbeennormalizedaccordingto C +Bmodelcomesclose.ThismotivatedtheZeldovich Dekel andRees1987).Thus,biasedCDMpredictsabulk since inthatcasethetotalmassdensityiscomparableto approximation calculationsdiscussedinthenextsection. observations. TheDressieretal(1987)valueisshownon Figure 2,forcomparison.ButitisinterestingthattheQ=0.2 curves arenormalizedsothatÔM/M=latR8/iMpc;thustheyall“unbiased.” labeled C),andQ=0.20.4withequalmatterdensityincolddarkbaryons(C+B).TheHubbleconstanthasbeenassumed to beh=0.5.The min min We calculatebulkvelocitiesusingallofthepoints,i.e., of For givenspectraoffluctuationswegenerateparticlereali- Fig. 1.—FluctuationamplitudeÖM/Mforfourmodels:fí=1andQ0.2with90%ofthematterdensityincolddark10%baryons (theseare III. VERYLARGESCALESTRUCTUREINZELDOVICH © American Astronomical Society • Provided by the NASA Astrophysics Data System 1 ÖM/M =1at8/i"Mpc(8) REALIZATIONS BLUMENTHAL, DEKEL,ANDPRIMACK the scaleR.Incaseswherethereissignificantpoweronscales wavelengths totheintegralinequation(4).ComparingFigures mass, Vandthedispersionaboutit.Wethencalculaterms we calculatethethree-dimensionalbulkmotionofcenter is givenbytheZeldovichapproximation(AppendixA).We in theprevioussection.ThelinearcalculationofÔM/Mnear but aresystematicallyhigher,~10%higheronthesmaller values andasimilartrendwithRasthelinearapproximation, 2 and3,itisseenthattheZeldovichapproximationresults all scaleswithinR,reflectingthestrongcontributionoflarge larger thanRthebulkvelocityandscatterremainhigh on realization whichiscomparabletothetypicalbulkvelocityon approximation resultsshowabigscatterfromrealizationto realization. TheresultsarepresentedinFigures3and4.Figure value ofVoverallwindowsthesamesizewithingiven the realizationexcluding0.05Routershell.Ineachvolume various radiisuchthattheyareencompassedbythevolumeof linear correctionusingZeldovich realizations(e.g.,Hoffman scales increasingto~20%higheronthelargestscaleplotted. averaged overtherealizationshaveapproximatelysame over severalrealizations.AsFigure4shows,theZeldovich tion andconsideringconcentricspheresaboutthesecentersof then apply“top-hat”windowsinposition-space,byselecting definiteness wehavenormalized allthefluctuationspectra appears thattherenormalization factorisonly~15%.For overestimated bulkvelocity.One maytrytoestimatethenon- ôM/M «1maybeanunderestimate.Thiswouldresultin an 3 summarizestheresults;eachcurverepresentsanaverage But fromtheAT-bodysimulations ofDavisetal(1985),it Mpc, whereÔM/Miscalculatedinthelinearapproximation as 100 centersatrandominsidethe0.825Rradiusofeachrealiza- 1988), butsuchanestimatesuffers fromsevereuncertainties. 1 We havenormalizedthevelocitiesusingÖM/M=1at8~ 198 8ApJ. . .326. .539B next figurerepresentsthemeasurement reportedbyBursteinetal.(1986)andDressier(1987). approximation realizationsasexplained inthetext.Errorbarsrepresentstandarddeviationofrmsvelocity ofseveralrealizations.Thestarinthisandthe Fig. 3—Root-mean-squarebulkvelocity forvariousmodelsasafunctionoftheradiusRspherical windowfunction,calculatedfromZeldovich p 2.Root-mean-squarebulkvelocityforvariousmodelsasafunctionoftheradiusRsphericalwindowfunction,calculatedusingeq.(4) IG © American Astronomical Society • Provided by the NASA Astrophysics Data System £ > £ 1000 800 600 400 200 0 0 4080120 160 200240280320 1 r 1 JT[I< i I»1^i—I^^—i—I—^—i—i—I—i—^—i—I—i—i—i— n=ic R (Mpc) 198 8ApJ. . .326. .539B 1 544 61-3 when comparedtotheobservations(eq.[1]),whileinallthree nonlinear renormalizationfactor.Itshouldbekeptinmind according toequation(8)using(3)andignoringthis considering thefairlylargescatterbetweendifferentsimula- because ofthepossibilitybiasing.Recallthatmodel that thereisinanycasesomeuncertaintythenormalization different realizations. (Schectman 1985)andofsuperclusters(BahcallBurgett regarded asconsistentwithobservations. observed correlationfunction,allthreeopenmodelsshould be tudes inthehybridC+Bmodelsarealittletoohigh. But open modelsthetwoagreeverywell.Ifanything,ampli- which correspondston/n«200andiV=3.Theresulting each casethemeanseparation oftheobjectsconsidered.We tions illustratedinFigure6,andtheuncertainty the cluster-cluster correlationfunctionisplottedasaof ÔM/M smallerbyafactor~2.5(Davisetal.1985),sothe CDM withQ=1requires“biasedgalaxyformation” calculations. Figure7shows thecorrelationfunctionfor the predicted£istoolowinamplitudebyafactorof ~3 Ir/d inFigure5.IntheCDMcasewithQ=1andnobiasing, predicted bulkvelocityshouldbesmallerbythesamefactor.In attempted toseeifasimilar scalingrelationobtainsinour and theadoptednormalizationisappropriate. the modelswithQ~0.2,however,thereisnoneedforbiasing, “ smallclusters,”definedsothat theirnumberdensityishalf that ofAbell’srichness>1 (i.e.,fortheseobjects,d=AAh~ number densitytoAbellclustersofrichness>1,i.e.,n=6_ <5 =1todefine“galaxies.”Toobtainclustersofcomparable similarity oftheresultsinFigure 7tothoseinFigure5suggests Mpc). Intheclusterfinderfor thiscaseweusediV=2.The 1986) canalsobeapproximatedbyequation(1),wheredis in min c A1 min min x 10~(/TMpc)(BahcallandSoneira1983),weused/d p 4BulkvelocityfromseveralrealizationsoftheQ=0.2C+Bmodel.HeavylineanderrorbarsaresameasonFig.3.Notelargespread between The observedcorrelationfunctionsoflessrichclusters Clusters areidentifiedasdescribedinAppendixB.Weuse IG © American Astronomical Society • Provided by the NASA Astrophysics Data System £ 1000 400 800 600 200 _l IL1— -| 1[IT" 40 BLUMENTHAL, DEKEL,ANDPRIMACK 80 120 160200240280 R (Mpc) 2 I ■ Zeldovich approximationdoesindeedscalewithr/d. that thecorrelationfunctionofclusterscalculatedfrom linear Gaussiannoise.Oneofthemodelstheyconsider(their thiou (1987)havestudiedalargevarietyofmodelsusingthe “BCO”) isthesameasourQ=0.2C+B.Theagreement statistical formalismofBardeenetal.(1986)andassuming between theirresultsfor^andoursisgood. that thegalaxy-galaxyandcluster-clustercorrelationfunctions is notatrivialconsequenceofmodellikeC+B,sincethey with timeasaresultoftheactiongravity.Indeed,thisfact arise quitedifferently.Themattertwo-pointcorrelationfunc- can beusedtoidentifythepresentepochinAT-bodycalcu- tion isinitiallyratherflatatearlytimes,butitsteepensrapidly £(r) and¿observationallyhavesimilarlogarithmicslopes mass oratleast£(r)oc£(r).However,the~r~slopeofÇ in lations (e.g.,Davisetal.1985),assumingthatgalaxiestrace but insteadreflectstheinitialfluctuations.Itisthusinteresting gravity, whichisnotincludedintheZeldovichapproximation, right inthemodelsconsidered. and perhapssignificantthattheslopeof£comesoutroughly hybrid opencolddarkmatterplusbaryons(C+B)model can Figure 6doesnotrepresenttheresultofnonlinearaction of c models withequalQand , itisclearthatsimilarresults indicated bytherecentobservations. Althoughweconsidered velocity Vandtheenhanced cluster-clustercorrelationsÇ reproduce withhighprobabilitythelarge-scalestreaming will obtainifthesequantities areonlyroughlyequal.More ian, scale-invariant(Harrison-Zeldovich) fluctuationspectrum generally, thesecalculationssuggest thatanadiabatic,Gauss- gc g c c bcdm c In paralleltoourcalculations,Bardeen,Bond,andEfsta- One issuethatseemsworthcommentingonisthefact The resultspresentedinthelastsectionareevidencethat I ■i IV. DISCUSSION 320 198 8ApJ. . .326. .539B -1 approximate fittotheobservationaldata,eq.(1). where d=55/iMpcisthemeanseparationbetweenclusters.Errorbarsarestandarddeviationofseveralrealizations.Heavy solidlineisan realizations fortheothermodelsstudied. Fig. _5.—Cluster-clustercorrelationfunctionsforclustersequivalenttoAbellR>lselectedfromZeldovichrealizationsfourmodels,plotted vs.x=2r/d, Fig. 6.—Cluster-clustercorrelationfunctions asinFig.5,fromeightrealizationsoftheÍ2=0.2C+Bmodel. Therewasasimilarspreadamongtheindividual © American Astronomical Society .1 1 .1 1 Provided bythe NASA Astrophysics Data System x x 198 8ApJ. . .326. .539B 5 calculated tobe8x10“for h=0.75,andtheUson even thelocalIMFisnotreallyunderstood,aprimordial mass functionfortheseobjectsisnotasimpleextrapolation of Wilkinson (1984)95%confidence levelfieldupperlimitof the 4Í5scale,whichBardeen, Bond, andEfstathiou(1987)have between thepredictionof Q =0.2C+BmodelforÔT/Ton peaked eitheratsmallorlarge massesisnotatallimpossible. either case,theopenquestionistounderstandwhyinitial forms thatthesebaryonsmighttakewithoutrunningafoul of matter. (Orelse,asmallerfractionofthebaryonicmaterial model likeC+Bmustsomehowbeconvertedintodark (stars andgas)ingalaxiesclusters,mostofthebaryonsa model onthescaleofgalaxiesandclusters. calculations, itwillbeimportanttoverifythatthemodelalso more elaboratecalculations,presumablywithAf-bodycodes. that ofthevisiblestarsinourvicinity.Butsinceorigin of “brown dwarfs”;seeDekelandShaham1979)compact retains thedesirablefeaturesoforiginalcolddarkmatter can possiblyleadtoverylargescalestructureconsistentwith Bond, andArnett1984)orofordinarystars(Larson1987). In objects arisingastheremnantsofverymassivestars(Carr, standard limitsaresmall,unevolvedstars(i.e.,“Jupiters” or galaxies andclusters,whichseemsunlikely.)Thetwoknown than thecolddarkmatterparticipatesinformation of of magnitudemoredarkmatterthanvisiblebaryonic In additiontocheckingsuchquantitiesasVandÇin than definitive;inparticular,theysuggestthedesirabilityof approximation (GrinsteinandWise1987),assuggestiverather tuation spectrum. observations ifthereisenhancedlarge-scalepowerinthefluc- plotted vs.x—Ir/d. 546 c 1 It isnotclearhowseriously totaketheapparentconflict We regardourresultswhicharebasedontheZeldovich Since observationsindicatethatthereisroughlyoneorder Fig. 7.—Cluster-clustercorrelationfunctionsfor“smallclusters”withmeanseparationd=44h~MpcselectedfromZeldovichrealizations fourmodels, © American Astronomical Society • Provided by the NASA Astrophysics Data System .1 i BLUMENTHAL, DEKEL,ANDPRIMACK 1 5 5 ={C(1+QZi)3/21] -3(1_n)C(1+ßoZi)1/2 ]} is theThomsoncrosssection,and where Q =0.2,thegrowthfactorfor linearfluctuationsfromz=34 Zi andremainedionizedthereafter.Theopticaldepthisthen enough towashoutthesmall-anglefluctuationsincosmic nation, 3M/M=8.5(withÔM/M =1at8/i“Mpc).For ground radiation,reionization musthaveoccurredbefore» washing outsmall-anglefluctuations inthecosmicback- background radiation. The simplestistosupposethatreionizationoccursearly M ä10,justabovethe Jeansmassrightafterrecombi- as significant,thereareatleastthreepossibleexplanations. quoted upperlimitimplies.If,however,theconflictisregarded that theUsonandWilkinsondataarenotasrestrictivetheir 2.9 x10“.KaiserandLasenby(1987)haverecentlyshown 34 fortheC+BmodelwithQ =Q0.1andh0.5.For Setting t(z)equaltounity,itfollowsthat,beeffective in X 0 bcdm ¿ ih °-~ • 0 p(1+zfdz = Suppose thattheuniversewasentirelyreionizedataredshift 21 Jo (1+z)(l+Qz) 0 1(2,.)= ondl=—I0.072Qh,(9) Teb Q0 Jo **0 ri() Zic 242 10 s)mayincreasethedeuteriumabundanceenough(via annihilation intoenergeticnucleon-antinucleonpairswhich fraction oftheCDMparticleswhichannihilateatlatetimes 1988) .TheLiconstraintisapparentlyalsorelaxed,sotheonly sists ofweaklyinteractingparticlesmass~3GeV,thesmall cdm Applegate andHogan1985;Applegate,Hogan,Scheerer bfccdm nucleosynthesis constraints(Audouze,Lindley,andSilk1985; mena havebeenstudiedrecentlywhichcouldmodifythe synthesis constraints,ofcourse;butseveralphysicalpheno- cold andbaryonicmatterisnotinseriousconflictwiththe 1987) .Inparticular,assumingthatthecolddarkmattercon- observed structureeither. b 1987; Alcock,Fuller,andMatthewsDimopoulosetal. Presumably, anQ=1modelwithcomparableamountsof angle ÔT/T.Aswehaveseenintheprevioussection,an There isalsoageometriceffectinthesamedirectionforsmall- recombination, andhencethepredictionforÔT/T,issmaller. from recombinationtothepresentepoch,soamplitudeat either case,thereismoregrowthofthefluctuationamplitude increase Qabove0.2ortoaddacosmologicalconstant.In for example,itignoresthepossibleeffectsofreionization on thevelocitiesofgasparticles. more fluctuationsthaniterases,especiallyinanopenuniverse. But thisanalysisshouldnotberegardedasconclusivebecause, has recentlyarguedthatreionizationcanactuallyintroduce help resolvethesmall-anglebT¡Tproblem.Vishniac(1987) reionization: itisfairlymarginalbutnotimpossible. ated inaresultingearlygenerationofstarstocause lapsed bythistime,butenoughenergymayhavebeenliber- Only positivefluctuationsstrongerthan~3crwillhavecol- to z=0is13.2;thus1afluctuationshadamplitude0.65at. No. 2,1988 f We wanttorepresentarandom-phase realizationofagivenpower-spectrumsmalldensity fluctuations, There isnoastrophysicalevidencetoprecludeadding a Such modelsareinconflictwiththestandardnucleo- = 0.4C+Bmodelproducesamplelarge-scalestructure. The othertwomodificationstothemodelareeither It isnotatallclear,however,thatreionizationcanactually © American Astronomical Society • Provided by the NASA Astrophysics Data System ZELDOVICH REALIZATION OFGAUSSIANFLUCTUATIONS VERY LARGESCALESTRUCTUREINOPENCOSMOLOGY P(k) =<|^)|%, (Al) |=fc APPENDIX A -1 niac, andSimonWhite. Hoffman, NickKaiser,DavidKoo,Seckel,EthanVish- or correspondencewithDickBond,SandraFaber,Yehuda and financialsupport.Wegratefullyacknowledgediscussions and A.D.wouldliketothanktheLickObservatory the would liketothankTheHebrewUniversityforitshospitality, Santa CruzInstituteforParticlePhysicstheirhospitality by afacultyresearchgrantatUCSC.G.R.B.andJ. P. ation areonthevergeofdetection. natively, perhapsfluctuationsinthecosmicbackgroundradi- with thisC+Bscheme,buttherearepotentialsolutions.Alter- ble tothoseobserved.Theremaybe<577Tproblemsassociated streaming velocitiesandcluster-clustercorrelationscompara- baryons inanopencosmology.Thisresultslarge-scale ing thefluctuationspectrumonlargescales;namely,addi- tion tothecolddarkmatterofacomparabledensityin We haveconsideredaplausiblephysicalmechanismforboost- evidence forverylargescalestructurethatwehavediscussed. mistic orpessimisticaboutcolddarkmatterinthelightof communication). and A.SzalaysuggestalowervalueforQ(D.Koo,private function ofredshiftLohandSpillar(1986).Althoughthese ment thatbearsonthevalueofQisgalaxycountsasa Other recentgalaxycountmeasurementsbyD.Koo,R.Kron, observations suggestthatQ>0.4,theyneedtobeconfirmed. Í2 ä0.2(Peebles1986).Themostinterestingrecentmeasure- straints. Asiswellknown,themoststraightforwardinterpreta- itself enhancesthelarge-scalepowerand,incontextof tion oftheobservationaldataongalaxiesandclusterssuggests C +Bmodel,isrequiredbytheusualnucleosynthesiscon- matter. lates comparabledensitiesoftwotypesnonbaryonicdark scales. Butthismodelisindeedanadhochybridwhichpostu- have enhancedpowerinthefluctuationspectrumonverylarge probably littleeffectfromtheCDM.Thusthismodelwillalso ing lengthÀ=120(m/10eV)Mpcissolargethatthere and Szalay1983).Butwithverylightneutrinosthefreestream- tion spectrumassociatedwithneutrinofreestreaming(Bond metto 1985;BonomettoandValdarniniAchilli, Occhionero, andScaramella,1985)thefeatureinfluctua- cold darkmatterisknowntomodify(ValdarniniandBono- with Q«0.1(seeShafiandStecker1984).Thepresenceof flavors ofverylight(feweV)neutrinosinanopencosmology worth studyingistheadditiontoQ»0.1ofoneormore cation ofthefluctuationspectrum.Anothermodelthatmaybe Bardeen, Bond,andEfstathiou(1987),includingadhocmodifi- scales butaddlargescalepower.Manywereconsideredby here thatmightpreserveitsgoodbehaviorongalaxytocluster dard colddarkmattermodelbesidesthosewehaveconsidered v v cdm This workwassupportedinpartbyNSFgrant8415444and In conclusion,itisnotclearwhetheroneshouldbeopti- As wehavediscussed,consideringanopencosmologyboth Finally, thereareofcourseothermodificationsthestan- 547 198 8ApJ. . .326. .539B /2 2/3_1 (the Nyquistfrequency), random. Theiramplitudes,karechosenatrandomwithin(k,)suchthatthenumberdensityofwavesisw(/c). In practice,we phases ^arechosenuniformlyatrandomintheinterval(0,2n).Thedirectionsofwavevectors,k uniformlyat which, inthelinearapproximation,issimply radius Ras the powerspectrumfromit.A CDMspectrumwasreproducedtoanaccuracybetterthan 20%overtherange(k<2n/R,/c). tend togenerateaGaussiandistributionbasedonthecentral limittheorem. amplitudes werechosenfromaGaussiandistribution;therandom phasesandthelargenumberofwavesineverysmallInkinterval N ä1000pereachdecadeofk,indeedapproximatesa normaldistributionverywell.Itisnotdueonlytothefactthat select thekvaluesviaafunctionu(k)whichsatisfies The amplitudesP¡arechosenatrandomfromaGaussiandistributioninwhichthevarianceispowerspectrum, P(k).The The correspondingdensityfluctuationis representing onlythegrowingmodes.Withapproximation(A3)onehastoday each particleisassignedacorrespondingpeculiarvelocityrelativetotheHubbleflowof which correspondsto logarithmic intervalink,i.e., is forcedwhenperiodicboundaryconditionsareimposedandFouriertransformscalculated(Efstathiouetal1985). Butthen would giveauniformcoverageofthethree-dimensional/c-space.Thischoiceisequivalenttousecubicgridin /c-space, which used inthesuperposition. The valuesofu(k)arechosenuniformlyatrandomintheintervalu(k),) i 3 E w(k) = w(k)dk = 3 K dqj i= 1i dij/j N u k =1/3 b(t =t)bÙH.(A5) 2w/fe=2(K/JV). (A 12) 4rc(ln k-In) 0 max maxmin : b(t)¿cos(k¡-q+(p^l -a(t)b(t)y\r(q), (A4) -a(t)bm(g), (A2) u(k) =\nk,(A10) “(kmax) -«(kmin) i=l N k N k N k Jfcmin 0023 P(k)W(kR)dk, du(k). ( Vol. 326 (A 13) (All) (A8) (A9) (A7) (A6) 198 8ApJ. . .326. .539B 3 ea 1 .1987,inNearlyNormalGalaxies, ed.S.Faber(NewYork:Springer- Bond, J.R.,andEfstathiou,G.1984,Ap. J.(Letters),285,L45. Bond, J.R.1986,inGalaxyDistances andDeviationsfromUniversalExpansion, Blumenthal, G.R.,Faber,S.M.,Primack,J.andRees,M.1984,Nature, Audouze, J.,Lindley,D.,andSilk,J.1985,Ap.(Letters),293,L53. Applegate, J.H.,Hogan,C,andScheerer,R.1987,P/rys.Rev.D,35,1151. Applegate, J.H.,andHogan,C.1985,Phys.Rev.D,31,3037. Boesgaard, A.M.,andSteigman,G.1985, Ann.Rev.Astr.Ap.,23,319. Blumenthal, G.R.,Faber,S.M.,Flores,andPrimack,J.R.Í986,Ap. J., Barnes, J.,Dekel,A.,Efstathiou,G.,andFrenk,C.1985,Ap.295,368. Bardeen, J.,Bond,J.R.,Kaiser,N.,andSzalay,A.Í986,Ap.304,15. Bardeen, J.,Bond,J.R.,andEfstathiou,G.1987,Ap.321,28. Bahcall, N.,andSoneira,R.1983,Ap.J.,270,20. Bahcall, N.,andBurgett,W.1986,Ap.J.(Letters),300,L35. Alcock, C.A.,Fuller,G.M.,andMatthews,J.1987,preprint. Achilli, S.,Occhionero,F.,andScaramella,R.1985,Ap.J.,299,577. Bonometto, S.,andValdarnini,R.1985, Ap.J.(Letters),299,L71. Bond, J.R.,andSzalay,A.1983,Ap.J., 276,443. No. 2,1988 cluster members. comparable tothegivennumberdensityofclustersinterest.Thecenterclusterisdefinedascenter-of-mass ofthe clusters.” Theparameterdisthenchosenbytrialanderrorsuchthatthemeannumberdensityofresultant clustersis (Bl). Eachclustermemberisrequiredtobeseparatedfromitsnearestneighbor(whoalsoamember)byless thanacritical neighbors. WealsoapplyaminimumnumberrequirementN,whichwaschosentobe3forAbellclustersand 2for“small separation d.Theparameterdisrelatedtoadensitythresholdbyp/p=(d/d)~,wherethemean between where “filaments”intersect).Analternativenecessaryconditionforcandidatesgalaxiesthatmaybelongto rich clustersis filaments whereæ2>À(attheintersectionsofsheets),andcompactspheroidalobjects^ the“knots” corresponds toacollapsealongthelocalprincipalaxisj.Iffluctuationingivenregionisdominatedby wavelength, maximum ofÀy.thedensitythereincreasesasb(t)grows,approachinginfinityatsomecriticaltimeinwhich bXj=1.This therefore bÀ>À,withsomechosenpositivevalueforA. the criticalformationepoch,isdeterminedbyratiosofeigenvalues.FlatsheetswillbeformedwhereÀ> /l,elongated then acoherentstructureisformedwithdimensionscomparabletothatwavelength.Thegeometryofthestructures, atleastnear As anecessaryselectioncriterionforparticlesthatmayrepresent“galaxies”onecanuseöp/p><5,withsomechosen valuefor ^min (-g*>in'top-hat”model,thelinearvalues<5=1.06and1.7correspondtoturnaroundcollapse,respectively). In thelinearregime,b^<1,localdensityfluctuationcanbeapproximatedby assumption ofnorotation),witheigenvaluesÀ^q)definedsuchthat^>A/I3,thedensitycanbewrittenas where pisthemeandensity.Afterdeformationtensordiagonalizedlocally(thesymmetric underthe normalized suchthatonacertainscale,R, factor Iisduetothefactthatweusesine’sratherthanexponentialsinFourieranalysis.)ThefunctionsP(k)andb(t)are where W(kR)istheFouriertransformofwindowinposition-space,chosenheretobea“tophat”radiusR(eq.[5]).(The which isafunctionoftheLagrangianpositionqandtimei,givenby For example,ifthereisnobias,onewouldchooseb=1atR$h~Mpcbasedonthepresentdistributionofbrightgalaxies. described inAppendixA. min 231 3min 1 2 min min x 2 u 0u ed. B.F.MadoreandR.Tully(Dordrecht :Reidel),p.255. 311, 517(BFPR). 301,27. Verlag), p.388. In ordertoassign“galaxies”clustersweapplyaminimumpairseparationcriteriontheirpositionsasgiven by equation We canrequirefurtherthatthecollapseisthree-dimensionaltoacertainextent.Considerpointwhichlocal positive According totheZeldovichapproximation,localdensityatanEulerianposition We explainherehowwedefineclustersof“galaxies”inarealizationthematterdistributionwhichhasbeengeneratedas © American Astronomical Society • Provided by the NASA Astrophysics Data System CLUSTER FINDINGINTHEZELDOVICHAPPROXIMATION VERY LARGESCALESTRUCTUREINOPENCOSMOLOGY 1 p =p(l-b^y^lbly\lb^y. 2 det r(q,t) =a(t)lq-bm(q)'] <5p/p äb(Ài-f^2+^3)•(B4) 2 ({ÔM/M)y =b{t).(A14) Ru APPENDIX B REFERENCES .1984h,Ap.J.,284,445. Dimopoulos, S.,Esmailzadeh,R.,Hall, L.J.,andStarkman,G.D.1987,pre- Dekel, A.,andSilk,J.1986,Ap.J.,303,39. Dekel, A.,andShaham,J.1979,Astr.Ap.,74,186. Dekel, A,andRees,M.J.1987,Nature,326,455. Efstathiou, G.,Davis,M.,Frenk,C.S., andWhite,S.D.M.1985,Ap.J.Suppl, Efstathiou, G.,andBond,J.R.1987,M.N.R.A.S., 227,35. Dressier, A.,Faber,S.M.,Burstein, D., Davies,R.L.,Lynden-Bell,Terle- Dekel, A.1984a,in8thJohnsHopkinsWorkshoponCurrentProblems in Davis, M.,andPeebles,P.J.E.1983,Ap.J.,267,465. Davis, M.,Efstathiou,G.,Frenk,C.S.,andWhite,S.D.M.1985,Ap.J., 292, Collins, A.,Joseph,R.D.,andRobertson,N.A.1986,Nature,320,506. Carr, B.J,Bond,J.R,andArnett,W.D.1984,Ap.J.,277,445. Burstein, D.,Davies,R.,Dressier,A.,Faber,S.M.,Lynden-Bell,Terlevich, print. 57,241. vich, R.J.,andWegner,G.1987,Ap.J. (Letters),313,L37. World Scientific),p.191. Particle Theory,ed.G.DomokosandS.Koveski-Domokos,(Singapore: Expansion, ed.B.F.MadoreandR.Tully(Dordrecht:Reidel),p.123. 371. R., andWegner,G.1986,inGalaxyDistancesDeviationsfromUniversal (B3) (B2) (Bl) 549 198 8ApJ. . .326. .539B 5 Joel R.Primack:SantaCruzInstituteofParticlePhysics,BoardStudiesinUniversityCalifornia, SantaCruz, CA 95064 Avishai Dekel:RacahInstituteofPhysics,TheHebrewUniversity,Jerusalem91904,Israel indicates thatitiscapableofsubstantiallyreducingÔT/Tonascalefewarcminutes.Ifsuchreionizationoccurs, ourB+C Cruz, CA95064 George R.Blumenthal:LickObservatoryandBoardofStudiesinAstronomyAstrophysics,UniversityCalifornia, Santa model maybeconsistentwithobservationsevenwithoutacosmologicalconstant. ÍVQ0.3. RaisingQallows/ismallerand/or/Qlarger. limit isconsistentwithoutQ=0.2C+Bmodel,acosmologicalconstantsuchthattheuniverseflat,only if/i«1and private communication,1987December3).AccordingtorecentcalculationsbyJ.HoltzmanofUCSC(1988,inpreparation), this announced newupperlimitsonÔT/Tat7.15;thelatest(95%confidencelevel)boundisST/T<1.7x10"(A.C. S.Readhead, Evolution ofLarge-ScaleStructuresintheUniverse,ed.J.AudouzeandA.Szalay(Dordrecht:Reidel),press [1988])have .1986,Nature,321,27. .1984,Ap.J.,284,439. .19826,Ap.J.(Letters),263,LI. .1982a,Ap.J.,258,415. .1987,preprint. Primack, J.R.1986,2dESO-CERNSymposium:Cosmology,Astronomy,and Politzer, D.H.,andWise,MB.1984,Ap.J.(Letters),285,LI. Jensen, L.G.,andSzalay,A.S.1986,Ap.J.(Letters),305,L5. 550 Peebles, P.J.E.1980,TheLarge-ScaleStructureoftheUniverse(Princeton: Loh, E.D.,andSpillar,J.1986,Ap.(Letters),307,LI. Klypin, A.A.,andKopylov,1.1983,SovietAstr.Letters,9,41. Kaiser, N.1984,Ap.J.(Letters),284,L9. Melchiorri, F.,B.O.,Ceccarelli,C,andPietranera,L.1981,Ap.J. Ling, E.N.,Frenk,C.S.,andBarrow,J.D.1986,M.N.R.A.S.,223,21P. Larson, R.B.1987,CommentsAp.,11,273. Kaiser, N.,andSilk,J.1987,Nature,inpress. Kaiser, N.,andLasenby,A.1987,Nature,inpress. Hoffman, Y.1987,Ap.J.(Letters),submitted. Grinstein, B.,andWise,M.J.1987,preprintCALT-68-1418. Gott, J.R.,andRees,M.1975,Astr.Ap.,45,365. ft Frenk, C.S.,White,S.D.M.,Efstathiou,G.,andDavis,M.1985,Nature,317, Faber, S.M.,andGallagher,J.1979,Ann.Rev.Astr.Ap.,17,135. Fundamental ,ed.G.SettiandL.VanHove(GarchingbeiMünchen: ESO), p.193. (Letters), 250,LI. Princeton UniversityPress). 595. A recentreconsiderationoftheeffectreionizationatz»100(G.EfstathiouandJ.R.Bond,M.N.R.A.S.,227, 33p [1987]) Note AddedinProof—A.C.S.Readhead,R.Lawrence,T.Myers,andW.L.Sargent(inIAUSymposium 130, © American Astronomical Society BLUMENTHAL, DEKEL,ANDPRIMACK Provided bythe NASA Astrophysics Data System .19846,Nature,312,427. Zeldovich, Ya.B.1970,Astr.Ap.,5,84. Yang, J.,Turner,M.S.,Steigman,G.,Schramm,D.N.,andOlive,K.A.1984, White, S.D.M.,Frenk,C.S.,Davis,andEfstathiou,G.1987,Ap.J.,313, Vittorio, N.,andTurner,M.S.1986,preprint. Vittorio, N.,andSilk,J.1985,Ap.(Letters),297,LI. Vittorio, N.,Juzskiewicz,R.,andDavis,M.1986,Nature,323,132. Vishniac, E.1987,preprint. Valdarnini, R.,andBonometto,S.A.1985,Astr.Ap.,146,235. Uson, J.M.,andWilkinson,D.T.1984a,Nature,146,235. Turner, M.S.,Steigman,G.,andKrauss,L.1984,Phys.Rev.Letters,52, Silk, J.,andVittorio,N.1987,Ap.317,564. Shafi, Q.,andStecker,F.W.1984,Phys.Rev.Letters,53,1292. Schechtman, S.1985,Ap.J.Suppl.57,77. Rubin, V.C,Ford,W.K.,Thonnard,N.,Roberts,M.S.,andGraham,J.A. Reno, H.,andSeckel,D.1988,inpreparation. Primack, J.R.,Blumenthal,G.andDekel,A.1986,inGalaxyDistances Ap. J.,281,493. 505. 2090. (Dordrecht :Reidel),p.265. Deviations fromUniversalExpansion,ed.B.F.MadoreandR.Tully 1976, Ap.J.,81,687.