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Mon.Not.R.Astron.Soc.000,000–000(0000) Printed28January2013 (MNLATEXstylefilev2.2) Using galaxy-galaxy weak lensing measurements to correct the Finger-of-God Chiaki Hikage1,2, Masahiro Takada3, David N. Spergel1,3 1DepartmentofAstrophysicalSciences,PrincetonUniversity,PeytonHall,PrincetonNJ08544,USA 2 2Kobayashi-MaskawaInstitutefortheOriginofParticlesandtheUniverse(KMI),NagoyaUniversity,Aichi464-8602,Japan 1 3InstituteforthePhysicsandMathematicsoftheUniverse(IPMU),TheUniversityofTokyo,Chiba277-8582,Japan 0 2 28January2013 n a J 5 ABSTRACT For decades, cosmologists have been using galaxies to trace the large-scale distribution of ] matter.Atpresent,thelargestsourceofsystematicuncertaintyinthisanalysisisthechallenge O of modeling the complex relationship between galaxy redshift and the distribution of dark C matter.Ifallgalaxiessatinthecentersofhalos,therewouldbeminimalFinger-of-God(FoG) h. effectsandasimplerelationshipbetweenthegalaxyandmatterdistributions.However,many p galaxies,evensomeoftheluminousredgalaxies(LRGs),donotliein thecentersofhalos. - Becausethegalaxy-galaxylensingisalsosensitivetotheoff-centeredgalaxies,weshowthat o r wecanusethelensingmeasurementstodeterminetheamplitudeofthiseffectandtodeter- t mine the expectedamplitude of FoG effects. We developan approachforusing the lensing s a datatomodelhowtheFoGsuppressesthepowerspectrumamplitudesandshowthatthecur- [ −1 rentdataimpliesa30%suppressionatwavenumberk = 0.2hMpc .Ouranalysisimplies 2 thatitisimportanttocomplementaspectroscopicsurveywithanimagingsurveywithsuffi- v cientdepthandwidefieldcoverage.Jointimagingandspectroscopicsurveysallowarobust, 0 unbiaseduseofthepowerspectrumamplitudeinformation:itimprovesthemarginalizederror 4 ofgrowthratef ≡dlnD/dlnabyuptoafactorof2overawiderangeofredshiftsz <1.4. 6 g 1 Wealsofindthatthedarkenergyequation-of-stateparameter,w0,andtheneutrinomass,fν, . canbeunbiasedlyconstrainedbycombiningthelensinginformation,withanimprovementof 6 10–25%comparedtoaspectroscopicsurveywithoutlensingcalibration. 0 1 Keywords: cosmology:theory–galaxyclustering–darkenergy 1 : v i X 1 INTRODUCTION r a Overthepast threedecades, astronomershavebeen conducting everlargerredshift surveysintheireffortstoprobethe large-scalestructure of the universe (Davis&Huchra 1982; deLapparentetal. 1986; Kirshneretal. 1987; Yorketal. 2000; Peacocketal. 2001). In the comingdecade,weareembarkingonevenlargersurveys:BOSS1,WiggleZ2(Blakeetal.2011),Vipers3,FMOS4,HETDEX5,BigBOSS6 (Schlegeletal.2009),LAMOST7,SubaruPFS8,Euclid9,andWFIRST10.Thisupcominggenerationofsurveysaremotivatedbyourdesire tounderstandcosmicaccelerationandtomeasurethecompositionoftheuniversebysimultaneouslymeasuringgeometryanddynamics.The combinationofcosmicmicrowavebackground(CMB)dataandlargeredshiftsurveystracethegrowthofstructureformationfromthelast- scatteringsurface(z 1100)tolowredshiftsanddeterminecosmologicalparameterstohighprecision(Wangetal.1999;Eisensteinetal. ≃ 1 http://cosmology.lbl.gov/BOSS/ 2 http://wigglez.swin.edu.au/site/ 3 http://vipers.inaf.it/ 4 http://www.naoj.org/Observing/Instruments/FMOS/ 5 http://hetdex.org/ 6 http://bigboss.lbl.gov/ 7 http://www.lamost.org/website/en 8 http://sumire.ipmu.jp/en/ 9 http://sci.esa.int/euclid 10 http://wfirst.gsfc.nasa.gov/ (cid:13)c 0000RAS 2 Hikage, Takada& Spergel 1999;Tegmarketal.2004;Coleetal.2005).Measurements of thebaryon acousticoscillation(BAO)scaleprovideuswitharobustgeo- metricalprobeoftheangulardiameterdistanceandtheHubbleexpansionrate(Eisensteinetal.2005;Percivaletal.2007).Observationsof redshift-spacedistortionmeasurethegrowthrateofstructureformation(Zhangetal.2007;Guzzoetal.2008;Wang2008;Guziketal.2010; Whiteetal.2009;Percival&White2009;Song&Percival2009;Song&Kayo2010;Yamamotoetal.2010;Tangetal.2011).Combining measurementsofthegrowthofstructureformationandthegeometryoftheuniverseprovidesakeycluetounderstandingthenatureofdark energy,propertiesofgravityoncosmologicalscales,orthenatureofcosmicacceleration(Albrechtetal.2006;Peacocketal.2006). Thegalaxypowerspectruminredshiftspace,adirectobservablefromaredshiftsurvey,isatwo-dimensionalfunctionofwavelengths perpendicularandparalleltotheline-of-sightdirection (Peacocketal.2001;Okumuraetal.2008;Guzzoetal.2008).Whilegalaxyclus- teringinreal spaceisstatisticallyisotropicinan isotropicand homogeneous universe, theline-of-sight components of galaxies’ peculiar velocitiesaltergalaxyclusteringinredshiftspace(Kaiser1987).Forreview,seeHamilton(1998).Theamplitudeofthedistortiondepends bothongeometryanddynamics(Alcock&Paczynski1979;Seo&Eisenstein2003). For the surveys to achieve their ambitious goals for precision cosmology, we will need a detailed understanding of the underlying systematics.Oneofthemajorsystematicuncertaintiesinredshift-spacepowerspectrummeasurementsisnon-linearredshiftdistortiondue to the internal motion of galaxies within halos, the so-called Finger-of-God (FoG) effect (Jackson 1972; Scoccimarro 2004). Since it is sensitive to highly non-linear physics as well as difficult to model galaxy formation/assembly histories, the FoG effect is the dominant systematicinredshiftsurveys. Reidetal.(2009)advocatedusinghalosratherthanLuminousRedGalaxies(LRGs;Eisensteinetal.2001)totracelarge-scalestructure. In an analysis of LRGs sampled with the Sloan Digital Sky Survey (SDSS)11, Reidetal. (2010) implemented this scheme by removing satelliteLRGsfromthesamehalowiththeaidofthemockcatalogandthehalomodelprescription.FromtheSDSSLRGdataset,Reidetal. (2010)foundthatabout6%ofLRGsaresatellitegalaxies,whiletheremaining94%arecentralgalaxiesofhaloswithmasses > 1013M . ⊙ ∼ Oncesuchahalocatalogisconstructed,clusteringpropertiesofhalosareeasiertomodel,becausehaloshaveonlybulkmotionsinlarge-scale structure,andthereforehavethereducedFoGeffect.Despitethiseffort,theremainingFoGeffectisadominantsystematicuncertainty. FoG effects are just one of the non-linear systematics. Future analysis of the redshift-space power spectrum of halos will need to model non-linear clustering, non-linear bias, and non-linear redshift distortion effect due to their bulk motions. Recent simulations and refined perturbation theory suggest that halo clustering based approach seems avery promising probe of cosmology (Scoccimarro 2004; Crocce&Scoccimarro2006;Matsubara2008;Saitoetal.2011;Taruyaetal.2010;Tangetal.2011;Reid&White2011;Sato&Matsubara 2011). For a halo-based catalog, a significant source of uncertainty is the position of the galaxies in the halos. Hoetal. (2009) compared LRGpositionswiththeX-raysurface-brightnesspeak,reportingasizablepositionaldifference.FortheLRGanalysis,thisisthedominant uncertainty(seeReidetal.2010,fortheusefuldiscussioninAppendixC).Thevirialtheoremimpliesthatoff-centeredLRGsaremoving relativetothehalocenterthusproducinganFoGeffect. Inthispaper,weproposeanovelmethodofusingacross-correlationofspectroscopicgalaxies(e.g.,LRGs)withbackgroundgalaxy imagestocorrecttheFoGcontaminationtotheredshift-spacepowerspectrum.Darkmatterhaloshostingspectroscopic galaxiesinducea coherentlensingdistortioneffectonbackgroundgalaxyimages,andthesignalsaremeasurableusingthecross-correlationmethod–theso- calledgalaxy-galaxyorcluster-galaxyweaklensing.Thelensingsignalshavebeennowmeasuredatahighsignificancebyvariousgroups (Mandelbaumetal. 2006; Sheldonetal. 2009; Leauthaudetal. 2010; Okabeetal. 2010). If we include off-centered galaxies and use the galaxypositionasahalocenterproxyofeachhalointhelensinganalysis,thelensingsignalsatangularscalessmallerthanthetypicaloffset scalearediluted(seeOguri&Takada2011,forausefulformulationoftheoff-centeringeffectoncluster-galaxyweaklensing).Thusthe galaxy-galaxylensingsignalscanbeusedtoinfertheamountoftheoff-centeredgalaxycontamination(Johnstonetal.2007;Leauthaudetal. 2010;Okabeetal.2010).Furthermore,sincelensingisauniquemeansofreconstructingthedarkmatterdistribution,itmayallowustoinfer the halo center on individual halo basis if a sufficiently high signal-to-noise ratio isavailable (Ogurietal. 2010). Hence, a weak-lensing basedcalibrationoftheFoGeffectinredshift-spacepowerspectrummeasurementsmaybefeasibleifspectroscopicand imagingsurveys observethesameregionofthesky.Fortunately,manyupcomingsurveyswillsurveythesameregionofthesky:theBOSSandSubaruHyper SuprimeCam(HSC)Survey(Miyazakietal.2006),theSubaruPFSandHSCsurveys(SubaruMeasurementsofImagesandRedshifts:the SuMIReproject),EuclidandWFIRSToracombinationofLSST(LSSTScienceCollaborationsetal.2009)withspectroscopicsurveys. InSection2,wewillfirstdevelopamodelofcomputingtheredshift-spacepowerspectrumofLRGsbasedonthehalomodelapproach (seeCooray&Sheth2002,forathorough review). Extending White(2001)and Seljak(2001),wemodel thedistributionof off-centered LRGsasasourceofFoGdistortions.FollowingthemethodinOguri&Takada(2011),wealsomodelthedistributionofoff-centeredLRGs asasourceofsmoothingoftheLRG-galaxylensingsignal.AssumingsurveyparametersoftheSubaruHSCimagingsurveycombinedwith theBOSSand/orSubaruPFSspectroscopic surveysaswellastheEuclidimagingandspectroscopic surveys, westudytheimpactof the FoGeffectonparameter estimations.Wealsostudytheabilityof thecombined imaging andspectroscopic surveysforcorrectingforthe FoGeffectcontaminationbasedontheoff-centeringinformationinferredfromtheLRG-galaxylensingmeasurements. Fortheparameter forecast,wepayparticularattentiontothedarkenergyequation-of-state parameter, w0,theneutrinomassparameter, fν,andthegrowth rateateachredshiftslice.UnlessexplicitlystatedwewillthroughoutthispaperassumeaWMAP-normalizedΛCDMmodelasourfiducial 11 http://www.sdss.org/ (cid:13)c 0000RAS,MNRAS000,000–000 Usinggalaxy-galaxyweak lensingmeasurementstocorrect theFinger-of-God 3 Real space Redshift space r || halo bulk vel. r gal axy 2pt correlation internal vel. DM halo Figure1.Aschematicillustrationoftheredshift-distortioneffectonredshift-spaceclusteringofdominantluminousredgalaxies(DLRGs;seetextfordetails). WeassumethatacatalogofDLRGsisconstructedsothateachhalocontainsoneDLRG.Theredshiftdistortioneffectontheredshift-spacepowerspectrumof DLRGsarisesfromtwocontributions:thebulkmotionofhalosthathosteachDLRG,andtheinternalmotionofDLRGwithinahalo,theFinger-of-God(FoG) effect.IfsomeoftheLRGsarenotinthecenteroftheirhalos,thentheirmotionsproducesignificantFoGeffects.Thehalobulkmotioncausesadisplacement ofhalopositioninredshiftspace,whiletheinternalmotionstretchesthedistributionregionofDLRGswithinahaloalongtheline-of-sightdirection. cosmological model (Komatsuetal. 2009):Ωbh2 = 0.0226, Ωcdmh2 = 0.1109, ΩΛ = 0.734, respectively, τ = 0.088, ns = 0.963, A(k = 0.002Mpc−1) = 2.43 10−9,whereΩb,Ωcdm andΩΛ aretheenergydensityparametersofbaryon,CDManddarkenergy(the × cosmologicalconstantwithw0 = 1here),τ istheopticaldepthtothelastscatteringsurface,andns andAarethetiltandamplitudeof − theprimordialcurvaturepowerspectrum. 2 FORMULATION:REDSHIFT-SPACEPOWERSPECTRUM Inthissection,wegiveaformulationformodelingtheredshift-spacepowerspectrumofluminousredgalaxies(LRGs)basedonthehalo modelapproach(White2001;Seljak2001). 2.1 DominantLuminousRedGalaxies(DLRGs) Weak lensing studies (Mandelbaumetal. 2006; Johnstonetal. 2007) and clustering analyses (Rossetal. 2007, 2008; Wakeetal. 2008; Zhengetal. 2009; Reid&Spergel 2009; Whiteetal. 2011) find that most LRGsreside inmassive halos. Whilethetypical massive halo contains only one LRG, roughly 5-10% of all LRGs are satellite galaxies in a halo containing multiple LRGs. These satellite galaxies contributealargeonehalotermthatisanadditionalsourceofshotnoiseandnon-linearityinpowerspectrumestimation.Reidetal.(2009) outlineaprocedureofidentifyingthesesatelliteLRGsthroughfindingmultiplepairsthatlieincommonhalos(orthesmallspatialregion) and then using only the brightest luminous red galaxies in each halo as a tracer. We call these galaxies dominant luminous red galaxies (DLRGs).TheseDLRGsaremorelineartracersoftheunderlyingmatterfieldthantheLRGsare.Reidetal.(2010)andPercivaletal.(2010) adoptthisproceduretodeterminetheSDSSLRGpowerspectrum.Inthispaper,wefocusontheseDLRGssothateachhalocontainseither zerooroneDLRG. 2.2 HaloModelApproachforDLRGs SincethereisonlyoneDLRGperhalo,thetwo-halotermdeterminestheclusteringofthesegalaxiesinthehalomodelpicture(Cooray&Sheth 2002;Takada&Jain2003).IftheDLRGssatinthecenterofeachhalo,thentheDLRGpowerspectrumwouldbelinearlyrelatedtothehalo powerspectrum.However,sincetheDLRGsdonotalwayslieinthecenterofthehalo,thepowerspectrumisgivenas 1 dn dn PDLRG(k)= n¯2 Z dMZ dM′ dMNHOD(M)põff(k;M)dM′NHOD(M′)põff(k;M′)Phh(k;M,M′), (1) DLRG wheredn/dMisthehalomassfunction,NHOD(M)isthehalooccupationnumber(noteNHOD 61asdescribedbelow),andPhh(k;M,M′) isthecross-powerspectrumofhalosofmassesMandM′.Numericalsimulationsshowthatthehalocross-powerspectrumisapproximately (cid:13)c 0000RAS,MNRAS000,000–000 4 Hikage, Takada& Spergel alinearlybiasedversionofthematterpowerspectrum(Reidetal.2009):P (k;M,M′) b(M)b(M′)PNL(k),whereb(M)isthehalo hh ≃ m bias,andPNL(k)isthenon-linearmatterpowerspectrum. Thisapproximation simplifiestherelationshipbetweentheDLRGandmatter m powerspectrum: 2 1 dn PDLRG(k)=(cid:20)n¯DLRG Z dM dMb(M)NHOD(M)põff(k;M)(cid:21) PmNL(k). (2) Thequantityn¯DLRGisthemeannumberdensityofDLRGsdefinedas dn n¯DLRG dM NHOD(M). (3) ≡Z dM ThemeanbiasofhaloshostingDLRGsisdefinedas 1 dn ¯b dM b(M) NHOD(M). (4) ≡ n¯DLRG Z dM ThemeanmassofhaloshostingDLRGsissimilarlyestimatedasM¯h (1/n¯DLRG) dM M(dn/dM)NHOD(M). ≡ Thecoefficientpõff(k;M)inEq.(2)istheFouriertransformoftheaverageradiaRlprofileofDLRGswithinahalowithmassM: rvir sin(kr) p˜ (k;M)=4π r2drp (r) , (5) off Z off kr 0 wherepoff(r)isnormalizedsoastosatisfy 0rvir4πr2dr poff(r) = 1,andrvir isthevirialradiusofahalowithmass M,whichcanbe definedoncethevirialoverdensityandthebaRckgroundcosmologyarespecified.Notethat,sincethepowerspectrumisastatisticalquantity, wejustneedtheaveragedDLRGdistributionwithinahalo,whichisthereforeaone-dimensionalfunctionofradiusrwithrespecttothehalo centerinastatisticallyhomogeneousandisotropicuniverse.Inthefollowing,quantitieswithtildesymboldenotetheirFourier-transformed coefficientsforournotationalconvention. TheterminthesquarebracketinEq.(2)describesthehaloexclusioneffect.Becausehaloshavefinitesizes,roughlytheirvirialradius, thereisonlyonedominantgalaxyinthisregion(seeFig.1).Ifweareimplementinganalgorithmthateliminatesmultiplegalaxiesinthe finiteregion, thenweimposeanexclusionregionaround each galaxy. ThetwohalotermdescribesthecorrelationsbetweentwoDLRGs intwodifferent halos. The DLRGpower spectrum at small scales (large k’s) isthus suppressed compared to thematterpower spectrum multipliedwith¯b2(seeFig.11inCooray&Sheth2002). IfeachDLRGresidesatthecenterofeachhalo(e.g.,thecenterofmass),p˜ (k) = 1(orp δ (r)).Howeversomefractionof off off D ∝ DLRGsinthesampleareexpectedtohaveanoffsetfromthehalocenter(Skibbaetal.2011).Duetothecollision-lessnatureofdarkmatter, darkmatterhaloslackclearboundarywithsurroundingstructuresanddonothaveasphericallysymmetricmassdistribution.Thusthehalo centerisnotawell-definedquantity.WhileDLRGs,themostmassivegalaxyinthehalo,willeventuallysinktowardthecenterofthehalo throughdynamicalfriction,manyclustersaredynamicallyyoungandhaveexperiencedrecentinteractions.Thus,weexpectthatDLRGsare notallinthecentersofhalosandthatthedistributionofthetheirpositionsinthehalosevolvewithredshift. Howdoesthishalomodelpictureneedtobechangedinredshiftspace?Tomodeltheredshift-spacepowerspectrum,weneedtoproperly takeintoaccounttheredshiftdistortioneffectduetopeculiarvelocitiesofDLRGs.IfallDLRGsarelocatedatthecenterintheirhosthalos, DLRGsmovetogetherwiththeirhosthaloshavingcoherent,bulkvelocitiesinlarge-scalestructure,andtheredshift-spaceclusteringisnot affectedbytheFoGeffect.However,asillustratedinFig.1,ifsomeDLRGsareoffsetfromthecenter,theywillhaveinternalmotionswithin theirhosthalos,whichcausestheFoGeffect.ThevirialtheoremimpliesthattheamplitudeofthedisplacementoftheDLRGfromthecenter ofitshaloisdirectlyrelatedtotheDLRGvelocitydispersionwithinthehalo. Inthehalomodelpicture,theFoGeffectcanbeincorporatedbystretchingtheaverageradialprofileofDLRGsalongtheline-of-sight directionbytheamount of theinternal motion, asillustratedintheright panel of Fig.1. Thisstretchenhances thehalo exclusion effect, whichsuppressesthepowerspectrumamplitudes.Thustheredshift-spacedistributionofDLRGswithinahalobecomestwo-dimensional, givenasafunctionoftworadii,r andr ,perpendicularandparalleltotheline-of-sightdirectionwithrespecttothehalocenter.Alsonote ⊥ k thattheinternalvelocitydistributionofDLRGswithinahaloisconsideredtobestatisticallyisotropicandthereforeitdependsontheradius rfromthehalocenter,halomassM andredshiftz(seeSection2.3.3fordetails).Theaveragedredshift-spacedistributionofDLRGswithin ahalo,denotedasp (r ,r ),canbegivenasasmearingofthereal-spacedistributionwiththedisplacementfunction: s,off ⊥ k ∞ p (r ,r ;M)= dr′ R(r r′;r′,M)p r2 +r′2 , (6) s,off ⊥ k Z−∞ k k− k off(cid:16)q ⊥ k (cid:17) whereR(∆r ;r,M)isthedisplacement functionofDLRGsduetothevelocitydistributioninsideahaloandsatisfiesthenormalization k condition: d(∆r )R(∆r ) = 1.AssumingthattheinternalmotionofDLRGsismuchsmallerthanthespeedoflight,thedisplacement k k oftheradiaRlpositionofagivenDLRGisdirectlyrelatedtotheline-of-sightcomponentoftheinternalvelocityv as k v ∆r = k , (7) k aH(z) whereH(z)istheHubbleexpansionrateattheredshiftofDLRG. (cid:13)c 0000RAS,MNRAS000,000–000 Usinggalaxy-galaxyweak lensingmeasurementstocorrect theFinger-of-God 5 Hence,assumingadistantobserverapproximation,theredshift-spacepowerspectrumofDLRGscanbegivenintermsoftheFourier- transformofp (r ,r ;M)as s,off ⊥ k 2 1 dn Ps,DLRG(k,µ)=(cid:20)n¯DLRG Z dM dMb(M)NHOD(M)p˜s,off(k,µ;M)(cid:21) PsN,mL(k,µ), (8) whereµisthecosineanglebetweentheline-of-sight directionand thewavevector k,i.e.µ k /k,andPNL(k ,k )isthenon-linear ≡ k s,m ⊥ k redshift-spacepowerspectrum.Inthispaper,wesimplyassumethattheredshiftdistortioneffectduetothecoherentbulkmotionofhalosis describedbylineartheory(Kaiser1987): PNL(k,µ)=PNL(k)[1+2βµ2+β2µ4], (9) s,m m whereβ f /¯b,f isthelineargrowthrate,f dlnD/dlna,and¯bistheeffectivebiasofhaloshostingtheDLRGs(Eq.[4]).Asgiven g g g ≡ ≡ bytheterminthesquarebracketinEq.(8),theFoGeffectduetooff-centeredDLRGscausesscale-dependent,angularanisotropiesinthe redshiftpowerspectrumamplitudes. InthelimitthatallDLRGsareatthetruecenterofeachhalo,theredshift-spacepowerspectrum(Eq.[8])isreducedtothehalopower spectruminredshiftspace: 1 poff(r)= 4πr2δD(r)→Ps,DLRG(k,µ)=¯b2PsN,mL(k,µ)≃Ps,halo(k,µ), (10) whereP (k,µ)isthehalopowerspectruminredshiftspace.Morerigorouslyspeaking,haloclusteringisaffectedbynon-linearitiesin s,halo gravitationalclustering,redshiftdistortionandbiasingatscalesevenintheweaklynon-linearregimeweareinterestedin(Scoccimarro2004; Taruyaetal.2009,2010;Saitoetal.2011;Tangetal.2011;Reid&White2011;Sato&Matsubara2011).Forthesewecanuseanaccurate modeloftheredshift-spacespectrumofhalosbyusingrefinedperturbationtheoryand/orN-bodyandmocksimulations(Taruyaetal.2009, 2010; Sato&Matsubara 2011).Hence wecan extendtheformulation aboveinordertoinclude thesenon-lineareffects, simplybyusing amodelofnon-linear,redshift-spacehalopowerspectrumforP (k,µ;M,M′),insteadofb(M)b(M′)PNL(k,µ)inEq.(8).However, s,hh s,m thisisbeyondthescopeofthispaper,andweherefocusontheFoGeffectbyassumingtheKaiserformula(9)forthesakeofclarityofour discussion. 2.3 Modelingredients Tocomputetheredshift-spacepowerspectrum(Eq.[8]),weneedtospecifythemodelingredients:halooccupationdistributionofDLRGs, theoff-centereddistributionofDLRGsandthevelocitydistributioninsidehalos.Inthissubsection,wewillgivethesemodelingredients adoptedinthispaper. 2.3.1 HODandthehalomodelingredients Firstweneedtospecifythehalomassfunctionandthehalobias.WeusethefittingformuladevelopedinSheth&Tormen(1999)tocompute thehalomassfunctionandthehalobiasinourfiducialcosmologicalmodel(alsoseeTakada&Jain2003). Auseful,empiricalmethodfordescribingclusteringpropertiesofgalaxiesisthehalooccupationdistribution(HOD)(Scoccimarroetal. 2001;Zhengetal.2005,alsoseereferencestherein).TheHODgivestheaveragenumberofgalaxiesresidinginhalosofmassM andat redshiftz.ThepreviousworkshaveshownthatthehalomodelpredictionusingtheHODmodelingcanwellreproducetheobservedproperties ofLRGclusteringoverwiderangesoflengthscalesandredshifts(0 < z < 0.5)(Zhengetal.2009;Reid&Spergel2009;Whiteetal.2011). ∼ ∼ SinceweassumethatsatelliteLRGscanberemovedbasedonthemethodofReidetal.(2010),weusetheHODforcentralLRGsthatis foundinReid&Spergel(2009): NHOD(M) Ncen(M)= 1 1+erf log10(M)−log10(Mmin) , (11) ≃ 2(cid:20) (cid:18) σlogM (cid:19)(cid:21) whereerf(x)istheerrorfunction,andweadoptMmin =8.05 1013M⊙andσlogM =0.7.Wedonotconsiderapossibleredshiftevolution × oftheHOD,becauseanystrongredshiftdependencehasnotbeenfoundfromactualdata.NoteNHOD(M)61.Alsonotethatweusethe HODmodelfor“central”galaxies,butthisdoesnotmeanthatallDLRGsunderconsiderationarecentralgalaxies,buteachhalohas one DLRGatmost. 2.3.2 RadialprofileofDLRGs Theradial profileof DLRGsisnot well known, asthetrue centerof ahalo isnot easy toestimateobservationally. Several studies, both observationalandnumerical,suggestthattheDLRGaremorecentrallyconcentratedthanthedarkmatter,butdonotalllieinthebottomof thedarkmatterpotentials(Lin&Mohr2004;Koesteretal.2007;Johnstonetal.2007;Hoetal.2009;Hilbert&White2010;Okabeetal. 2010; Ogurietal. 2010; Skibbaetal. 2011). Hoetal. (2009) compared the LRG positions with X-ray peak positions for known X-ray (cid:13)c 0000RAS,MNRAS000,000–000 6 Hikage, Takada& Spergel Figure2.ThisfigureshowsthestatisticallyaveragedradialprofileofdarkmatterandDLRGsinhaloswithmassM =1014h−1M⊙andhaloconcentration cvir = 4.3,andatredshiftz = 0.45.ThesolidcurvedenotesaGaussianradialprofile(oroff-centered)modelwiththewidththatistakentoberoff = 0.3rs(∼ 100kpc),wherers isthescaleradiusofthedarkmatterNFWprofile.ThedottedcurveisanNFWradialprofilemodel,whichisgivenbythe concentrationparameterofcoff =20. clusters,andfoundthattheLRGradialdistributioncanbefittedwithanNFWprofilewithhighconcentrationparameter(c 20).Usingthe ∼ Subaruweaklensingobservationsforabout20X-rayluminousclusters,Ogurietal.(2010)fitanellipticalNFWmodeltothedarkmatter distribution.Theyfoundthat,formostclusters,thepositionaldifferencebetweenthelensing-inferredmasscenterandthebrightestcluster galaxyiswellfittedbyaGaussiandistributionwithwidthof 100h−1kpc,ascalecomparabletothepositionaluncertaintiesinthelensing ∼ analysis.Fortheseclusters,thelensingdataisconsistentwiththeDLRGslyinginthecenterofmassoftheirhosthalos.However,inafew clusters,theDLRGsareclearlyoffsetfromthecenterofthe potentialwithcharacteristicdisplacementsof 400h−1kpc.Johnstonetal. ∼ (2007)reachedasimilarconclusion:mostDLRGsareinthecentersoftheirhalo;however,ahandfularesignificantdisplaced.However,the resultsarenotyetconclusiveduetothelimitedstatistics.Inthispaper,weemploythefollowingtwoempiricalmodelsforaradialprofileof DLRGsbasedontheseobservationalimplications: 1 r2 exp , (Gaussianoffsetmodel), p (r;M)= (2π)3/2ro3ff(M) (cid:18)−2ro2ff(M)(cid:19) (12) off  c3off f 1 , (NFWoffsetmodel), 4πrv3ir (coffr/rvir)(1+coffr/rvir)2  wheref 1/[ln(1+c ) c /(1+c )]andtheprefactorofeachmodelisdeterminedsoastosatisfy thenormalizationcondition off off off 0rvir4πr≡2drpoff(r)=1.Th−eseprofilesarespecifiedbyoneparameter(roff orcoff),butdifferintheshape.Notethatrvirisspecifiedasa Rfunctionofhalomassandredshift,andcoff differsfromtheconcentrationparameterofdarkmatterprofile. Asa working example, we willemploy roff(M) = 0.3rs(M) = 0.3rvir(M)/cvir(M) for Gaussian DLRGradial distributionand coff =20forNFWDLRGradialdistributionasourfiducialmodels.HerersisthescaleradiusofdarkmatterNFWprofile,andcvir isthe concentrationparameter.ForthefollowingresultswewillusethesimulationresultsinDuffyetal.(2008)tospecifycvirasafunctionofhalo massandredshift.Ourfiducialmodelofr =0.3r givesr 100kpcforhalosof1014M ,consistentwiththeresultsinOgurietal. off s off ⊙ ≃ (2010).NotethatthetypicaloffsetoftheDLRGfromthecenterofthehalopotentialvarieswithhalomass. Fig.2showstheGaussianandNFWmodelsfortheradialprofileofDLRGsinsideahaloofmassM =1014h−1M andatz=0.45. ⊙ Notethatforourfiducialmodel,r 250h−1kpc.Theseprofilesaremuchmorecentrallyconcentratedthanatypicaldarkmatterhaloof s ≃ thismassscale,representedbyanNFWprofilewithc=4.3. TheFouriertransformsoftheseradialprofilesareanalyticfunctions: exp[ r2 (M)k2/2], (Gaussianmodel), − off põff(k;M)= f sinη Si(η(1+c )) Si(η) +cosη Ci(η(1+η)) Ci(η) sin(ηcoff) , (NFWmodel), (13)  off (cid:20) { − } { − }− η(1+coff)(cid:21)  whereη = krvir/coff,andSi(x)andCi(x)arethesineandcosineintegralfunctions.TheFouriertransformpõff(k;M) 1onrelevant ≃ scales. (cid:13)c 0000RAS,MNRAS000,000–000 Usinggalaxy-galaxyweak lensingmeasurementstocorrect theFinger-of-God 7 Figure3.Leftpanel:VelocitydispersionofDLRGsaveragedoverDLRGradialprofile,asafunctionofhosthalomassatz=0.45,assumingtheGaussian (solidcurve)andNFW(dotted)radialprofileprofilesasinFig.2.Forcomparisonthedashedcurveshowsthevelocitydispersionofdarkmatter.Rightpanel: RedshiftdependenceofthevelocitydispersionsofDLRGsandDMforahalowithmassM =1014h−1M⊙. 2.3.3 VelocitydispersionofDLRGs Inthispaper,wesimplyassumethatgalaxiesatagivenradiusrhavethefollowing1Dvelocitydispersionthatisdeterminedbythemass enclosedwithinthesphere: GM(<r) σ2(r;M)= . (14) v 2r Atvirialradiusrvir,thevelocitydispersionisdeterminedbyvirialmass: 1/3 1/6 M ∆(z) σv(r=rvir;M)=472km/s(cid:18)1014h−1M⊙(cid:19) (cid:18)18π2(cid:19) (1+z)1/2, (15) where we use the definition of virial mass given in terms of the overdensity ∆(z) at redshift z (we use the fitting formulae given in Nakamura&Suto1997).SinceanNFWprofilehasanasymptoticbehaviorof M(< r) r2 asr 0, thevelocitydispersion hasthe ∝ → limitσ (r;M) 0asr 0. v → → ThevelocitydispersionofLRGsispoorlyknown(seeSkibbaetal.2011,forthefirstattempt).InAppendixA,wegiveanalternative model of computing thevelocitydispersion byassuming an isothermal distributionforthephase spacedensity of DLRGswithinahalo, whereweproperlytakeintoaccountthedifferentradialprofilesofDLRGsanddarkmatter. TheaveragedvelocitydispersionofDLRGswithinhalosofagivenmassscaleM canbeobtainedbyaveragingthevelocitydispersion (Eq.[14])withtheradialprofileofDLRGs: rvir σ2 (M) 4πr2drp (r;M)σ2(r;M). (16) v,off ≡Z off v 0 ThisvelocitydispersionhasanasymptoticlimitwhenalltheDLRGsareatthecenterofeachhalo:σ 0whenp (r) δ (r).Fig.3 v,off off D → ∝ plotsthevelocitydispersionofDLRGs,σ (M),asafunctionofhalomassMforafixedredshift(leftpanel),andasafunctionofredshift v,off zforafixedhalomass(right),respectively.ThevelocitydispersionofDLRGsislargerinmoremassivehalosandathigherredshifts.Within thesamehalo,thevelocitydispersionofDLRGsissmallerthanthatofdarkmatterby10-20%intheamplitudes,becauseDLRGsaremore centrallyconcentrated.Eq.(15)impliesthatthevelocitydispersionofboththeDLRGsandthedarkmatterscalesasσ (M) M1/3. v,off ∝ 2.4 Redshift-spacepowerspectrumofDLRGandthecovariancematrix WeuseourmodelfortheradialdistributionoftheDLRGsandtheirvelocitydistributiontoestimatetheeffectoftheoffsetonthepower spectrum.WeassumethevelocitydistributionofDLRGswithinhalosisGaussian,wherethewidthofthedistributionisgivenbythevelocity dispersion(Eq.[14]): R(∆r ;r,M)d(∆r )= 1 exp vk2 dv , (17) k k √2πσ (r,M) (cid:20)−2σ2 (r,M)(cid:21) k v,off v,off (cid:13)c 0000RAS,MNRAS000,000–000 8 Hikage, Takada& Spergel Figure4.TheFoGsuppressionscale(seeEq.[21]),σv,off/aH(z),asafunctionofthemodelparametersofDLRGradialprofiles,fortheGaussianradial profilemodel(leftpanel)andtheNFWmodel(rightpanel).TheFoGscaleiscomputedbyaveragingthevelocitydispersionofDLRGsoverthehalomass functionweightedbytheDLRGhalooccupationdistribution(Eq.[11]).Thedifferentcurvesarefordifferentredshifts. where∆r =v /aH(z)andtheprefactorisdeterminedsoastosatisfy ∞ d∆r′ R(∆r ;M)=1. k k −∞ k k TheFouriertransformoftheredshift-spaceradialprofile(seeEq.[6R])canbeexpressedas rvir σ2 (r,M)k2µ2 p˜ (k ,k ;M) 4πr2drp (r;M)exp v,off , (18) s,off ⊥ k ≃Z off (cid:20)− 2a2H2(z) (cid:21) 0 wherek = k2 +k2andtheexponentialfunctionaboveistheFouriertransformofEq.(17).Weagainnotethat,exactlyspeaking,p˜ ⊥ k s,off q alsodependsontheFouriertransformofthereal-spaceradialprofilep˜ asimpliedbyEq(6),butweusep˜ 1atlargelengthscalesof off off ≃ interest,muchlargerthanthecharacteristicoffsetoftheDLRGfromthehalocenter.Alsonotethat,forthelimitp δ (r),p˜ 1 off D s,off → → asσ 0atr 0. v,off → → Hencetheredshift-spacepowerspectrumofDLRGs(seeEq.[8])canbecomputedforagivencosmologicalmodelbyinsertingEq.(18) into 2 1 dn Ps,DLRG(k,µ) = (cid:20)n¯DLRG Z dM dMb(M)NHOD(M)p˜s,off(k,µ;M)(cid:21) PsN,mL(k,µ). (19) Atverylargelengthscales(orverysmallk’s)theredshift-spacepowerspectrumcanbeapproximatedas σ2 k2µ2 Ps,DLRG(k,µ)≈¯b2(cid:20)1− av2,oHff2(z) (cid:21)PsN,mL(k,µ), (20) whereσ2 isthevelocitydispersionaveragedoverthehalomassfunctionweightedwiththeDLRGHOD: v,off 1 dn σ2v,off ≡ ¯bn¯DLRG Z dMdMb(M)NHOD(M)σv2,off(M). (21) Aswewillshowbelow,theapproximation(20)isnotaccurateatk > 0.15h/Mpc. ThusthekeyquantitycharacterizingtheFoGeffectonDLRG∼powerspectrumisthehalo-massaveragedvelocitydispersion, σ2 . v,off Table1givesthevaluesfordifferentredshiftsassumingourfiducialmodelparameters.TheDLRGvelocitydispersionisalsocomparedwith thatofdarkmatterwithinthesamehaloshostingDLRGs.ItcanbefoundthatthetypicalFoGsuppressionscale,estimatedasσ /(aH), v,off isofscalesof5h−1Mpc.Therefore,evenatlargelengthscalesk 0.1hMpc−1,whichisemployedintheliteratureinordertoextract ≃ cosmologicalinformation,theFoGeffectsuppressesthepowerspectrumamplitudesbyafactorof0.75(1 [0.1 5]2 0.75)accordingto − × ≃ Eq.(20),asystematiccorrectionthatismuchlargerthanthereportedstatisticalerrorsinmanysurveys(seeAppendix CinReidetal.2010, fordiscussion). Fig.4plotshowthetypicalFoGsuppressionscalechangeswithchangingparametersoftheGaussianandNFWDLRGradialprofiles. FortheGaussianradialprofile,theFoGdisplacementscalehasamaximumscaleofσ /(aH) 5h−1Mpcaroundr r ,asthe v,off off s ∼ ∼ velocitydispersionpeaksatthescaleradius r foranNFWprofile.Ontheotherhand,fortheNFWradialprofile,thedisplacementscale s (cid:13)c 0000RAS,MNRAS000,000–000 Usinggalaxy-galaxyweak lensingmeasurementstocorrect theFinger-of-God 9 Figure5.ThisfigureshowshowdifferentmasshaloscontributetothepowerspectrumsuppressionshowninFig.4.WhilethetypicalDLRGsitsinahalo withmass∼1013M⊙,theFoGinthemoremassivehalosplaysthemostimportantroleinsuppressingthegalaxypowerspectrum.Theleftandrightpanels aretheresultsfortheGaussianandNFWradialprofileprofiles,respectively,whereweassumethefiducialmodelparametersroff =0.3rsandcoff =20, respectively,asinFig.2. hasaweakdependenceonc ,andleadsto 5h−1Mpcoveralltherangeofc .TheredshiftdependenceoftheFoGscaleisweakand off off ∼ changesbyonly 10%. ∼ Fig.5showsthatthemoremassivehalosareresponsibleformostoftheFoGsuppression.WhilethetypicalDLRGsitsinhalosofmass 5 1013h−1M ,theFoGarisesprimarilyfrommoremassivehalosasthesehaloshavelargervelocitydispersions.Here,weagainassume ⊙ × thefiducialmodelparametersoftheFoGeffectsasinFig.2.TheplotshowsthathaloswithmassesM 1014 h−1M haveadominant ⊙ ∼ contributiontotheFoGeffectatredshiftsz =0.35and0.7forboththeGaussianandNFWradialprofiles,whilelessmassivehalosbecome moreimportantathigherredshifts. Thecovariancematrixdescribesstatisticaluncertaintiesinmeasuringtheredshift-spacepowerspectrumfromagivensurvey,andthe correlationsbetween thepower spectra of different wavenumbers. Takahashietal.(2009) show that theassumption of Gaussian errorsis validonscalesofinterest.InthislimitofGaussianerrors,thecovariancematrixhasasimpleform: 2δKδK 1 2 Cov[Ps,DLRG(ki,µa)Ps,DLRG(kj,µb)]= Nmodiej(kaib,µb)hPs,DLRG(ki,µb)+ n¯DLRGi , (22) wherek andµ arethei-thanda-thbinsofwavenumberandcosineangle,respectively,andδK andδK aretheKroneckerdeltafunction: i a ij ab δK =1ifi=jwithinthebinwidth,otherwiseδk =0andsoon.TheKroneckerdeltafunctionsimposethatthepowerspectrumofdifferent ij ij wavenumberbinsareindependent.ThequantityNmode(k ,µ )isthenumberofindependentFouriermodesaroundthebincenteredatk i a i andµ withwidths∆kand∆µ,whichcanberesolvedforagivensurveyvolumeV :Nmode(k ,µ ) = 2πk2∆k∆µV /(2π)3.Herewe a s i a i s assumethefundamentalFouriermodeisdeterminedbythesurveyvolumeask =2π/L,areasonableapproximationforasimplesurvey f geometry. 3 ANGULARPOWERSPECTRUMOFDLRG-GALAXYWEAKLENSING ObservationsofDLRG-galaxylensingmeasuretheradialdistributionofDLRGsinthehalo.Inthissubsection,webrieflyreviewOguri&Takada (2011)discussionofhowtheradialdistributionoftheDLRGsintheirhaloaffectsthegalaxy-galaxylensingobservables. The halos hosting DLRGs distort background galaxy images. By cross-correlating positions of DLRGs on the sky with tangential ellipticitycomponentofbackgroundgalaxyimageswithrespecttothelineconnectingDLRGandbackgroundgalaxy,wecanmeasurethe radiallyaveragedmassdistributionaroundaDLRG(Mandelbaumetal.2006).Whilethisstackinganalysisisusuallydoneinrealspace,we willdescribetheresultsinFourierspaceastheeffectoftheDLRGoffsetsareconvolutioninrealspaceandmultiplicationinFourierspace (Oguri&Takada2011). Since we are interested in small angular scales, we can use the flat-sky approximation (Limber 1954) and use the halo model in Oguri&Takada(2011)tocomputetheangularpowerspectrumofDLRG-galaxylensing(alsoseeTakada&Bridle2007): (cid:13)c 0000RAS,MNRAS000,000–000 10 Hikage,Takada &Spergel C (l)=C1h(l)+C2h(l), (23) γg γg γg whereC1h andC2h arethe1-and2-halotermspectradefinedasfollows.Forthefull-skyexpressionofthelensingpowerspectrum,see γg γg dePutter&Takada(2010).The1-halotermcontributiontogalaxy-galaxylensingarisesfromthemassdistributionwithinonehalothathosts DLRGsandgivesdominantcontributiontothesignalonsmallangularseparations: 1 d2V dn 1 Cγ1gh(l)≡ n¯2DDLRG Z dχdχdΩSDLRG(z)WGL(χ)χ−2Z dM dMNHOD(M)ρ¯m0 (cid:2)MuÑFW(k;M,z)põff(k;M)|k=l/χ+msh,DLRG(cid:3), (24) whereρ¯m0 isthemeanmassdensitytoday, χisthecomoving angulardiameterdistance(whichisgivenas afunctionofredshiftviathe distance-redshiftrelation),WGL(χ)isthelensingefficiencyfunctionforagivensourcegalaxypopulation(seeEq.19inOguri&Takada 2011),andd2V/dχdΩisthevolumeelementintheunitcomovingintervalandtheunitsolidangle;d2V/dχdΩ=χ2foraflatuniverse.The functionSDLRG(z)istheredshiftselectionfunctionofDLRGs.Forsimplicity,weassumeacompleteselectionfunction:SDLRG(z) = 1 withintheredshiftrangeofthesurvey,andotherwiseSDLRG =0.Thequantityn¯2DDLRGisthemeanangularnumberdensityofDLRGinthe redshiftslice:n¯2DDLRG ≡ dχ(d2V/dχdΩ) dM(dn/dM)NHOD(M)SDLRG(z).Thetermdenotedbymsh,DLRG givesthecontribution arisingfromasubhalohosRtingDLRG,andwRewillthroughoutthispaperassumethesubhalomassmsh,DLRG =0.32 1012h−1M⊙asour × fiducialvalue,impliedfromtheresultsinJohnstonetal.(2007). Therearetwocontributionstothe1-haloterminEq.(24):thefirstterminthebracketdescribesthecontributionofthehaloofmass M and the second term describes the contribution of a subhalo hosting the DLRG. For the first term, we assume an NFW profile char- acterizingthedarkmatterdistributionwithinahalo,and uÑFW istheFourier-transform(seeEq.29inOguri&Takada2011). Including off-centeredDLRGsinthegalaxy-galaxylensinganalysisdilutesthemeasuredlensingsignalamplitudesatthesmallscales(Johnstonetal. 2007;Oguri&Takada2011).Thisoff-centeringeffectonthelensingpowerspectrumcanbeincludedbysimplyreplacingthedarkmatter profilewithu˜p˜ ,where p˜ istheFourier-transformedcoefficientsof theDLRGradialprofile(seeEq.[13]).Forthesubhalo contribu- off off tionwesimplyassumethedeltafunctionforthemassprofile,agoodapproximationattherelevantangularscales.Inthislimit,thepower spectrumbehaveslikeawhiteshotnoise. Similarly,the2-halotermcontribution,whichdominatesatlargescales,isgivenas 1 d2V dn l Cγ2gh(l)≡ n¯2D Z dχ dχdΩSDLRG(z)WGL(χ)χ−2(cid:20)Z dMdMb(M)NHOD(M)(cid:21)PmL(cid:18)k= χ;z(cid:19), (25) DLRG wherePL(k)isthelinearmasspowerspectrum.This2-halotermscaleswithhalobias:ifDLRGsareresidingonmoremassivehalosor m equivalentlymorebiasedhalos,the2-halotermhasgreateramplitudes. Toperformparameterforecastsforplannedlensingsurveys,wealsoneedtomodelthelensingpowerspectrumcovariance.Following Takada&Jain (2009), we assume Gaussian errors so that the covariance matrix of the lensing power spectrum is given by a product of samplingvarianceandshotnoiseterms(seeOguri&Takada2011,forthedefinitionofthecovariancematrix). Fig. 6 shows the angular power spectrum of DLRG-galaxy weak lensing expected when cross-correlating DLRGs in redshift slice 0.3 < z < 0.4withbackground galaxyimagesthathaveatypicalredshiftofz 1asexpectedforaSubaru-typeimagingsurvey. The ∼ topdottedcurveshowsthepowerspectrumwhenallDLRGsareateachhalo’scenter,whiletheboldsolidcurvesshowthespectrumwith theoff-centering effectassuming ourfiducial modelsofthe Gaussian(leftpanel) and NFW(rightpanel) radialprofilesasinFig.2.The figure shows that the off-centering effect significantly dilutes the lensing power spectrum amplitudes at angular separations smallerthan the projected offset scale. The dashed curve shows the contribution from DLRG subhalo assuming m = 0.32 1012h−1M . sh,DLRG ⊙ × TheDLRGsubhalocontributionbecomessignificantatl > 2 104,whereanangularscaleofl 104 correspondstotheprojectedscale ∼ × ∼ 300kpcfortheDLRGredshiftofz=0.35foraΛCDMmodel. ∼ Theboxesaroundthecurveshowstatisticaluncertaintiesinmeasuringbandpowersateachmultipolebins,expectedwhenmeasuring theDLRG-galaxylensingforanoverlappingareaof2000squaredegreesbetweenspectroscopicandimagingsurveys.Hereweassumethe depthexpectedforSubaruHSCsurveythatprobesgalaxiesattypicalredshiftsofz 1(seeSection4.1fordetails).Theplotclearlyimplies s ∼ thatcombiningsuchimagingandspectroscopicsurveysallowsustoinfertheoff-centeringeffectatahighsignificance.Wewillbelowgive amorequantitativeestimate. 4 RESULTS Inthissection,weestimatetheabilityofongoingandplannedsurveysforusingtheDLRG-galaxyweaklensingmeasurementstocorrectthe FoGeffectonredshift-spacepowerspectrumofDLRGs. 4.1 Surveyparameters TomodeltheDLRGpowerspectrumavailablefromongoingandupcomingspectroscopicsurveysweassumesurveyparametersthatresemble BOSS,SubaruPFS,andEuclidsurveys.TheexpectedstatisticaluncertaintiesinmeasuringtheDLRGpowerspectrumineachredshiftslice depend on the area coverage (or equivalently the comoving volume) and the number density and bias parameter of DLRGs. For the sky (cid:13)c 0000RAS,MNRAS000,000–000