Mon.Not.R.Astron.Soc.000,1–??(2008) Printed15January2009 (MNLATEXstylefilev2.2) Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups Ran Li1,2⋆, H.J. Mo2, Zuhui Fan1, Marcello Cacciato3, Frank C. van den Bosch3, Xiaohu Yang4, Surhud More3 9 1Department of Astronomy, Peking University,Beijing 100871, China 0 2Department of Astronomy, Universityof Massachusetts, Amherst MA 01003, USA 0 3Max-Planck Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany 2 4Shanghai Astronomical Observatory, the Partner Group of MPA, Nandan Road 80, Shanghai 200030, China n a J 5 1 ABSTRACT We use galaxy groups selected from the Sloan Digital Sky Survey (SDSS) together ] with mass models for individual groups to study the galaxy-galaxy lensing signals h expected from galaxies of different luminosities and morphological types. We com- p - pare our model predictions with the observational results obtained from the SDSS o by Mandelbaum et al. (2006) for the same samples of galaxies. The observational re- r sults are well reproduced in a ΛCDM model based on the WMAP 3-year data, but t s a ΛCDM model with higher σ8, such as the one based on the WMAP 1-year data, a significantly over-predicts the galaxy-galaxylensing signal. We model, separately,the [ contributions to the galaxy-galaxylensing signals from different galaxies: central ver- 3 sussatellite,early-typeversuslate-type,andgalaxiesinhaloesofdifferentmasses.We v also examine how the predicted galaxy-galaxy lensing signal depends on the shape, 4 density profile, and the location of the central galaxy with respect to its host halo. 3 9 Key words: dark matter - large-scale structure of the universe - galaxies: haloes - 4 methods: statistical . 7 0 8 0 1 INTRODUCTION needs to be tested against observations. More recently, the : v halooccupation modelhasopenedanotheravenuetoprobe According to the current paradigm of structure formation, i the galaxy-dark matter halo connection (e.g. Jing, Mo & X galaxies form and reside inside extended cold dark haloes. Börner 1998; Peacock & Smith 2000; Berlind & Weinberg r While the formation and evolution of dark matter haloes 2002; Cooray & Sheth 2002; Scranton 2003; Yang, Mo & a in the cosmic density field is mainly determined by grav- vandenBosch2003; vandenBosch,Yang&Mo2003; Yan, itational processes, the formation and evolution of galax- Madgwick & White 2003; Tinker et al. 2005; Zheng et al. iesinvolvesmuchmorecomplicated,andpoorlyunderstood 2005; Cooray 2006; Vale & Ostriker 2006; van den Bosch processes, such as radiative cooling, star formation, and et al. 2007). This technique uses the observed galaxy lumi- all kinds of feedback. One important step in understand- nosity function and clustering properties to constrain the ing how galaxies form and evolve in the cosmic density average number of galaxies of given properties that occupy fieldisthereforetounderstandhowthegalaxies ofdifferent adarkmatterhaloofgivenmass.Althoughthemethodhas physical properties occupy dark matter haloes of different theadvantagethat it canyield muchbetterfitstothedata masses. Theoretically, the connection between galaxies and than the semi-analytical models or numerical simulations, dark matter haloes can be studied using numerical simula- onetypicallyneedstoassumeasomewhatad-hocfunctional tions (e.g., Katz, Weinberg & Hernquist 1996; Pearce et al. form to describe the halo occupation model. 2000; Springel2005; Springelet al.2005) orsemi-analytical A more direct way of studyingthegalaxy-halo connec- models (e.g. White & Frenk 1991; Kauffmann et al. 1993, tionistousegalaxygroups1,providedthattheyaredefined 2004; Somerville & Primack 1999; Cole et al. 2000; vanden assetsofgalaxies thatresideinthesamedarkmatterhalo. Bosch 2002; Kang et al. 2005; Croton et al. 2006). These Recently, Yang et al. (2005; 2007) have developed a halo- approaches try to model the process of galaxy formation based group finder that is optimized for grouping galaxies from first principles. However, since our understanding of thatresideinthesamedarkmatterhalo.Usingmockgalaxy therelevantprocessesisstillpoor,thepredictedconnection between the properties of galaxies and dark matter haloes 1 In this paper, we refer to a system of galaxies as a group re- gardless of its richness, including isolated galaxies (i.e., groups ⋆ E-mail:ranl@astro.umass.edu withasinglemember)andrichclustersofgalaxies. 2 Ran Li et. al redshiftsurveysconstructedfromtheconditionalluminosity obtained by Mandelbaum et al. (2006) for the same galax- functionmodel(e.g.Yangetal.2003) andasemi-analytical ies.Ourgoalisthreefold.First,wewanttotestwhetherthe model (Kang et al. 2005), it is found that this group finder method of halo-mass assignment to groupsadopted by Y07 isverysuccessfulin associating galaxies with theircommon isreliable.Sincethemethodprovidesapotentiallypowerful darkmatterhaloes(seeYangetal.2007;hereafterY07).The way to obtain the halo masses associated with the galaxy groupfinderalsoperformsreliablyforpoorsystems,includ- groups, the test results have general implications for the ingisolated galaxies in small mass haloes, makingit ideally study of the relationship between galaxies and dark matter suitedforthestudyoftherelationshipbetweengalaxiesand haloes. Second, we want to examine in detail the contribu- dark matter haloes over a wide range of halo masses. How- tions to the galaxy-galaxy lensing signal from different sys- ever, in order to interpret the properties of the galaxy systems,suchascentralversussatellitegalaxies,early-typever- temsintermsofdarkmatterhaloes,oneneedstoknowthe sus late-type galaxies, and groups of different masses. Such halomassassociatedwitheachofthegroups.Oneapproach analysis can help us interpreting the observational results. commonlyadoptedistousesomehalomassindicator(such Finally, we would like to study how the predicted galaxy- asthetotalstellarmassorluminositycontainedinmember galaxy lensing signal depends on model assumptions, such galaxies) torank thegroups.With theassumption that the as the cosmological model and the density profiles of dark corresponding halo masses have the same ranking and that matter haloes. In a companion paper (Cacciato et al. 2008, themassfunctionofthehaloesassociatedwithgroupsisthe hereafterC08),weusetherelationshipbetweengalaxiesand same as that given by a model of structure formation, one darkmatterhaloesobtainedfromtheconditionalluminosity canassign ahalomasstoeachoftheobservedgroups.This function(CLF)modeling(Yangetal.2003;vandenBoschet approach was adopted byY07for thegroup catalogue used al. 2007) to predict the galaxy-galaxy cross correlation and in this paper. There are three potential problems with this tocalculatethelensingsignal,whileherewedirectlyusethe approach. First, the approach is model-dependent, in the observed galaxy groups and theirgalaxy memberships. sense that the assumption of a different model of structure This paper is organized as follows. In Section 2 we de- formation will lead to a different halo mass function, and fine the statistical measure that characterizes the galaxy- hence assign different masses to the groups. Second, even galaxy lensing effect expected from the mass distribution if the assumed model of structure formation is correct, it associated with the galaxy groups. We provide a brief de- is still not guaranteed that the mass assignment based on scription of the galaxy group catalogue and the models the ranking of group stellar mass (or luminosity) is valid. of the mass distribution associated with galaxy groups in Finally, even if all groups are assigned with accurate halo Section 3. We present our results in Section 4 and con- masses, the question how dark matter is distributed within clude in Section 5. Unless specified otherwise, we adopt a the galaxy groups remains open. Clearly, it is important to ΛCDM cosmology with parameters given by the WMAP have independent mass measurements of the haloes asso- 3-year data (Spergel et al. 2007, hereafter WMAP3 cos- ciated with galaxy groups to test the validity of the mass mology) in our analysis: Ω = 0.238, Ω = 0.762, and m Λ estimates based on the stellar mass (luminosity) ranking. h H /(100kms−1Mpc−1)=0.73, σ =0.75 . 0 8 ≡ Gravitational lensing observations, which measure the imagedistortionsofbackgroundgalaxiescausedbythegrav- itational field of the matter distribution in the foreground, 2 GALAXY-GALAXY LENSING provide a promising tool to probe the dark matter distri- Galaxy-galaxy lensing provides a statistical measure of the bution directly. In particular, galaxy-galaxy weak lensing, profile of the tangential shear, γ (R), averaged over a thin whichfocusesontheimagedistortionsaroundlensinggalax- t annulus at the projected radius R around the lens galaxies, can be used to probe the distribution of dark mat- ies. This quantity is related to the excess surface density ter around galaxies, hence their dark matter haloes. The (hereafter ESD) around thelens galaxy, ∆Σ, as galaxy-galaxy lensing signal produced by individual galaxies is usually very weak, and so one has to stack the signal ∆Σ(R)=γ (R)Σ =Σ¯(0.7, are used in thegroup identi- on the other hand, consists of two parts: one from the sub- fication.ThemagnitudCesandcolorsofallgalaxiesarebased halo associated the satellite, which contributes to the inner onthestandardSDSSPetrosian technique(Petrosian 1976; part, and the other from the host halo, which dominates Strauss et al. 2002), and have been corrected for galactic atlargerR.Thissimplemodeldemonstratesclearlythat,in extinction (Schlegel, Finkbeiner & Davis 1998). All mag- ordertomodelthegalaxy-galaxylensingsignalproducedby nitudes have been K-corrected and evolution-corrected to a population of lens galaxies, one needs to model carefully z = 0.1 following the method described in Blanton et al. thedistributionofmatteraroundbothhosthaloesandsub- (2003). In Y07, three group samples were constructed us- haloes.Todothis,weneednotonlytoidentifythehaloesin ing galaxy samples of different sources of galaxy redshifts. which each lens galaxy resides, but also to model the mass Our analysis is based on Sample II, which includes 362,356 and density profile of each host halo and subhalo. In addi- galaxieswithredshiftsfromtheSDSSand7091galaxieswith tion we also need to model the distribution of dark matter redshiftstakenfromalternativesurveys:2dFGRS(Collesset relativetogalaxies.Inthefollowingsection,wedescribeour al. 2001), PSOz (Saunders et al.2000) or from the RC3 (de modeling with theuse of observed galaxy groups. Vaucouleurset al.1991). Therearein total301,237 groups, including those with only one member galaxy. The group finder has been applied to mock catalogue to test the com- 3 MODELING THE MASS DISTRIBUTION pleteness and purity of the groups in Y07. About 90% of ASSOCIATED WITH THE SDSS GROUPS the groups have a completeness f > 0.6 and 80% groups c with f > 0.8, where f is defined as the ratio between the 3.1 The SDSS Group Catalogue c c number of true members that are selected as the members Our analysis is based on the SDSS galaxy group catalogue of the group and the number of the total true members of constructedbyY07.Thegroupsareselectedwiththeadap- thegroup. 4 Ran Li et. al Figure2.ThehalomassMS (estimatedusingstellarmass),versusM∗(lowerpanels)andL(upperpanels)ofthegalaxiesinthehaloes. Theleftpanels areforcentralgalaxiesandtherightpanelsareforsatellitegalaxies. 3.2 Halo Mass Assignment its absolute magnitude and color using the fitting formula given by Bell et al. (2003). An important aspect of the group catalog construction is thedetermination of thehalo mass, M , of each group. In The basic assumption of the ranking method is that vir Y07,twoestimatorsareadopted.Thefirst,M ,isestimated thereisaone-to-onerelationbetweenM (orL )and L stellar 19.5 usingtherankingofthecharacteristicluminosityofagroup, the group mass. Using the dark matter halo mass function which is the total luminosity of all member galaxies in the predictedbyamodelofstructureformation, onecanassign group with M 5logh 6 19.5 (hereafter referred to as ahalomasstoeachgroupaccordingtoitsM -ranking r stellar − − L ). The second, M , is estimated using the ranking of (orL -ranking).Inthispaper,weusethemassfunction 19.5 S 19.5 thecharacteristicstellarmass,M whichisdefinedtobe obtainedbyWarrenetal.(2006). Notethatthisone-to-one stellar thetotalstellarmassofgroupmemberswithM 5logh6 mapping is applicable only when the group sample is com- r − 19.5. For each galaxy the stellar mass is estimated from plete. In Y07, three complete samples are constructed in − Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups 5 Figure3.Thedistributionofthehosthalomassesforthecentralandsatellitegalaxiesindifferentluminositybins,asindicatedbythe r-bandabsolute-magnituderangeineachpanel. three redshift ranges. Only groups in the complete samples thanthatbetweenL andhalomass,weadoptM asour 19.5 S are used in the ranking. The mass of other groups are esti- fiducial halo mass throughout. As we demonstrate in 4.3, § mated by a linear interpolation based on the M -M using M instead yields results that are fairly similar. stellar vir L relation (or the L - M ) obtained from the complete 19.5 vir sample. Detailed tests using mock galaxy redshift samples Fig. 2 shows the relation between the host halo mass, have shown that the 1-σ error of the estimated halo mass MS, and thegalaxy stellar mass M∗ (thelower two panels) is 0.3 dex (Y07). In addition, the two mass estimators, or the galaxy luminosity L (the upper two panels). Results ∼ M andM ,agree remarkablywell with each other,with a are shown separately for central galaxies (left panels) and L S scatter that decreases from about 0.1 dex at the low-mass satellite galaxies (right panels). As one can see, the stellar end to about 0.05 dex at the high-mass end. Since the cor- mass (luminosity) of central galaxies is quite tightly cor- relationbetweenM andhalomassissomewhattighter related with their host halo masses. However, for satellite stellar galaxies of a given stellar mass (or luminosity), their host 6 Ran Li et. al halo mass covers a verylarge range, reflecting thefact that a set of masses for each group mass. We then set the mass manylow-massgalaxiesaresatellitesinmassivehaloes.The originallyassociatedwithasatellitegalaxyaccordingtothe distributions of host halo masses, M , for central or satel- stellar mass ranking of the satellites in the group. Here we S lite galaxies in different luminosity bins are shown in Fig. implicitly assume that the initial subhalo mass function is 3.Onaverage,brightercentralgalaxies resideinmoremas- the same as the mass function of the subhalos that host sive haloes. For faint galaxies, the halo-mass distribution is satellites. This assumption is not proved by any observa- broader, again because many faint galaxies are satellites in tions,andwehavetolivewithitsinceamorerealisticmodel massive systems. is not currently available. Fortunately, subhalos only con- In the group catalogue, the mass assignment described tribute a small fraction to the total lensing signal on small above is used only for groups where the brightest galaxy scales. The uncertainty here will not have a significant im- is brighter than M 5logh = 19.5. This is because the pact on any of our conclusions. To obtain the final mass in r − − massrankingusedinthegroupcatalogisbasedonthetotal the subhalo at the present time, the evolution of the sub- stellar mass (or total luminosity) of the member galaxies haloes needs to be taken into account. In other words, we that are brighter than M 5logh = 19.5. The groups need to know the fraction of the mass that is stripped and r − − with nogalaxies brighterthanthis magnitudethushaveno how the structure of a subhalo changes after the stripping. assigned rank.Asdescribedin Y07,thereason forchoosing Here it is convenient to introduce a parameter f which is m thismaginitudeisacompromisebetweenhavingacomplete theretained mass fraction of thesubhalo. Gao et al. (2004) sampleinarelatively largevolumeandhavingmoregroups studiedtheradial dependenceof theretained mass fraction that are represented by a number of member galaxies. For f from alarge sampleofsubhaloesinalarge cosmological m groups in which all member galaxies have M 5logh > simulation. In their work, f is considered as a function of r m − 19.5,adifferentmethodhastobeadopted.Inmodelingthe r /r , where r is the distance of the subhalo from the s vir,h s − luminosity function and stellar mass function of thecentral center of the host halo and r is the virial radius of the vir,h galaxies based on the same SDSS group catalogue as used host halo. The simulation of Gao et al. gives here, Yanget al. (2008) obtain an average relation between f =0.65(r /r )2/3. (4) m s vir,h the luminosity (or stellar mass) of the central galaxy and the halo mass down to M 5logh 17. We adopt this Wewilladoptthisinourmodelingofthemassesassociated r relation toassign halomasse−stoallg∼rou−ps(includingthose with subhaloes. However, in the group catalogue, only the containingonlyoneisolated galaxy)representedbycentrals projected distance, rp, from the group center is available. withMr 5logh> 19.5.Forconvenience,thehalomasses The 3D-distance, rs, is obtained by randomly sampling the obtained−in this way−are also referred to as M (based on NFWprofileofthehosthalowiththegivenprojectedradius S the stellar mass of central galaxies) and ML (based on the rp. r-bandluminosity of thecentral galaxies), respectively. Thus, the mass assigned to a subhalo is determined by thefollowing threefactors: (1) thestellar mass of thesatel- litegalaxy;(2)thehosthalomass;(3)thedistancebetween 3.3 Mass Distribution in Haloes and Subhaloes the satellite and the center of the host. Here the host halo mass comes into our calculation in two ways. It not only With the group catalogue described above, we can model determines the subhalo mass function, but also affects the the dark matter distribution by convolving the halo distri- parameter f in Eqs.4. Note that the accretion history of bution with thedensity profiles of individualhaloes. In our m the host halo may also affect the value of f . We have to modeling of the density profiles, thehost halo of a group is m neglect such effect because it is unclear how to model the assumedtobecenteredonthecentralgalaxy.Therearetwo accretion histories for individualgroups. waystodefineacentralgalaxy:oneistodefinethecentralin For host haloes, we use the following NFW profile a group to bethegalaxy with thehighest stellar mass, and (Navarro,Frank&White1997)tomodelthemassdistribu- theotheristodefinethecentraltobethebrightestmember. tion: Formostgroupsthesetwodefinitionsgivethesameresults, butthereareveryfewcases(lessthan∼2%)wheredifferent ρ(r)= δ0ρ¯ . (5) centralgalaxiesaredefined.Inourfiducialmodel,wedefine (r/rc)(1+r/rc)2 themostmassivegalaxies(intermofstellarmass)tobethe where ρ¯ is the mean density of the universe, r is a scale c central galaxies. radius,relatedtovirialradiusr bytheconcentration,c= vir In a hierarchical model, a dark matter halo forms r /r ,andδ isacharacteristicover-densityrelatedtothe vir c 0 through a series of merger events. During the assembly of average over-density of a virialized halo, ∆ , by vir a halo, most of the mass in the merging progenitors is ex- ∆ c3 pected to be stripped. However, some of them may survive δ = vir . (6) assubhaloes,althoughthetotalmasscontainedinsubhaloes 0 3 ln(1+c) c/(1+c) − issmall,typically 10%(vandenBosch,Tormen&Giocoli We adopt the value of ∆ given by the spherical collapse vir ∼ 2005). Some of the subhaloes are associated with ‘satellite model (see Nakamura & Suto 1997; Henry 2000). Numeri- galaxies’ in a halo. In our modeling of the galaxy-galaxy calsimulationsshowthathaloconcentrationsarecorrelated weak lensing, we only take into account subhaloes associ- with halo mass, and we use the relations given by Macciò atedwithsatellitegalaxies,treatingothersubhaloesaspart et al.(2007), convertedtoourdefinition ofhalo mass. Note of the host halo. Giocoli, Tormen & van den Bosch (2008) thathereweuser ,insteadoftheconventionalnotationr , c s provide a fitting function of the average mass function for todenotethescaleradiusoftheNFWprofile,asr hasbeen s subhaloesatthetimeoftheiraccretionintotheparenthalo used to denote the distance of a subhalo from the center of of a given mass. Using this mass function, we first sample its host. Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups 7 Forsub-haloes,wemodeltheirdensityprofilesusingthe Table 1. The properties of galaxy samples. In each case, the resultsobtainedbyHayashietal.(2003),whofoundthatthe absolute-magnituderange,themeanredshift,themeanluminos- densityprofilesof strippedsub-haloescan beapproximated ity, and the fraction of late-types are listed. Note that L∗ = as 1.2×1010h−2L⊙ f ρs(r)= 1+(r/tr )3ρ(r), (7) Sample Mr hzi hL/L∗i flate t,eff where f is a dimensionless factor describing the reduction L1 −180.1h−1Mpc∼. F , where F and F are th×e dedeVVaucou−leurs and ex×- exp deV exp ponentialprofiles,respectively.AsinMandelbaumetal.,we usetheaverageoffrac devintheg,randibands.Galaxies withfrac dev>0.5areclassified asearly-type,whilethose 4 RESULTS with frac dev<0.5 as late-type. It should be pointed out that we did not carry a ray- 4.1 SDSS Lensing Data tracing simulation to predict the galaxy-galaxy lensing re- Before presenting our model predictions, we first describe sults. Instead, we directly calculate the excess surface den- the observational results that we will use for comparison. sity around SDSS galaxies. Thus, our calculation doesn’t TheobservationalresultstobeusedwereobtainedbyMan- dealwithsourcegalaxies.Ontheotherside,inMandelbaum delbaumetal.(2006),whoanalyzedthegalaxy-galaxylens- et al. (2006) thesourcegalaxies arecarefully weighted, and ing effects using galaxies in a sample constructed from the lensing signals are calibrated, to reduce any bias in the ob- SDSS DR4 spectroscopic sample. Their sample of lensing servational measurements (see Mandelbaum et al. 2005 for 8 Ran Li et. al Figure 4. Comparison of the lensing signal predicted by the fiducial model with the observational results. Here the ESD is plotted as the function of the transverse distance R for lensing galaxies in different luminosity bins. Data points with error bars are the observationalresultsofMandelbaumetal.,whilethelinesarethemodelpredictions.Thedotted,dashedanddot-dashedlinesrepresent the contributions of the ‘1-halo term’ of central galaxies, the ‘1-halo term’ of satellite galaxies, and the ‘2-halo term’ (of both centrals and satellites), respectively. The solid lines show the predicted total ESD. The r-band magnitude range for each case can be found in Table1. details).Thus,weassumetheobservationalresultsareunbi- parison of the lensing signal for galaxies divided according ased,andcomparethemdirectlywithourmodelpredictions. to luminosity. Theobservationdatausedherewaskindlyprovidedby Theerrorbarsontheobservationalpointsare1σ statis- R.Mandelbaum.ThedatausedinFig.6hasbeenpublished ticalerror.Thesystematicerrorofthegalaxy-galaxylensing in Mandelbaum et al. (2006) where lensing signal is calcu- has been discussed in detail in Mandelbaum et al. (2005). latedforearlyandlatetypegalaxiesseparately.R.Mandel- The test has been carried out for three source samples: baum also provided the lensing data combining early and r < 21, r > 21, and high redshift LRGs. The overall sys- latetypegalaxiesforustomakethecomparisons presented tematicerrorisfoundtobecomparabletoorslightlylarger in all other figures. Note that here we only show the com- than thestatistical error shown here. Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups 9 Figure 5. The contribution to the ESD plotted separately for dark matter haloes of different masses. In each panel, the dotted line shows the contribution from haloes with MS >1014h−1M⊙. The dashed line shows the contribution from haloes with 1013h−1M⊙ 6 MS <1014h−1M⊙,andthedot-dashedlineshowsthecontributionfromhaloeswithMS <1013h−1M⊙.Thesolidlineshowsthetotal lensingsignalpredictedbythefiducialmodel.Forcomparison,theobservational dataareincludedasdatapointswitherror-bars. 4.2 The Fiducial Model vational results obtained by Mandelbaum et al. (2006) for thesameluminositybins.Overall,ourfiducialmodelrepro- InFig.4weshowthelensingsignalaroundgalaxiesindiffer- duces the observational data reasonably well, especially for entluminositybinsobtainedfromourfiducialmodel,which bright galaxy bins where the observational results are the has model parameters as described in the last section and most reliable. The reduced χ2 is 3.2 combining all the lu- assumes the WMAP3 cosmology. Here the ESD is plotted minosity bins. The best match is for L5f, with a reduced as a function of the projected distance R from galaxies. χ2 of 0.9. Given that we do not adjust any model parame- The solid line shows the averaged ESD of all galaxies in ters, the χ2 indicates a good agreement. For the three low- thecorrespondingluminositybin.Theamplitudeofthepre- luminosity bins, the predicted ESD is lower than the corre- dicted ESD increases with galaxy luminosity, reflecting the sponding observational result. For L1 and L2, the observa- fact that brighter galaxies on average reside in more mas- tional data are very uncertain. For L3, if we take the ob- sive haloes, as shown in Figs. 2 and 3. These results are to servational data points at face-value, the discrepancy with be compared with the data points which show the obser- 10 Ran Li et. al Figure 6. The right panels show the ESD of early galaxies in different luminosity bins, while the left panels show the results for late galaxies. The data points with error-bars show the observational results. The model predictions of the ESD using stellar mass as halo massindicatorareshownasthesolidlines.Forcomparison,thedashedlinesshowthecorrespondingmodelpredictionsusingMLasthe halomasses. the model prediction is significant. As described in 3.2, and about 70% in L2 and L1. It is possible that this rela- § for groups which do not contain any member galaxies with tionunderestimatesthehalomass.Inordertoseetheeffect M 5logh < 19.5, their halo masses are not obtained caused by such uncertainties, we have used a set of param- r − − from the ranking of M , but from the average stellar eters from Yang et al. (2008) that are still allowed by the stellar mass-halo mass relation of central galaxies that is required observed stellar mass function but give larger halo masses tomatchtheobservedstellarmassfunctionofcentralgalax- tothehostsoffaintcentralgalaxies. Thisincreasesthepre- ies.Whileallthegalaxiesinthebrightluminositybinshave dicted ESD for L3 by 20%, not sufficient to explain the ∼ their host halo mass assigned by ranking method, the frac- discrepancy. Indeed, this discrepancy is not easy to fix. In tion of the galaxies in halos that have masses assigned ac- the observational data, the amplitudes of the ESD for L3 cording to the mass-halo mass relation is about 30% in L3 at R 0.3 - 0.9h−1Mpc are actually slightly higher than ∼

Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups

Similar Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups