A Galaxy-Weighted Measure of the Relative Peculiar

arXiv:astro-ph/9705224v128May1997AGalaxy-WeightedMeasureoftheRelativePeculiarVelocityDispersionMarcDavisDepts.ofAstronomyandPhysics,Univ.ofCalifornia,[e-mail],[e-mail],[e-mail]rdispersionofgalaxieshasforthepastdecadebeenthestandardmeasureofthethermalenergyoffluctuationsintheobservedgalaxydistribution.Thisstatisticisknowntobeunstable,sinceitisapair-weightedmeasurethatisverysensitivetorare,richclustersofgalaxies.Asamorestablealternative,weherepresentasingle-particle-weightedstatisticσ1,whichcanbeconsideredasanestimateoftheone-dimensionalrmspeculiarvelocitydispersionofgalaxiesrelativetotheirneighbors,andwhichcanbeinterpretedbymeansofafilteredversionoftheCosmic-Energyequation.Wecalculatethisstatisticfortheall-skysurveyofIRASgalaxies,findingσ1=95±16km/sec.TheUGCcatalogyieldsahighervalue,σ1=130±15km/s.WecalibrateourprocedurebymeansofmockcatalogsconstructedfromN-bodysimulationsandfindthatourmethodisstableandhasmodestbiaseswhichcaneasilybecorrected.Weusethemeasuredvaluesofσ1inafilteredLayzer-Irvineequationtoobtainanestimateof˜Ω≡Ω/b2.Wefindthat˜Ω≈0.14±0.05forboththeIRASandUGCcatalogs,whichisslightlylowerthanotherrecentdeterminations,butisconsistentwithatrendofaneffectiveΩthatincreasesgraduallywithscale.11.IntroductionConsiderableefforthasbeendevotedinthepastdecadetothemeasurementofthequantityσ12(r),therelativepeculiarvelocitydispersionofpairsofgalaxiesasafunctionoftheirseparation(Davisetal1978;Peebles1979,1981;DavisandPeebles1983,hereafterDP83;Beanetal1983;deLapparentetal1988;Hale-Suttonetal1989;Moetal1993;Zureketal1994;Fisheretal1994a;Marzkeetal1995;Brainerdetal1996,Somervilleetal.1996).Therelativepairvelocitydispersionismosteasilyextractedfromthetwo-pointcorrelationfunctioninredshiftspace,ξ(rp,π).Sincethiscorrelationisapair-weightedquantity,soisthermspeculiarvelocitydispersion.Denseclustersofgalaxiescontainmanypairsandhavehighinternalvelocitydispersion.Consequentlytheycandominatethemeasureddispersion,asshownbyMoetal.(1993),Zureketal.(1994),andSomervilleetal.(1996),whofoundthattheeliminationofthecoresofdenseclustersleadstoasignificantlyreductioninthemeasuredσ12andarguedthatthe“true”valuemightbesubstantiallylargerthanearlierestimates.Forthesereasons,manyauthors(Hale-Suttonetal1989;Moetal.1993;Zureketal.1994;Guzzoetal.1996)haveconcludedthatthisstatisticisunreliable.ItispossiblethatareliableestimatehasfinallybeenachievedwiththenearlycompleteCfA2survey(Marzkeetal1995),butuntiltheseresultsareconfirmedbyevenlargersurveys,acloudofuncertaintywillremain.Therelativepairdispersionσ12(r)iswelldefinedfromthepointofviewofkinetictheory(DavisandPeebles1977,Peebles1980)andisanessentialingredientinthecosmicvirialtheorem(CVT),whichisitselfastatementofequilibriumbetweenkineticpressureandgravitationalaccelerationaveragedacrossallvirializedsystems.Theproperevaluationofthecosmicvirialtheoremincludesanearlydivergentintegraloverthepoorlyconstrained3-pointcorrelationfunctionofgalaxiesζ(seePeebles1980,eq.75.10).Comparisonofσ12tothisintegralyieldsameasureofthecosmicdensityparameterΩ,subjecttoalltheusualargumentsaboutbiasinthegalaxydistribution.Thesensitivityofσ12tothetreatmentoftheraredenseclustersisverylikelymatchedbysimilarsensitivityintheintegraloverζ,butthishasneverbeencheckeddirectly.JuszkiewiczandYahil(1989)haveshownhowthestandardCVT,whichappliestononlinearclusteringonsmallscales,canbereadilyextendedtothelinearregimevalidonlargescales.Moetal.(1996)haveshownhowanalyticalapproximationstoσ12andtootherloworderstatisticscanbederivedfromtheinitiallinearpowerspectrumP(k).BartlettandBlanchard(1993,1996)havepointedoutthattheintegralofthethree-pointfunctionζshouldbetakenoverthegalaxy-galaxy-masscorrelations,andthattheusualestimateofζbasedongalaxy-galaxy-galaxycorrelationscanseriouslyunderestimatetheinferredΩ,ifgalaxieshaveextendedmassivehalos.Kepneretal.(1996)havesuggestedthatmuchofsamplingvarianceofσ12islikelytobereducedifitistabulatedasafunctionofthegalaxydensitysurroundingeachpair.AlthoughtheCVTtestforΩisrarelyperformedwithanyrigor,σ12(r)forthemassdistributioniseasilymeasuredinhighresolutionN-bodysimulations,andisoftenusedtocomparethemwithobservation(e.g.Davisetal1985).ThefactthattheobservedpairdispersioninthegalaxydistributioniswellbelowthatexpectedforthemassdistributioninanΩ=1cosmologywasthemajorfactormotivatingtheconceptofbiasinthegalaxydistribution.Considerableattentionhasbeengiventothedistinctionbetweenσ12(r)measuredforgalaxiesandthesamequantitymeasuredforthemassinsimulations(e.g.Davisetal1985;Couchmanetal1990;GelbandBertschinger1994;Zureketal1994).Theresultsare,notsurprisingly,verysensitivetothespatialresolutionofthesimulationandtothedegreetowhichclumpsofdarkmatterinthesimulationsareseparatedintodistinct“galaxies”.2Thereisclearlyaneedforasimplestatistictoevaluatethekineticenergyofthegalaxydistribution,preferablyonethatisbotheasilyevaluatedandstable.Inthispaper,wedevelopastatisticwhichco