The stellar-to-halo mass relation of GAMA galaxies from 100 square degrees of KiDS weak lensing data

0.64 MB·

by  Edo van Uitert

MNRAS000,000–000(2016) PreprintJanuary27,2016 CompiledusingMNRASLATEXstylefilev3.0
The stellar-to-halo mass relation of GAMA galaxies from 100 square
degrees of KiDS weak lensing data
Edo van Uitert 1,2⋆, Marcello Cacciato3, Henk Hoekstra3, Margot Brouwer3, Cristo´bal Sifo´n3,
Massimo Viola3, Ivan Baldry4, Joss Bland-Hawthorn5, Sarah Brough6, M. J. I. Brown7,
Ami Choi8, Simon P. Driver9,10, Thomas Erben2, Catherine Heymans8, Hendrik Hildebrandt2,
Benjamin Joachimi 1, Konrad Kuijken3, Jochen Liske11, Jon Loveday12, John McFarland13,
Lance Miller14, Reiko Nakajima2, John Peacock8, Mario Radovich15, A. S. G. Robotham16,
6
Peter Schneider2, Gert Sikkema13, Edward N. Taylor17, Gijs Verdoes Kleijn13
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0
2 1UniversityCollegeLondon,GowerStreet,LondonWC1E6BT,UK
2Argelander-Institutfu¨rAstronomie,AufdemHu¨gel71,53121Bonn,Germany
n
3LeidenObservatory,LeidenUniversity,NielsBohrweg2,NL-2333CALeiden,TheNetherlands
a
J 4AstrophysicsResearchInstitute,LiverpoolJohnMooresUniversity,IC2,LiverpoolSciencePark,146BrownlowHill,LiverpoolL35RF,UK
5SydneyInstituteforAstronomy,SchoolofPhysicsA28,UniversityofSydney,NSW2006,Australia
5
6AustralianAstronomicalObservatory,POBox915,NorthRyde,NSW1670,Australia
2
7SchoolofPhysicsandAstronomy,MonashUniversity,Clayton,Victoria3800,Australia

8ScottishUniversitiesPhysicsAlliance,InstituteforAstronomy,UniversityofEdinburgh,RoyalObservatory,BlackfordHill,EdinburghEH93HJ,UK
]
A 9InternationalCentreforRadioAstronomyResearch(ICRAR),TheUniversityofWesternAustralia,35StirlingHighway,Crawley,WA6009,Australia
10ScottishUniversitiesPhysicsAlliance,SchoolofPhysics&Astronomy,UniversityofStAndrews,NorthHaugh,StAndrewsKY169SS,UK
G 11HamburgerSternwarte,Universita¨tHamburg,Gojenbergsweg112,21029Hamburg,Germany
. 12AstronomyCentre,DepartmentofPhysicsandAstronomy,UniversityofSussex,Falmer,BrightonBN19QH,UK
h 13KapteynAstronomicalInstitute,UniversityofGroningen,POBox800,NL-9700AVGroningen,theNetherlands
p 14DepartmentofPhysics,OxfordUniversity,KebleRoad,OxfordOX13RH,UK
o- 15INAF-OsservatorioAstronomicodiPadova,viadell‘Osservatorio5,I-35122Padova,Italy
r 16InternationalCentreforRadioAstronomyResearch,UniversityofWesternAustralia,35StirlingHighway,Crawley,WA6009,Australia
t 17SchoolofPhysics,TheUniversityofMelbourne,Parkville,VIC3010,Australia
s
a
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January27,2016
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ABSTRACT
7
We study the stellar-to-halo mass relation of central galaxies in the range
6
0 9.7<log10(M∗/h−2M⊙)<11.7 and z < 0.4, obtained from a combined analysis of
. the Kilo Degree Survey (KiDS) and the Galaxy And Mass Assembly (GAMA) survey.
1 We use ∼100deg2 of KiDS data to study the lensing signal around galaxies for which
0
spectroscopicredshiftsandstellarmassesweredeterminedbyGAMA.Weshowthatlensing
6
alone results in poor constraints on the stellar-to-halo mass relation due to a degeneracy
1
between the satellite fraction and the halo mass, which is lifted when we simultaneously
:
v fit the stellar mass function. At M∗ > 5 × 1010h−2M⊙, the stellar mass increases with
Xi halo mass as ∼Mh0.25. The ratio of dark matter to stellar mass has a minimum at a halo
r mass of 8× 1011h−1M⊙ with a value of Mh/M∗ = 56+−1160 [h]. We also use the GAMA
a groupcataloguetoselectcentralsandsatellitesingroupswithfiveormoremembers,which
trace regionsin space where the local matter density is higher than average, and determine
for the first time the stellar-to-halo mass relation in these denser environments.We find no
significantdifferencescomparedtotherelationfromthefullsample,whichsuggeststhatthe
stellar-to-halomassrelationdoesnotvarystronglywith localdensity.Furthermore,we find
that the stellar-to-halo mass relation of central galaxies can also be obtained by modelling
the lensing signal and stellar mass function of satellite galaxies only, which shows that the
assumptionstomodelthesatellitecontributioninthehalomodeldonotsignificantlybiasthe
stellar-to-halomassrelation.Finally,weshowthatthecombinationofweaklensingwiththe
stellarmassfunctioncanbeusedtotestthepurityofgroupcatalogues.
Keywords: gravitationallensing-darkmatterhaloes
(cid:13)c 2016TheAuthors
⋆ [email protected]
2 Edo vanUitertet al.
1 INTRODUCTION isthemain driver behind thevariationof thestellar-to-halomass
relation, and that the change in colour is simply a consequence
Galaxiesformandevolveindarkmatterhaloes.Largerhaloesat- of quenching, as was already hypothesized in Mandelbaumetal.
tractonaveragemorebaryonsandhostlarger,moremassivegalax- (2006). This scenario could be verified by measuring the stellar-
ies.Theexactrelationbetweenthebaryonicpropertiesofgalaxies to-halo mass relationfor galaxiesin different environments. This
andtheirdarkmatterhaloesiscomplex,however,asvariousastro- requiresagalaxy catalogue including stellar massesand environ-
physicalprocessesareinvolved.TheseincludesupernovaandAGN mentalinformation,plusamethodtomeasuremasses.Weakgrav-
feedback(seee.g.Benson2010),whoserelativeimportancesgen- itational lensing offers a particularly attractive way of measuring
erallydependonhalomassinawaythatisnotaccuratelyknown, averagehalomassesofsamplesofgalaxies,asitmeasuresthetotal
but environmental effects also play an important role. Measuring projected matter density along the line of sight, without any as-
projections of these relations, such as the stellar-to-halomass re- sumptionabout thephysical stateof thematter,out toscalesthat
lation,helpsustogaininsightintotheseprocessesandtheirmass areinaccessibletoothergravitationalprobes.
dependences,andprovidesvaluablereferencesforcomparisonsfor Theseconditionsareprovidedbycombiningtwosurveys:the
numerical simulationsthatmodel galaxyformationand evolution KiloDegreeSurvey(KiDS)andtheGalaxyAndMassAssembly
(e.g.Munshietal.2013;Kannanetal.2014). (GAMA) survey. GAMA is a spectroscopic survey for galaxies
Thestellar-to-halomassrelationhasbeenstudiedwithava- with r < 19.8 that is highly complete (Driveretal. 2009, 2011;
rietyofmethods,includingindirecttechniquessuchasabundance Liskeetal.2015),facilitatingtheconstructionofareliablegroup
matching(e.g.Behroozietal.2010;Mosteretal.2013)orgalaxy catalogue (Robothametal. 2011). GAMA is completely covered
clustering(e.g.Wakeetal.2011;Guoetal.2014),whichcanonly bytheKiDSsurvey(deJongetal.2013),anongoingweaklensing
be interpreted within a cosmological framework (e.g. ΛCDM). surveywhichwilleventuallycover1500deg2 ofskyintheugri-
Satellitekinematicsofferadirectwaytomeasurehalomass(e.g. bands.Inthiswork,westudythelensingsignalaround 100000
Norbergetal. 2008; Wojtak&Mamon 2013), but this approach GAMAgalaxiesusingsourcesfromthe 100deg2ofKi∼DSimag-
∼
is relatively expensive as it requires spectroscopy for large sam- ing data overlapping with the GAMA survey from the first and
plesofsatellites.Weakgravitationallensingoffersanotherpower- secondpubliclyavailableKiDS-DR1/2datarelease(Kuijkenetal.
ful method that enables average halo mass measurements for en- 2015).
sembles of galaxies (e.g. Mandelbaumetal. 2006; Velanderetal. The outline of this paper is as follows. In Sect. 2 we
2014). Recently, various groups have combined different probes describe the data reduction and lensing analysis, and intro-
(e.g. Leauthaudetal. 2012; Couponetal. 2015), which enable duce the halo model that we fit to our data. The stellar-to-
more stringent constraints on the stellar-to-halo mass relation by halo mass relation of the full sample is presented in Sect. 3.
breakingdegeneraciesbetweenmodelparameters.Acoherentpic- In Sect. 4, we measure this relation for centrals and satel-
tureisemergingfromthese studies: thestellar-to-halomass rela- lites in groups with a multiplicity Nfof > 5. We conclude in
tion of central galaxies can be described by a double power law, Sect. 5. Throughout the paper we assume a Planck cosmology
with a transition at a pivot mass where the accumulated star for- (PlanckCollaborationetal.2014)withσ8 =0.829,ΩΛ = 0.685,
mationhasbeenmostefficient.Athighermasses,AGNfeedback ΩM = 0.315, Ωbh2 =0.02205 and ns = 0.9603. Halo masses
isthought to suppress star formation, whilst at lower masses, su- aredefinedasM ≡ 4π(200ρ¯m)R2300/3, withR200 theradiusof
pernovafeedbacksuppressesit.Thispivotmasscoincideswiththe aspherethatencompassesanaveragedensityof200timestheco-
locationwherethestellarmassgrowthingalaxiesturnsfrombeing movingmatterdensity,ρ¯m=8.74 1010h2M⊙/Mpc3,atthered-
·
in-situdominatedtomergerdominated(Robothametal.2014). shiftofthelens.Alldistancesquotedareincomoving(ratherthan
Most galaxies can be roughly divided into two classes, i.e. physical)unitsunlessexplicitlystatedotherwise.
red, ‘early-types’ whose star formation has been quenched, and
blue, ‘late-types’ that are actively forming stars. These are also
crudely related to different environments and morphologies. The 2 ANALYSIS
differences in their appearances point at different formation his-
tories. Their stellar-to-halo mass relations may contain informa- 2.1 KiDS
tionoftheunderlyingphysicalprocessesthatcausedthesediffer-
To study the weak-lensing signal around galaxies, we use the
ences. Hence it is natural to measure the stellar-to-halo mass re-
shape and photometric redshift catalogues from the Kilo Degree
lationsofredandbluegalaxiesseparately(e.g.Mandelbaumetal. Survey (KiDS; Kuijkenetal. 2015). KiDS is a large optical
2006;vanUitertetal.2011;Moreetal.2011;Velanderetal.2014; imaging survey which will cover 1500 deg2 in u, g, r and i to
Wojtak&Mamon 2013; Tinkeretal. 2013; Hudsonetal. 2015). magnitudelimitsof24.2,25.1,24.9and23.7(5σina2′′aperture),
The main result of the aforementioned studies is that at stellar respectively.Photometryin5infraredbandsofthesameareawill
massesbelow 1011M⊙,redandbluegalaxiesthatarecentrals become available from the VISTA Kilo-degree Infrared Galaxy
∼
(i.e.not asatelliteof alarger system) reside inhaloes withcom- (VIKING)survey(Edgeetal.2013).Theopticalobservationsare
parable masses. At higher stellar masses, the halo masses of red carriedoutwiththeVLTSurveyTelescope(VST)usingthe1deg2
galaxies are larger at low redshift, but smaller at high redshift at
imager OmegaCAM, which consists of 32 CCDs of 2048 4096
agivenstellarmass.Tinkeretal.(2013)interpretthissimilarityin pixels each and has a pixel size of 0.214′′. In this paper×, weak
halomassatthelow-massendasevidencethattheseredgalaxies lensing resultsare based on observations of 109 KiDS tiles1 that
haveonlyrecentlybeenquenched;thedifferenceatthehigh-mass
overlapwiththeGAMAsurvey,andhavebeencoveredinallfour
end isinterpreted asevidence that bluegalaxies havearelatively
optical bands and released toESOaspart of the firstand second
largerstellarmassgrowthinrecenttimes,comparedtoredgalax-
KiDS-DR1/2 data releases. The effective area after accounting
ies.
As red galaxies preferentially reside in dense environments
such as galaxy groups and clusters, it appears that local density 1 Atileisanobservationofapointingonthesky
MNRAS000,000–000(2016)
Thestellar-to-halomassrelationofGAMAgalaxiesfrom100squaredegreesofKiDSweaklensingdata 3
for masks andoverlaps between tilesis75.1 square degrees. The
image reduction and the astrometric and photometric reduction
usetheASTRO-WISEpipeline(McFarlandetal.2013);detailsof
the resulting astrometric and photometric accuracy can be found
in deJongetal. (2015). Photometric redshifts have been derived
withBPZ(Ben´ıtez2000;Hildebrandtetal.2012),aftercorrecting
themagnitudes intheoptical bands for seeing differencesby ho-
mogenisingthephotometry(Kuijkenetal.2015).Thephotometric
redshifts are reliable in the redshift range 0.005 < z < 1.2,
B
withz beingthelocationwheretheposteriorredshiftprobability
B
distribution has its maximum, and have a typical outlier rate of
<5% at z < 0.8 and a redshift scatter of 0.05 (see Sect. 4.4
B
in Kuijkenetal. 2015). In the lensing analysis, we use the full
photometricredshiftprobabilitydistributions.
Shearmeasurementsareperformedinther-band,whichhas
beenobservedunderstringentseeingrequirements(< 0.8′′).The
r-bandisseparatelyreducedwiththewell-testedTHELIpipeline
(Erbenetal. 2005, 2009), following procedures very similar
to the analysis of the CFHTLS data as part of the CFHTLenS
collaboration(Heymansetal.2012;Erbenetal.2013).Theshape
measurements are performed with lensfit (Milleretal. 2007;
Kitchingetal. 2008), using the version presented in Milleretal.
(2013). We apply the same calibration scheme to correct for Figure1.SpectroscopicredshiftversusstellarmassoftheGAMAgalaxies
multiplicative bias as the one employed in CFHTLenS; the intheKiDSoverlap.Thedensitycontoursaredrawnat0.5,0.25and0.125
accuracy of the correction is better than the current statistical timesthemaximumdensityinthisplane.ThetotaldensityofGAMAgalax-
uncertainties, as is shown by a number of systematics tests in iesasafunctionofredshiftandstellarmassareshownbythehistograms
Kuijkenetal.(2015).Shearestimatesareobtainedusingallsource onthex-andy-axes,respectively.Thedashedlinesindicatethemassbins
galaxiesinunmaskedareaswithanon-zerolensfitweightandfor ofthelenses.
whichthepeakoftheposteriorredshiftdistributionisintherange
0.005 < z < 1.2. The corresponding effective source number
B remainderofthiswork.Thedistributionofstellarmassversusred-
density is 5.98 arcmin−2 (using the definition of Heymansetal.
shiftofallGAMAgalaxiesintheKiDSfootprintisshowninFig.1.
(2012),whichdiffersfromtheoneadoptedinChangetal.(2013),
ThisfigureshowsthattheGAMAcataloguecontainsgalaxieswith
asdiscussedinKuijkenetal.2015).
redshiftsuptoz 0.5.Furthermore,bright(massive)galaxiesre-
≃
sideathigherredshifts,asexpectedforaflux-limitedsurvey.Note
that theapparent lack of galaxies moremassive than afew times
2.2 GAMA 1010h−2M⊙ atz < 0.2isaconsequence of plottingredshift on
thehorizontalaxisinsteadofbinsofequalcomovingvolume.The
The Galaxy And Mass Assembly (GAMA) survey (Driveretal. binsatlowredshiftcontainlessvolumeandthereforehavefewer
2009, 2011; Liskeetal. 2015) is a highly complete optical spec- galaxies(foraconstantnumberdensity).Itisnotaselectioneffect.
troscopicsurveythattargetsgalaxieswithr < 19.8overroughly We use the group properties of the G3C catalogue
286 deg2. In this work, we make use of the G3Cv7 group cata- (Robothametal. 2011) to select galaxies in dense environments.
logueandversion16ofthestellarmasscatalogue,whichcontain Groups are found using an adaptive friends-of-friends algorithm,
180000objects,dividedintothreeseparate12 5deg2 patches linkinggalaxiesbasedontheirprojectedandline-of-sightsepara-
∼ ×
thatcompletelyoverlapwiththenorthernstripeofKiDS.Weuse tions.Thealgorithmhasbeentestedonmockcatalogues, andthe
the subset of 100000 objects that overlaps with the 75.1 deg2 globalproperties,suchasthetotalnumberofgroups,arewellre-
∼
fromtheKiDS-DR1/2. covered. Version 7 of the group catalogue, which we use in this
Stellar masses of GAMA galaxies have been estimated in work, consists of nearly 24000 groups with over 70000 group
Tayloretal. (2011). In short, stellar population synthesis models members.Thecataloguecontainsgroupmembershi∼plistsandvari-
from Bruzual&Charlot (2003) are fit to the ugriz-photometry ousestimatesforthegroupcentre,aswellasgroupvelocitydisper-
fromSDSS.NIRphotometryfromVIKINGisusedwhentherest- sions, group sizesand estimatedhalo masses. Welimitourselves
framewavelength islessthan 11000 A˚.Toaccount for fluxout- togroupswithamultiplicityN >5,becausegroupswithfewer
fof
sidetheAUTOapertureusedfortheSpectralEnergyDistributions members are more strongly affected by interlopers, as a compar-
(SEDs), an aperture correction is applied using the fluxscale pa- isonwithmockdatahasshown (Robothametal.2011).Werefer
rameter.Thisparameterdefinestheratiobetweenr-band(AUTO) to these groups as ‘rich’ groups. We assume that the brightest2
aperture flux and the total r-band flux determined from fitting a group galaxy is the central galaxy, whilst fainter group members
Se´rsicprofileoutto10effectiveradii(Kelvinetal.2014).Thestel- arereferredtoassatellites.Analternativeprocedure toselectthe
lar masses do not include the contribution from stellar remnants.
Thestellarmasserrorsare 0.1dexandaredominatedbyamag-
∼
nitudeerrorfloorof0.05mag,whichisaddedinquadraturetoall 2 ThisisbasedonSDSSr-bandPetrosianmagnitudeswithaglobal(k+
magnitude errors, thus allowing for systematic differences in the e)-correction(seeRobothametal.2011).Duetovariationsinthemass-to-
photometrybetweenthedifferentbands.Therandomerrorsonthe lightratio,itoccasionallyhappensthatasatellitehasalargerstellarmass
stellarmassesarethereforeevensmaller,andweignoretheminthe thanthecentral.
MNRAS000,000–000(2016)
4 Edo vanUitertet al.
centralgalaxyistoiterativelyremovegroupmembersthatarefur- Sec.3.4ofViolaetal.(2015).Wehavealsocomputedthecovari-
thest away from the group centre of light. As the two definitions ancematrixusingbootstrappingtechniquesandfoundverysimilar
onlydifferforafewpercentofthegroupsandthelensingsignals resultsintheradialrangeofinterest.
are statistically indistinguishable (see Appendix A of Violaetal. Wegroup GAMA galaxies instellar massbins and measure
2015), we do not investigate this further and adopt the brightest theiraveragelensingsignals.Thebinrangeswerechosenfollowing
groupgalaxyasthecentralthroughout. Toverifythatthese‘rich’ twocriteria.Firstly,weaimedforaroughlyequallensingsignal-to-
groupstracedenseenvironments,wematchtheG3Ccatalogueto noiseratioof 15perbin.Secondly,weadoptedamaximumbin
∼
theenvironmentalclassificationcatalogueofEardleyetal.(2015), widthof0.5dex.Todeterminethesignal-to-noiseratio,wefitteda
whousesatidaltensorprescriptiontodistinguishbetweenfourdif- singularisothermalsphere(SIS)totheaveragelensingsignaland
ferentenvironments:voids,sheets,filamentsandknots.Usingthe determined the ratio of the amplitude of the SIS to its error. The
classificationthatisbasedonthe4h−1Mpcsmoothing scale,we adoptedbinrangesarelistedinTable1,aswellasthenumberof
findthat76%ofthecentralsofgroupswithN >5resideinfil- lensesandtheiraverageredshift;theaverage∆ΣisshowninFig.
fof
amentsandknots,comparedto49%ofthefullGAMAcatalogue, 2. We note, however, that our conclusions do not depend on the
whichshowstheN >5groupsformacrudetracerofdensere- choiceofbinning.
fof
gions.
Note that both the stellar mass catalogue and the GAMA
group catalogue were derived with slightly different cosmologi- 2.4 Thehalomodel
calparameters:Tayloretal.(2011)used(Ω ,Ω ,h)=(0.7,0.3,0.7)
Λ M The halo model (Seljak 2000; Cooray&Sheth 2002) has be-
and Robothametal. (2011) used (Ω ,Ω ,h)=(0.75,0.25,1.0) in
Λ M come a standard method to interpret weak lensing data. The im-
order to match the Millennium Simulation mocks. We accounted
plementation we employ here is similar to the one described
forthedifferenceinh,butnotinΩ andΩ ,becausethelensing
Λ M in vandenBoschetal. (2013) and has been successfully ap-
signalatlowredshiftdependsonlyweaklyontheseparametersand
plied to weak lensing measurements in Cacciatoetal. (2014);
thisshouldnotimpactourresults.
vanUitertetal. (2015), and to weak lensing and galaxy cluster-
The current lensing catalogues in combination with these
ingdatainCacciatoetal.(2013).Weprovideadescriptionofthe
GAMA catalogues have already been analysed by Violaetal.
modelhere,aswehavemadeanumberofmodifications.
(2015), where the main focus was GAMA group properties, and
Inthehalomodel,allgalaxiesareassumedtoresideindark
bySifo´netal.(2015),wherethemassesofsatellitesingroupswere
matter haloes. Using a prescription for the way galaxies occupy
derived.Here,weaimatabroaderscope,aswemeasurethestellar-
darkmatterhaloes,aswellasforthematterdensityprofile,abun-
to-halomassrelationover twoordersof magnitudeinhalomass.
dance and clustering of haloes, one can predict the surface mass
Studyingthecentralsandsatellitesin‘rich’groupssuppliesuswith
density(andthusthelensingsignal)aroundgalaxiesinastatistical
thefirstobservationallimitsonwhetherthestellar-to-halomassre-
manner:
lationchangesindenseenvironments.
ωS
Σ(R)=ρ¯ ξ (r)dω, (3)
m gm
Z0
2.3 Lensingsignal withξgm(r) the galaxy-matter cross-correlation, ω thecomoving
distance from the observer and ωS the comoving distance to the
Weak lensing induces a small distortion of the images of back-
source. For small separations, R ωLθ, with ωL the comoving
groundgalaxies.Sincethelensingsignalofindividualgalaxiesis ≈
distance to the lens and θ the angular separation from the lens.
generally tooweaktobedetected due tothelow number density
The three-dimensional comoving distance r is related to ω via
of background galaxiesinwide-fieldsurveys, it iscommon prac- r2 = (ωL θ)2+(ω ωL)2.Theintegraliscomputedalongthe
ticetoaverage the signal around many (similar)lens galaxies. In · −
lineofsight.
the regime where the surface mass density is sufficiently small,
Asthecomputationofξ (r)generallyrequiresconvolutions
gm
the lensing signal can be approximated by averaging the tangen-
in real space, it is convenient to express the relevant quantities
tialprojectionoftheellipticitiesofbackground (source)galaxies,
in Fourier-space where these operations become multiplications.
thetangentialshear:
ξ (r)isrelatedtothegalaxy-matterpowerspectrum,P (k,z),
gm gm
∆Σ(R) via
γ (R)= , (1)
h ti Σcrit ξ (r,z)= 1 ∞P (k,z)sinkrk2dk, (4)
with∆Σ(R)=Σ¯(<R) Σ¯(R)thedifferencebetweenthemean gm 2π2 Z0 gm kr
−
projectedsurfacemassdensityinsideaprojectedradiusRandthe withkthewavenumber.Onsmallphysicalscales,themaincontri-
surfacedensityatR,andΣ thecriticalsurfacemassdensity: butiontoP (k,z)comesfromthehaloinwhichagalaxyresides
crit gm
(theone-haloterm),whilstonlargephysicalscales,themaincon-
Σ = c2 DS , (2) tributioncomesfromneighbouringhaloes(thetwo-haloterm).Ad-
crit 4πGDLDLS ditionally,thehalomodeldistinguishesbetweentwogalaxytypes,
withDLandDStheangulardiameterdistancefromtheobserverto i.e.centrals and satellites.Centralsreside inthecentre of a main
thelensandsource,respectively,andDLSthedistancebetweenthe halo, whilst satellites reside in subhaloes that are embedded in
lensandsource.Foreachlens-sourcepairwecompute1/Σ by largerhaloes.Theirpowerspectraaredifferentandcomputedsep-
crit
integratingovertheredshiftprobabilitydistributionofthesource. arately.Henceonehas
The actual measurements of the excess surface density pro- P (k)=P1h(k)+P1h(k)+P2h(k)+P2h(k), (5)
files are performed using the same methodology outlined in Sec. gm cm sm cm sm
3.3of Violaetal.(2015).Thecovariance betweentheradial bins with P1h(k) (P1h(k)) the one-halo contributions from centrals
cm sm
ofthelensingmeasurementsisderivedanalytically,asdiscussedin (satellites), and P2h(k) (P2h(k)) the corresponding two-halo
cm sm
MNRAS000,000–000(2016)
Thestellar-to-halomassrelationofGAMAgalaxiesfrom100squaredegreesofKiDSweaklensingdata 5
Figure2.ExcesssurfacemassdensityprofileofGAMAgalaxiesmeasuredasafunctionofprojected(comoving)separationfromthelens,selectedinvarious
stellarmassbins,measuredusingthesourcegalaxiesfromKiDS.Thedashedredlineindicatesthebest-fithalomodel,obtainedfromfittingthelensingsignal
only.Thesolidgreenlineisthebest-fithalomodelforthecombinedfittotheweaklensingsignalandthestellarmassfunction,theorangeareaindicatesthe1σ
modeluncertaintyregimeofthisfit.Thestellarmassrangesthatareindicatedcorrespondtothelog10ofthestellarmassesandareinunitslog10(h−2M⊙).
Table1.Numberoflensesandmeanlensredshiftofalllenssamplesusedinthiswork.Thestellarmassrangesthatareindicatedcorrespondtothelog
10
ofthestellarmassesandareinunitslog10(h−2M⊙).The‘All’samplecontainsallGAMAgalaxiesthatoverlapwithKiDS-DR1/2,whilst‘Cen’and‘Sat’
referstothesamplesthatonlycontainthecentralsandsatellitesinGAMAgroupswithamultiplicityNfof >5.
M1 M2 M3 M4 M5 M6 M7 M8
[9.39,9.89] [9.89,10.24] [10.24,10.59] [10.59,11.79] [10.79,10.89] [10.89,11.04] [11.04,11.19] [11.19,11.69]
Nlens hzi Nlens hzi Nlens hzi Nlens hzi Nlens hzi Nlens hzi Nlens hzi Nlens hzi
All 15819 0.17 19175 0.21 24459 0.25 11475 0.29 3976 0.31 3885 0.32 1894 0.34 1143 0.35
Cen(Nfof >5) 15 0.08 55 0.12 185 0.16 242 0.18 185 0.19 276 0.21 241 0.23 209 0.26
Sat(Nfof >5) 1755 0.14 2392 0.18 3002 0.22 1267 0.26 388 0.27 343 0.27 138 0.29 65 0.32
terms. We follow the notation of vandenBoschetal. (2013) and describesthepowerspectrumofhaloesofmassM andM ,which
1 2
writethiscompactlyas: containsthelarge-scalehalobiasb (M )fromTinkeretal.(2010).
h h
Plin(k,z) is the linear matter power spectrum. We employ the
m
P1h(k,z)= (k,M ,z) (k,M ,z)n (M ,z)dM , (6) transfer function of Eisenstein&Hu (1998), which properly ac-
xy Hx h Hy h h h h
Z countsfortheacousticoscillations.Furthermore,weuse
Px2yh(k,z)=Z dM1Hx(k,M1,z)nh(M1,z) Hm(k,Mh,z)= Mρ¯mhuh(k|Mh,z), (8)
dM (k,M ,z)n (M ,z)Q(kM ,M ,z), (7)
2Hy 2 h 2 | 1 2 e
Z with Mh the halo mass, and uh(kMh,z) the Fourier transform
|
where x and y are either c (for central), s (for satellite), or m of the normalised density distribution of the halo. We assume
(for matter), nh(Mh,z) isthe halomass function of Tinkeretal. thatthedensitydistributionfolleowsaNavarro-Frenk-White(NFW;
(2010), and Q(kM ,M ,z)=b (M ,z)b (M ,z)Plin(k,z) Navarroetal. 1996) profile, with a mass-concentration relation
| 1 2 h 1 h 2 m
MNRAS000,000–000(2016)
6 Edo vanUitertet al.
fromDuffyetal.(2008): The contribution from the central galaxies is modelled as a
log-normaldistribution:
M −0.081
cwdhmer=e ffccoonncc×is1th0e.14no(cid:18)rmMapliizhvaotti(cid:19)on, whic(h1i+s za)−fr1e.e01p,arameter(9in) Φc(M∗|Mh)= exph−√(lo2gπ10lMn(∗1−0l2)oσgσc12c0MM∗c∗(Mh))2i, (15)
the fit, and Mpivot = 2 1012h−1M⊙. Note that the choice
forthisparticularparametr×isationisnotveryimportant,asessen- whereσcisthescatterinlogM∗atafixedhalomass.Forsimplic-
ity,weassumethatitdoesnotvarywithhalomass,assupported
tiallyallmass-concentrationrelationsfromtheliteraturepredicta
by the kinematics of satellite galaxies in the SDSS (Moreetal.
weakdependenceonhalomass.Furthermore,thescalingwithred-
shift c (1+z)−1 is motivated by analytical treatments of 2009,2011),bycombininggalaxyclustering,galaxy-galaxylens-
dm
∝ ingandgalaxyabundances (Cacciatoetal.2009;Leauthaudetal.
haloformation(seee.g.Bullocketal.2001).Itisworthmention-
2012) and by SDSS galaxy group catalogues (Yangetal. 2008).
ingthatmorecomplexredshiftdependencesareexpected(seee.g.
Mc representsthemeanstellarmassofcentralgalaxiesinahalo
Mun˜oz-Cuartasetal.2011)butthose deviationsareonly relevant ∗
ofmassM ,parametrisedbyadoublepowerlaw:
atredshiftslargerthanone,wellbeyondthehighestlensredshiftin h
thissFtuodryc.entralsandsatellites,wehave M∗c(Mh)=M∗,0[1+((MMhh//MMhh,,11))β]β11−β2 , (16)
N M
Hx(k,Mh,z)= hn¯xx|(z)hiux(k|Mh). (10) βwith(βM)ht,h1eapcohwaerracltaewrisstliocpmeaaststhsecalloew, M-(h∗i,g0h-a)nmoramssaelinzda.tioTnhiasnids
1 2
We set uc(kMh) = 1, i.ee., we assume that all central galax- thestellar-to-halomassrelationofcentralgalaxiesweareafter.
|
ies are located at the centre of the halo. We adopt this choice in For theCSMFof thesatellitegalaxies, weadopt amodified
order toelimit the number of free parameters in the model; addi- Schechterfunction:
tionally, lensing alone does not provide tight constraints on the
masiesdcennotrrimngaldisiasttiroibnutoiofnt.hTehmisamsso-cdoenllcinengtrcahtiooincercealantiloenad(steoeaeb.gi-. Φs(M∗|Mh)= Mφs∗s (cid:18)MM∗∗s(cid:19)αsexp"−(cid:18)MM∗∗s(cid:19)2#, (17)
vanUitertetal. 2015; Violaetal. 2015) but it does not bias the
whichdecreasesfasterthanaSchechterfunctionatthehigh-stellar
halomasses(vanUitertetal.2015)orthestellar-to-halomassrela-
massend. Galaxygroup catalogues show thatthesatellitecontri-
tion.Furthermore,weassumeu (kM ,z)=u (kM ,z),hence
s | h h | h bution to the total CSMF falls off around the mean stellar mass
thedistributionofsatellitesfollowsthedarkmatter.Thisisarea-
ofthecentralgalaxyforagivenhalomass(e.g.Yangetal.2008).
sonable assumption, given theelargediscrepanecies inthereported ThusoneexpectsthecharacteristicmassofthemodifiedSchechter
trends in the literature, which range from satellites being either function, Ms, to follow Mc. Inspired by Yangetal. (2008), we
∗ ∗
moreorlessconcentratedthanthedarkmatter(seee.g.Wangetal. assumethatMs(M ) = 0.56Mc(M ).Forthenormalizationof
∗ h ∗ h
2014,andthediscussiontherein).
Φs(M∗ Mh)weadopt
WespecifythehalooccupationstatisticsusingtheConditional |
StellarMassFunction(CSMF),Φ(M∗|Mh)dM∗,whichdescribes log10[φs(Mh)]=b0+b1×log10M13, (18)
the average number of galaxies with stellar masses in the range
M∗ ±dM∗/2 that reside inahalo of massMh. The occupation tweritsh.WMe13tes=tthMehs/en(1si0t1iv3ihty−o1fMou⊙r)r.ebsu0,ltbs1t,oatnhdeαloscaatrieonfroeefMparsa,maned-
numbersrequiredforthecomputationofthegalaxy-matterpower ∗
totheadditionofaquadraticterminEq.(18),inAppendixB.We
spectrafollowfrom
findthatourresultsarenotsignificantlyaffected.
M∗,2 Weassignamasstothesubhaloesinwhichthesatellitesre-
hNx|Mhi(M∗,1,M∗,2)= Φx(M∗|Mh)dM∗, (11) sideusingthesamerelationthatweuseforthecentrals(Eq.16).
ZM∗,1
For every stellar mass bin, we compute the average mass of the
where‘x’referstoeither‘c’(centrals)or‘s’(satellites),andM∗,1 main haloes in which the satelliteresides, and multiply this with
andM∗,2 indicatetheextremesofastellarmassbin.Theaverage aconstantfactor,f ,afreeparameterwhoserangeislimitedto
sub
numberdensityofthesegalaxiesisgivenby:
valuesbetween0and1.Inthisway,wecancrudelyaccountforthe
strippingofthedarkmatterhaloesofsatellites.Givenourlimited
n¯x(z)= Nx Mh (M∗,1,M∗,2)nh(Mh,z)dMh, (12) knowledgeofthedistributionofdarkmatterinsatellitegalaxies,we
h | i
Z assumethatitisdescribedbyanNFWprofile,whichprovideade-
andthesatellitefractionfollowsfrom
centdescriptionofthemassdistributionofsubhaloesintheMillen-
niumsimulation(PastorMiraetal.2011).Weusethesamemass-
Ns Mh (M∗,1,M∗,2)nh(Mh)dMh
fs(M∗,1,M∗,2)= h | i n¯ (z)+n¯ (z) . concentration relationasfor thecentrals.Thestatisticalpower of
R c s ourmeasurementsisnotsufficienttoadditionallyfitforatrunca-
(13)
tionradius(seeSifo´netal.2015).
Thestellarmassfunctionisgivenby: The above prescription provides us with the lensing signal
fromcentralsandsatellites,withseparatecontributionsfromtheir
ϕ(M∗,1,M∗,2)= [ Nc Mh + Ns Mh ]nh(Mh)dMh, (14) one-halo and two-halo terms. At small projected separations, the
h | i h | i contribution of the baryonic component of the lenses themselves
Z
where N M and N M arecomputed withEq. (11) using becomes relevant. We model this using a simple point mass ap-
c h s h
thebinhlimi|tsofithesthellar|masisfunction. proximation:
We separate the CSMF into the contributions of central and
satellite galaxies, Φ(M∗|Mh)=Φc(M∗|Mh)+Φs(M∗|Mh). ∆Σ1ghal(R)≡ hπMR∗2i, (19)
MNRAS000,000–000(2016)
Thestellar-to-halomassrelationofGAMAgalaxiesfrom100squaredegreesofKiDSweaklensingdata 7
3 STELLAR-TO-HALOMASSRELATION
Table2.Priorsadoptedinhalomodelfit
Parameter type range priormean priorsigma We start with an analysis of the lensing measurements to exam-
ine the stellar-to-halo mass relation of central galaxies, as was
log10(Mh,1) flat [9,14] - - done in several previous studies (e.g. Mandelbaumetal. 2006;
log10(M∗,0) flat [7,13] - - vanUitertetal.2011;Velanderetal.2014).Wefitthelensingsig-
β1 Gaussian - 5.0 3.0 nalsoftheeightlenssamplessimultaneouslywiththehalomodel.
log10β2 flat [−3,∞] - - Thebest-fittingmodelsfromthelensing-onlyanalysiscanbecom-
σc flat [0.05,0.5] - - paredtothedatainFig.2.Theresultingreducedchi-squared,χ2 ,
αs Gaussian - -1.1 0.9 red
b0 Gaussian - 0.0 1.5 hasavalueof1.0(with70degreesoffreedom),sothemodelspro-
b1 Gaussian - 1.5 1.5 videasatisfactoryfit.Intheleft-handpanelofFig.3weshowthe
fsub flat [0,1] - - constraintsonthestellar-to-halomassrelation.Abroadrangeofre-
fconc flat [0.2,2] - - lationsdescribethelensingsignalsequallywell.Furthermore,the
c0 flat [−5,5] - - right-handpanelofFig.3showsthattheuncertaintiesonthefrac-
c1 flat [9,16] - - tionofgalaxiesthataresatellitesisalsolarge.
vanUitertetal. (2011) pointed out that the uncertainties on
thesatellitefractionobtainedfromlensingonlyarelargeatthehigh
stellar mass end, and, even worse, that a wrongly inferred satel-
litefractioncanbiasthehalomassastheyareanti-correlated.The
with M∗ theaveragestellarmassofthelenssample. reasonforthisdegeneracy isthatloweringthehalomassreduces
h i
To summarise, the halo model employed in this paper themodelexcesssurfacemassdensityprofile,whichcanbepartly
has the following free parameters: (Mh,1,M∗,0,β1,β2,σc) and compensatedbyincreasingthesatellitefraction,assatellitesreside
(α ,b ,b ) to describe the halo occupation statistics of centrals onaverageinmoremassivehaloesthancentralsofthesamestel-
s 0 1
and satellites. f controls the subhalo masses of satellites, and larmass,therebyboostingthemodelexcesssurfacemassdensity
sub
f quantifies the normalization of the c(M) relation. We use profile at a few hundred kpc. This problem was partly mitigated
conc
non-informative flat or Gaussian priors, as listed in Table 2, ex- in vanUitertetal. (2011) and Velanderetal. (2014) by imposing
ceptforβ ,becauseourmeasurementsdonotextendfarbelowthe priors on the satellite fractions, which is not ideal, as the results
1
locationofthekinkinthestellar-to-halomassrelation.Asacon- aresensitivetothepriorsused.Asweemployamoreflexiblehalo
sequence,wearenotabletoprovidetightconstraintsontheslope modelhere,thisproblemisexacerbatedandadifferentsolutionis
at the low-mass end. Theother priors werechosen to generously required.
encapsulate previous literature results and they do not affect our Inordertotightentheconstraintsonthesatellitefractionand
results.Notethatforsomesamples,wehadtoadoptsomewhatdif- the stellar-to-halo mass relation, we either need to impose priors
ferentpriors;wecommentonthiswhereapplicable. inthehalomodel,orincludeadditional,complementarydatasets.
The parameter space is sampled with an affine invari- Since it is not obvious what priors to use, particularly since we
ant ensemble Markov Chain Monte Carlo (MCMC) sampler aimtostudyhowthestellar-to-halomassrelationdependsonen-
(Goodman&Weare2010).Specifically,weusethepubliclyavail- vironment,weoptforthesecondapproach. Themoststraightfor-
ablecode EMCEE(Foreman-Mackeyetal.2013).WerunEMCEE ward complementary data set is the stellar mass function, which
with four separate chains with 150 walkers and 4500 steps per constrainsthecentralandsatelliteCSMFsthroughEq.(14).Asa
walker.Thefirst1000steps(whichamountsto600000evaluations) result, the number of satellites cannot be scaled arbitrarily up or
are discarded as the burn-in phase. Using the resulting 2100000 downanymore,whichhelpstobreakthisdegeneracy.
modelevaluations,weestimatetheparameteruncertainties;thefit SinceGAMA isahighlycomplete spectroscopic survey, we
parameters that we quote in the following correspond to the me- can measure the stellar mass function by simply counting galax-
dian of the marginalized posterior distributions, the errors corre- ies, as long as we restrict ourselves to stellar mass and redshift
spondtothe68%confidenceintervalsaroundthemedian.Weas- ranges where the sample is volume limited. Hence we measure
sess the convergence of the chains with the Gelman-Rubin test the stellar mass function in three equally log-spaced bins be-
(Gelman&Rubin 1992) and ensure that R 6 1.015, withR the tween9.39<log10(M∗/h−2M⊙)<11.69andincludeallgalax-
ratiobetweenthevarianceofaparameterinthesinglechainsand iesfrom the G3Cv7 group catalogue withz < 0.15. The choice
thevarianceofthatparameterinallchainscombined.Inaddition, of the number of bins ismainly driven by the low number of in-
wecomputetheauto-correlationtime(seee.g.Akeretetal.2013) dependentbootstraprealisationswecanusetoestimatetheerrors
forourmainresultsandfindthatitisshorterthanthelengthofthe (discussed in Appendix A). We do not expect to lose much con-
chainsthatisneededtoreach1%precisiononthemeanofeachfit strainingpower fromthestellarmassfunction bymeasuringitin
parameter. threebinsonly.Notethatthemeanlensredshiftissomewhathigher
Forsomelensselections,wealsorunthehalomodelinan‘in- thantheredshiftatwhichwedeterminethestellarmassfunction,
formed’setting.WhenweusetheGAMAgroupcataloguetoselect buttheevolutionofthestellarmassfunctionisverysmalloverthe
andanalyseonlycentralsorsatellites,weonlyneedthepartofthe redshiftrangeconsideredinthiswork(see,e.g.,Ilbertetal.2013)
halomodelthatdescribestheirrespectivesignals.Hence,whenwe andhencecanbesafelyignored.Thestellarmassfunctionisshown
onlyselect centrals, weset theCSMFofsatellitestozero. When inFig.4,together withtheresultsfromBaldryetal.(2012),who
weselectsatellitesonly,wemodelboththeCSMFofthesatellites measured the stellar mass function for GAMA using galaxies at
and of the centrals of the haloes that host the satellites. Weneed z<0.06.Themeasurementsagreewell.
the latterto model the miscentred one-halo termand the subhalo We determine the error and the covariance matrix via boot-
massesofthesatellites. strapping, as detailed in Appendix A. We show there that 1) the
bootstrapsamplesshouldcontainasufficientlylargephysicalvol-
MNRAS000,000–000(2016)
8 Edo vanUitertet al.
Figure3.(left:)Stellar-to-halomassrelationofcentralgalaxiesfromKiDS+GAMA,determinedfromfittingthelensingsignalonly(dark/lightbrownindi-
cating1-/2-σmodeluncertaintyregime)orbycombiningthelensingsignalwiththestellarmassfunction(orange/yellowindicating1-/2-σmodeluncertainty
regime).Thecontoursarecutatthemeanstellarmassofthefirstandlaststellarmassbinusedinthelensinganalysis,toensureweonlyshowtheregime
wherethedataconstrainsit.(right:)ThefractionofgalaxiesthataresatellitesasafunctionofstellarmassforallGAMAgalaxies.Thecolouredcontours
showthe68%confidenceintervalforthefitstothelensingsignalonlyandtothecombinedfits,asindicatedinthepanel.Theupperthickblackdashedline
showsacrudeestimateofthesatellitefractionbasedontheGAMAgroupcatalogues(asdetailedinSect.3.3),thelowerthickdottedlineshowsalowerlimit.
Hatchedareasshowtheoverlapbetweenthe1-σlensing-onlyresultsandthecombinedanalysis.
ume.Ifthesamplevolumeistoosmall,theerrorswillbeunderes-
timated;2)themajorcontributiontotheerrorbudgetcomesfrom
cosmicvariance.ThecontributionfromPoissonnoiseistypically
of order 10-20%; 3) the stellar mass function measurements are
highlycorrelated.Smith(2012)showedthatthishasamajorimpact
ontheconfidencecontoursofmodelparametersfittedtothestellar
massfunction.Includingthecovarianceisthereforeessential,not
onlyforstudiesthatcharacterisethestellarmass/luminosityfunc-
tion(forexampleasafunctionofgalaxytype),butalsowhenitis
usedtoconstrainhalomodelfits.
To determine the cross-covariance between the shear mea-
surements and the stellar mass function, we measured the shear
of all GAMA galaxies with log10(M∗/h−2M⊙)>9.39 and
z<0.15ineachKiDSpointing,andusedthesameGAMAgalax-
ies to determine the stellar mass function. These measurements
wereusedasinput toour bootstrap analysis. Thecovariance ma-
trixofthecombinedshearandstellarmassfunctionmeasurements
revealedthatthecross-covariancebetweenthetwoprobesisnegli-
gibleandcanbesafelyignored.Thecovariancebetweenthelensing
measurementsandthestellarmassfunctionforsmallersubsamples
ofGAMAgalaxiesisexpectedtobeevensmallerbecauseoflarger
measurement noise.Therefore,wedonotrestrictourselvestothe
overlappingareawithKiDS,butusetheentire180deg2ofGAMA
areatodeterminethestellarmassfunctiontoimproveourstatistics.
Figure 4. Stellar mass function determined using all GAMA galaxies at
Wefitthelensingsignalofallbinsandthestellarmassfunc-
z < 0.15.Orangeregionsindicatethe68%confidenceintervalfromthe
tion simultaneously with the halo model. The best-fit models are halomodelfittothelensingsignalandthestellarmassfunction,linearly
shown inFig. 2 and 4, together withthe 1σ model uncertainties. interpolated between thestellar massbins.Thesolidgreenlineindicates
Thereducedχ2ofthebest-fitmodelis80/(83 10)=1.1(eight thebestfitmodel.ThestellarmassfunctionfromBaldryetal.(2012),de-
−
massbinstimestenangularbinsforthelensingsignal,plusthree terminedusingGAMAgalaxiesatz<0.06,isalsoshown.
massbinsforthestellarmassfunction),sothehalomodelprovides
anappropriatefit.Thelensingsignalofthebest-fitmodelisvirtu-
MNRAS000,000–000(2016)
Thestellar-to-halomassrelationofGAMAgalaxiesfrom100squaredegreesofKiDSweaklensingdata 9
allyindistinguishablefromthebest-fitmodelofthelensing-onlyfit. shouldenableustoderiverobustandphysicallymeaningful con-
Thestellar-to-halomassrelation,however,isbetterconstrained,as straintsonf .Thesubhalomassofsatellitesisnotconstrained
conc
isshowninFig.3.Therelationisflattertowardsthehighmassend, byour measurements, which iswhywe donot show it inFig.5.
as a result of a better constrained satellitefraction that decreases This is not surprising, given that most of our lenses are centrals,
withstellarmass(discussedinSect.3.3). and that the lensing signal is fairly noisy at small projected dis-
TheconstraintsontheparametersarelistedinTable3.The tancesfromthelens.
marginalised posteriors of the pairs of parameters are shown in
Fig.5.Thisfigureillustratesthatthemaindegeneraciesinthehalo
model occur between the parameters that describe the stellar-to- 3.1 Sensitivitytestsonstellar-to-halomassrelation
halo mass relation, and between the parameters that describe the
We have performed a number of tests to examine the robustness
CSMFofthesatellites.Thesedegeneraciesareexpected,giventhe
of our results.For computational reasons, welimitedthenumber
functional formsthatweadopted (seeEq.16,17,18).Forexam-
of model evaluations to 750000 (instead of 2100000), divided
ple, a larger value for Mh,1 would decrease the amplitude of the over two chains. We adopted a maximum value of R = 1.05 in
stellar-to-halomassrelation,whichcouldbepartlycompensatedby
theGelman-Rubinconvergencetesttoensurethatresultsaresuffi-
increasingM∗,0;hencetheseparametersarecorrelated.Similarly, cientlyrobusttoassesspotentialdifferences.
increasingb0wouldleadtoahighernormalisationofΦs(M∗|M), First,wetestifincompletenessinourlenssamplecanbiasthe
whichcouldbepartlycompensatedbydecreasingb ,hencethese
1 stellar-to-halo mass relation. As GAMA is a flux-limited survey,
twoparametersareanti-correlated.Furthermore,bycomparingthe
our lens samples miss the faint galaxies at a given stellar mass.
marginalisedposteriorstothepriors,weobserve thatall parame-
If these galaxies have systematically different halo masses, our
tersbutone,β ,areconstrainedbythedata.Wehaveverifiedthat
1 stellar-to-halomassrelationmaybebiased.Tocheckwhetherthis
varyingtheprioronβ doesnotimpactourresults.Comparingthe
1 isthecase,weselecteda(nearly)volume-limitedlenssampleusing
posteriorsofthecombinedfittotheanalysiswhereweonlyfitthe
the methodology of Langeetal. (2015). This method consists of
lensingsignalrevealsthatthestellarmassfunctionhelpsbycon-
determiningalimitingredshiftforgalaxiesinanarrowstellarmass
strainingseveralparameters;thosethatdescribethestellar-to-halo
bin,z ,whichisdefinedastheredshiftforwhichatleast90%of
massrelationandthosethatdescribethesatelliteCSMF. lim
thegalaxiesinthatsamplehavez <z ,withz themaxi-
InFig.6,wepresentthestellar-to-halomassrelationofcen- lim max max
mumredshiftatwhichagalaxycanbeobservedgivenitsrest-frame
tralgalaxiesandtheratioofhalomasstostellarmass.Therelation
spectralenergydistributionandgiventhesurveymagnitudelimit.
consistsof twoparts. Thisisnot simplyaconsequence of adopt-
z isdetermined iterativelyusing only galaxieswithz < z .
ing a double power law for this relation inthe halo model, since lim lim
Weremovedallgalaxieswithredshiftslargerthanz ( 60%of
thefithasthefreedomtoput thepivotmassbelow theminimum lim ∼
thegalaxiesinthefirststellarmassbin,fewerforthehighermass
massscaleweprobe,whichwouldeffectivelyresultinfittingasin-
gle power law. For M∗ < 5 1010h−2M⊙, the stellar-to-halo bins) and repeated thelensing measurements. Theresulting mea-
× surementsareabitnoisier,butdonotdiffersystematically.Wefit
massrelationisfairlysteepandthestellarmassincreaseswithhalo
ourhalomodeltothislensingsignalandthestellarmassfunction.
massasapowerlawofM withanexponent 7..Athigherstellar
masses,therelationflattenhsto M0.25.Thera∼tioofthedarkmatter Theresultingstellar-to-halomassrelationbecomesbroaderbyup
tostellarmasshasaminimum∼atahhalomassof8 1011h−1M⊙, to20%atthelowmassend,butisfullyconsistentwiththeresult
whereM∗c = (1.45 0.32) 1010h−2M⊙ and×thehalomassto showninFig.6.Henceweconcludethatincompletenessofthelens
stellar mass ratio ha±s a value×of Mh/M∗ = 56+−1160 [h]. The un- sampWleeishuanvleikaelslyottoesstiegdnitfihecainmtlpyabctiaosfovuarriroeususltass.sumptionsinthe
certaintyonthisratioreflectstheerrorsonourmeasurementsand
setupofthehalomodel.WegivedetailsofthesetestsinAppendix
does not account for theuncertainty of thestellar massestimates
B. None of the modifications led to significant differences in the
themselves,whicharetypicallyconsiderably smallerthanthebin
stellar-to-halomass relation, which shows that our resultsare in-
sizesweadoptedandhenceshouldnotaffecttheresultsmuch.The
sensitivetotheparticularassumptionsinthehalomodel.
locationoftheminimumisimportantforgalaxyformationmodels,
asitshowsthattheaccumulationofstellarmassingalaxiesismost
efficientatthishalomass.
3.2 Literaturecomparison
In the lower left-hand panel of Fig. 6, we also show the in-
tegrated stellar mass content of satellite galaxies divided by halo We limit the literature comparison to some of the most re-
mass.Inhaloeswithmasses&2 1013h−1M⊙,thetotalamount cent results, referring the reader to extensive comparisons be-
×
ofstellarmassinsatellitesislargerthanthatinthecentral.More tweenolderworksinLeauthaudetal.(2012);Couponetal.(2015);
stellarmassiscontainedinsatellitestowardshigherhalomasses;at Zu&Mandelbaum(2015).Ourmaingoalistoseewhetherourre-
5 1014h−1M⊙ 94%ofthestellarmassisinsatellitegalaxies. sultsareingeneral agreement. In-depthcomparisons betweenre-
× ∼
Notethatanotherconsiderablefractionofstellarmassiscontained sultsaregenerallydifficult,duetodifferencesintheanalysis(e.g.
inthediffuseintra-clusterlight(uptoseveraltensofpercents,see thedefinitionofmass,choicesinthemodelling)aswellasinthe
e.g.Lin&Mohr2004)whichwehavenotaccountedforhere. data(e.g.thecomputationofstellarmasses-note,however,thatthe
Thenormalisationofthemass-concentrationrelationisfairly stellarmassesusedintheliteraturestellar-to-halomassrelationswe
low,fconc =0.70+−00..1195.Anormalisationlowerthanunitywasan- comparetoareallbasedonaChabrierInitialMassFunction,asare
ticipated aswe didnot account for miscentring of centrals inthe ours).
halo model. Miscentring distributes small-scale lensing power to Leauthaudetal.(2012)measuredthestellar-to-halomassre-
largerscales,aneffectsimilartoloweringtheconcentration.Inour lation of central galaxies by simultaneously fitting the galaxy-
fits,itmerelyactsasanuisanceparameter, andshouldnot bein- galaxy lensing signal, the clustering signal and the stellar mass
terpretedasconflictingwithnumericalsimulations.Infuturework, function of galaxies in COSMOS. The depth of this survey al-
we will include miscentering of centrals in the modelling, which lowed them to measure this relation up to z = 1. In Fig. 6, we
MNRAS000,000–000(2016)
10 Edovan Uitertet al.
10(cid:0)✁✂1✄1☎✆✝12✞✟✄✠13
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10 11 12 13 10 11 120 4 8 12 0.00 0.25 0.50 0.75 0.1 0.2 0.3 0.4 1.2 0.8 0.4 1.5 0.0 1.5 0.8 1.6 2.4 0.4 0.8 1.2
(cid:0)✁✂✄☎✆✝✞✟✄✠ (cid:0)✁✂✄☎✆✝✡✟☎✠ ☛✄ ☛☞ ✌✍✎✏✑ ✒✓ ✔☎ ✔✄ ✕✖✎✗✖
Figure5.Posteriorsofpairsofparameters,marginalisedoverallotherparameters.DimensionsarethesameasinTable3.Solidorangecontoursindicate
the1-/2-σconfidenceintervalsofthelensing+SMFfit,whilstthebrowndashedcontoursindicatethe1-/2-σconfidenceintervalsofthelensing-onlyfit.Red
crossesindicatethebest-fitsolutionofthecombinedfit.Thepanelsonthediagonalshowthemarginalisedposterioroftheindividualfitparameters,together
withthepriors(bluedottedlines).IncludingtheSMFinthefitmainlyhelpstoconstrainthestellar-to-halomassrelationparameters(Eq.16)andαs.The
degeneraciesbetweenthestellar-to-halomassrelationparametersandthosethatdescribethesatelliteCSMF(Eq.17,18),followfromthefunctionalformwe
adopted.
showtheirrelationfortheirlow-redshiftsampleat0.22<z<0.48, (2012) are based on photometric redshifts, which generally in-
whichisclosesttoourredshiftrange.Therelationsagreereason- duce asmall Eddington biasin thestellar massestimates, partic-
ably well. We infer a slightly larger halo mass at a given stellar ularly at the high-stellar mass end where the stellar mass func-
mass, most noticeably at the high mass end. Their Mh/M∗ ratio tion drops exponentially (as illustrated in Fig. 4 in Droryetal.
reaches a minimum at a halo mass of 8.6 1011h−1M⊙ witha 2009).Incontrast,thestellarmassesinGAMAarecomputedus-
×
value of Mh/M∗ =38[h], 1.5σ below the minimum value of ing spectroscopic redshifts, and this bias does not occur. We at-
∼
the ratio we find. A systematic shift in stellar mass may explain temptedvariousshiftsinstellarmassandfoundthatthestellar-to-
muchofthedifference.ThestellarmassesusedinLeauthaudetal. halomassrelationsfullyoverlapifwedecreasethestellarmasses
MNRAS000,000–000(2016)