Using galaxy-galaxy weak lensing measurements to correct the Finger-of-God

0.9 MB·English

by  Chiaki Hikage

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
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2
28January2013
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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-scalestruc-
ture 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)p˜off(k;M)dM′NHOD(M′)p˜off(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)p˜off(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).
≡
Thecoefficientp˜off(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
p˜off(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.TheFouriertransformp˜off(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
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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)Mu˜NFW(k;M,z)p˜off(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 u˜NFW 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