Large-Scale Mass Power Spectrum from Peculiar Velocities

This is a brief progress report on a long-term collaborative project to measure the power spectrum (PS) of mass density fluctuations from the Mark III and the SFI catalogs of peculiar velocities. The PS is estimated by applying maximum likelihood analysis,

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aLARGE-SCALEMASSPOWERSPECTRUMFROMPECULIARVELOCITIESI.ZEHAVIRacahInstituteofPhysics,TheHebrewUniversity,Jerusalem91904,IsraelThisisabriefprogressreportonalong-termcollaborativeprojecttomeasurethepowerspectrum(PS)ofmassdensity uctuationsfromtheMarkIIIandtheSFIcatalogsofpeculiarvelocities.1,2ThePSisestimatedbyapplyingmaximumlikelihoodanalysis,usinggeneralizedCDMmodelswithandwithoutCOBEnormalization.Theapplica-tiontobothcatalogsyieldsfairlysimilarresultsforthePS,andtherobustresultsarepresented.1IntroductionInthestandardpictureofcosmology,structureevolvedfromsmalldensity uctua-tionsthatgrewbygravitationalinstability.Theseinitial uctuationsareassumedtohaveaGaussiandistributioncharacterizedbythePS.Onlargescales,the uc-tuationsarelinearevenatlatetimesandstillgovernedbytheinitialPS.ThePSisthusausefulstatisticforlarge-scalestructure,providingconstraintsoncosmol-ogyandtheoriesofstructureformation.Inrecentyears,thegalaxyPShasbeenestimatedfromseveralredshiftsurveys.3Inthiswork,wedevelopandapplylike-lihoodanalysis4inordertoestimatethemassPSfrompeculiarvelocitycatalogs.Twosuchcatalogsareused.OneistheMarkIIIcatalogofpeculiarvelocities,5acompilationofseveraldatasets,consistingofroughly3000spiralandellipticalgalaxieswithinavolumeof～80h 1Mpcaroundthelocalgroup,groupedinto～1200objects.TheotheristherecentlycompletedSFIcatalog,6ahomogeneouslyselectedsampleof～1300spiral eldgalaxies,whichcomplieswithwell-de nedcriteria.Itisinterestingtocomparetheresultsofthetwocatalogs,especiallyinviewofapparentdiscrepanciesintheappearanceofthevelocity elds.7,82Method

Givenadatasetd,thegoalistoestimatethemostlikelymodelm.InvokingaBayesianapproach,thiscanbeturnedtomaximizingthelikelihoodfunctionL≡P(d|m),theprobabilityofthedatagiventhemodel,asafunctionofthemodelparameters.UndertheassumptionthatboththeunderlyingvelocitiesandtheobservationalerrorsareGaussianrandom elds,thelikelihoodfunctioncanbewrittenasL=[(2π)Ndet(R)] 1/2exp 1

This is a brief progress report on a long-term collaborative project to measure the power spectrum (PS) of mass density fluctuations from the Mark III and the SFI catalogs of peculiar velocities. The PS is estimated by applying maximum likelihood analysis,

Figure1:Likelihoodanalysisresultsforthe atΛCDMmodelwithh=0.6.lnLcontoursinthe nplaneareshownforSFI(leftpanel)andMarkIII(middle).Thebest- tparametersaremarkedby‘s’and‘m’onboth,forSFIandMarkIIIrespectively.TherightpanelshowsthecorrespondingPSfortheSFIcase(solidline)andforMarkIII(dashed).TheshadedregionistheSFI90%con denceregion.ThethreedotsarethePScalculatedfromMarkIIIbyKolattand

Dekel(1997),10togetherwiththeir1σerror-bar.

maximumlikelihood.Con dencelevelsareestimatedbyapproximating 2lnLasaχ2distributionwithrespecttothemodelparameters.Notethatthismethod,basedonpeculiarvelocities,essentiallymeasuresf( )2P(k)andnotthemassdensityPSbyitself.Carefultestingofthemethodwasdoneusingrealisticmockcatalogs,9designedtomimicindetailtherealcatalogs.

WeuseseveralmodelsforthePS.Oneoftheseistheso-calledΓmodel,wherewevarytheamplitudeandtheshape-parameterΓ.ThemainanalysisisdonewithasuitofgeneralizedCDMmodels,normalizedbytheCOBE4-yrdata.Theseincludeopenmodels, atmodelswithacosmologicalconstantandtiltedmodelswithorwithoutatensorcomponent.Thefreeparametersarethenthedensityparameter ,theHubbleparameterhandthepowerindexn.TherecoveredPSissensitivetotheassumedobservationalerrors,thatgoaswellintoR.Weextendthemethodsuchthatalsothemagnitudeoftheseerrorsisdeterminedbythelikelihoodanalysis,byaddingfreeparametersthatgovernaglobalchangeoftheassumederrors,inadditiontomodelingthePS.We nd,forbothcatalogs,agoodagreementwiththeoriginalerrorestimates,thusallowingforamorereliablerecoveryofthePS.3Results

Figure1shows,asatypicalexample,theresultsforthe atΛCDMfamilyofmodels,withatensorcomponentintheinitial uctuations,whensettingh=0.6andvarying andn.TheleftpanelshowsthelnLcontoursfortheSFIcatalogandthemiddlepaneltheresultsforMarkIII.Ascanbeseenfromtheelongatedcontours,whatisdeterminedwellisnotaspeci cpointbutahighlikelihoodridge,constrainingadegeneratecombinationoftheparametersoftheform n3.7=0.59±0.08,inthiscase.Therightpanelshowsthecorrespondingmaximum-likelihoodPSforthetwocatalogs,wheretheshadedregionrepresentsthe90%con denceregionobtainedfromtheSFIhigh-likelihoodridge.

TheseresultsarerepresentativeforallotherPSmodelswetried.Foreach

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This is a brief progress report on a long-term collaborative project to measure the power spectrum (PS) of mass density fluctuations from the Mark III and the SFI catalogs of peculiar velocities. The PS is estimated by applying maximum likelihood analysis,

catalog,thedi erentmodelsyieldsimilarbest- tPS,fallingwellwithineachoth-ersformaluncertaintiesandagreeingespeciallywellonintermediatescales(k～0.1hMpc 1).Thesimilarity,seeninthe gure,ofthePSobtainedfromSFItothatofMarkIIIisillustrativefortheothermodelsaswell.Thisindicatesthatthepeculiarvelocitiesmeasuredbythetwodatasets,withtheirrespectiveerrorestimates,areconsistentwitharisingfromthesameunderlyingmassdensityPS.NotealsotheagreementwithanindependentmeasureofthePSfromtheMarkIIIcatalog,usingthesmootheddensity eldrecoveredbyPOTENT(thethreedots).10

Therobustresult,forbothcatalogsandallmodels,isarelativelyhighPS,withP(k) 1.2=(4.5±2.0)×103(h 1Mpc)3atk=0.1hMpc 1.Anextrapolationtosmallerscalesusingthedi erentCDMmodelsgivesσ8 0.6=0.85±0.2.Theerror-barsarecrude,re ectingthe90%formallikelihooduncertaintyforeachmodel,thevarianceamongdi erentmodelsandbetweencatalogs.Thegeneralconstraintofthehighlikelihoodridgesisofthesort h50µnν=0.75±0.25,whereµ=1.3andν=3.7,2.0forΛCDMmodelswithandwithouttensor uctuationsrespectively.ForopenCDM,withouttensor uctuations,thepowersareµ=0.9andν=1.4.Forthespanofmodelschecked,thePSpeakisintherange0.02≤k≤0.06hMpc 1.TheshapeparameteroftheΓmodelisonlyweaklyconstrainedtoΓ=0.4±0.2.Wecaution,however,thattheseresultsareasyetpreliminary,andmightdependontheaccuracyoftheerrorestimatesandontheexactimpactofnon-linearities.2Acknowledgments

IthankmyclosecollaboratorsinthisworkA.Dekel,W.Freudling,Y.Ho manandS.Zaroubi.Inparticular,IthankmycollaboratorsfromtheSFIcollaboration,L.N.daCosta,W.Freudling,R.Giovanelli,M.Haynes,S.SalzerandG.Wegner,forthepermissiontopresentthesepreliminaryresultsinadvanceofpublication.References

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This is a brief progress report on a long-term collaborative project to measure the power spectrum (PS) of mass density fluctuations from the Mark III and the SFI catalogs of peculiar velocities. The PS is estimated by applying maximum likelihood analysis,

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