Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups
Ran Li, H.J. Mo, Zuhui Fan, Marcello Cacciato, Frank C. van den Bosch, Xiaohu Yang, Surhud More
aa r X i v : . [ a s t r o - ph ] J a n Mon. Not. R. Astron. Soc. , 1– ?? (2008) Printed 24 November 2018 (MN L A TEX style file v2.2)
Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups
Ran Li , ⋆ , H.J. Mo , Zuhui Fan , Marcello Cacciato , Frank C. van den Bosch ,Xiaohu Yang , Surhud More Department of Astronomy, Peking University, Beijing 100871, China Department of Astronomy, University of Massachusetts, Amherst MA 01003, USA Max-Planck Institute for Astronomy, K¨onigstuhl 17, D-69117 Heidelberg, Germany Shanghai Astronomical Observatory, the Partner Group of MPA, Nandan Road 80, Shanghai 200030, China
ABSTRACT
We use galaxy groups selected from the Sloan Digital Sky Survey (SDSS) togetherwith mass models for individual groups to study the galaxy-galaxy lensing signalsexpected from galaxies of different luminosities and morphological types. We com-pare our model predictions with the observational results obtained from the SDSSby Mandelbaum et al. (2006) for the same samples of galaxies. The observational re-sults are well reproduced in a ΛCDM model based on the WMAP 3-year data, buta ΛCDM model with higher σ , such as the one based on the WMAP 1-year data,significantly over-predicts the galaxy-galaxy lensing signal. We model, separately, thecontributions to the galaxy-galaxy lensing signals from different galaxies: central ver-sus satellite, early-type versus late-type, and galaxies in haloes of different masses. Wealso examine how the predicted galaxy-galaxy lensing signal depends on the shape,density profile, and the location of the central galaxy with respect to its host halo. Key words: dark matter - large-scale structure of the universe - galaxies: haloes -methods: statistical
According to the current paradigm of structure formation,galaxies form and reside inside extended cold dark haloes.While the formation and evolution of dark matter haloesin the cosmic density field is mainly determined by grav-itational processes, the formation and evolution of galax-ies involves much more complicated, and poorly understoodprocesses, such as radiative cooling, star formation, andall kinds of feedback. One important step in understand-ing how galaxies form and evolve in the cosmic densityfield is therefore to understand how the galaxies of differentphysical properties occupy dark matter haloes of differentmasses. Theoretically, the connection between galaxies anddark matter haloes can be studied using numerical simula-tions (e.g., Katz, Weinberg & Hernquist 1996; Pearce et al.2000; Springel 2005; Springel et al. 2005) or semi-analyticalmodels (e.g. White & Frenk 1991; Kauffmann et al. 1993,2004; Somerville & Primack 1999; Cole et al. 2000; van denBosch 2002; Kang et al. 2005; Croton et al. 2006). Theseapproaches try to model the process of galaxy formationfrom first principles. However, since our understanding ofthe relevant processes is still poor, the predicted connectionbetween the properties of galaxies and dark matter haloes ⋆ E-mail:[email protected] needs to be tested against observations. More recently, thehalo occupation model has opened another avenue to probethe galaxy-dark matter halo connection (e.g. Jing, Mo &B¨orner 1998; Peacock & Smith 2000; Berlind & Weinberg2002; Cooray & Sheth 2002; Scranton 2003; Yang, Mo &van den Bosch 2003; van den Bosch, Yang & Mo 2003; Yan,Madgwick & White 2003; Tinker et al. 2005; Zheng et al.2005; Cooray 2006; Vale & Ostriker 2006; van den Boschet al. 2007). This technique uses the observed galaxy lumi-nosity function and clustering properties to constrain theaverage number of galaxies of given properties that occupya dark matter halo of given mass. Although the method hasthe advantage that it can yield much better fits to the datathan the semi-analytical models or numerical simulations,one typically needs to assume a somewhat ad-hoc functionalform to describe the halo occupation model.A more direct way of studying the galaxy-halo connec-tion is to use galaxy groups , provided that they are definedas sets of galaxies that reside in the same dark matter halo.Recently, Yang et al. (2005; 2007) have developed a halo-based group finder that is optimized for grouping galaxiesthat reside in the same dark matter halo. Using mock galaxy In this paper, we refer to a system of galaxies as a group re-gardless of its richness, including isolated galaxies (i.e., groupswith a single member) and rich clusters of galaxies.c (cid:13)
Ran Li et. al redshift surveys constructed from the conditional luminosityfunction model (e.g. Yang et al. 2003) and a semi-analyticalmodel (Kang et al. 2005), it is found that this group finderis very successful in associating galaxies with their commondark matter haloes (see Yang et al. 2007; hereafter Y07). Thegroup finder also performs reliably for poor systems, includ-ing isolated galaxies in small mass haloes, making it ideallysuited for the study of the relationship between galaxies anddark matter haloes over a wide range of halo masses. How-ever, in order to interpret the properties of the galaxy sys-tems in terms of dark matter haloes, one needs to know thehalo mass associated with each of the groups. One approachcommonly adopted is to use some halo mass indicator (suchas the total stellar mass or luminosity contained in membergalaxies) to rank the groups. With the assumption that thecorresponding halo masses have the same ranking and thatthe mass function of the haloes associated with groups is thesame as that given by a model of structure formation, onecan assign a halo mass to each of the observed groups. Thisapproach was adopted by Y07 for the group catalogue usedin this paper. There are three potential problems with thisapproach. First, the approach is model-dependent, in thesense that the assumption of a different model of structureformation will lead to a different halo mass function, andhence assign different masses to the groups. Second, evenif the assumed model of structure formation is correct, itis still not guaranteed that the mass assignment based onthe ranking of group stellar mass (or luminosity) is valid.Finally, even if all groups are assigned with accurate halomasses, the question how dark matter is distributed withinthe galaxy groups remains open. Clearly, it is important tohave independent mass measurements of the haloes asso-ciated with galaxy groups to test the validity of the massestimates based on the stellar mass (luminosity) ranking.Gravitational lensing observations, which measure theimage distortions of background galaxies caused by the grav-itational field of the matter distribution in the foreground,provide a promising tool to probe the dark matter distri-bution directly. In particular, galaxy-galaxy weak lensing,which focuses on the image distortions around lensing galax-ies, can be used to probe the distribution of dark mat-ter around galaxies, hence their dark matter haloes. Thegalaxy-galaxy lensing signal produced by individual galax-ies is usually very weak, and so one has to stack the signalfrom many lens galaxies to have a statistical measurement.The first attempt to detect such galaxy-galaxy lensing signalwas reported by Tyson et al. (1984). More recently, with theadvent of wide and deep surveys, galaxy-galaxy lensing canbe studied for lens galaxies of different luminosities, stellarmasses, colors and morphological types (e.g. Brainerd et al.1996; Hudson et al. 1998; McKay et al. 2001; Hoekstra etal. 2003; Hoekstra 2004; Sheldon et al. 2004; Mandelbaumet al. 2005, 2006; Sheldon et al. 2007a; Johnston et al. 2007;Sheldon et al. 2007b; Mandelbaum et al. 2008). Given thatgalaxies reside in dark matter haloes, these results provideimportant constraints on the mass distribution associatedwith galaxies in a statistical way.In this paper, we use the galaxy groups of Y07 selectedfrom the Sloan Digital Sky Survey (SDSS), together withmass models for individual groups, to predict the galaxy-galaxy lensing signal expected from SDSS galaxies. We com-pare our model predictions with the observational results obtained by Mandelbaum et al. (2006) for the same galax-ies. Our goal is threefold. First, we want to test whether themethod of halo-mass assignment to groups adopted by Y07is reliable. Since the method provides a potentially powerfulway to obtain the halo masses associated with the galaxygroups, the test results have general implications for thestudy of the relationship between galaxies and dark matterhaloes. Second, we want to examine in detail the contribu-tions to the galaxy-galaxy lensing signal from different sys-tems, such as central versus satellite galaxies, early-type ver-sus late-type galaxies, and groups of different masses. Suchanalysis can help us interpreting the observational results.Finally, we would like to study how the predicted galaxy-galaxy lensing signal depends on model assumptions, suchas the cosmological model and the density profiles of darkmatter haloes. In a companion paper (Cacciato et al. 2008,hereafter C08), we use the relationship between galaxies anddark matter haloes obtained from the conditional luminosityfunction (CLF) modeling (Yang et al. 2003; van den Bosch etal. 2007) to predict the galaxy-galaxy cross correlation andto calculate the lensing signal, while here we directly use theobserved galaxy groups and their galaxy memberships.This paper is organized as follows. In Section 2 we de-fine the statistical measure that characterizes the galaxy-galaxy lensing effect expected from the mass distributionassociated with the galaxy groups. We provide a brief de-scription of the galaxy group catalogue and the modelsof the mass distribution associated with galaxy groups inSection 3. We present our results in Section 4 and con-clude in Section 5. Unless specified otherwise, we adopt aΛCDM cosmology with parameters given by the WMAP3-year data (Spergel et al. 2007, hereafter WMAP3 cos-mology) in our analysis: Ω m = 0 . Λ = 0 . h ≡ H / (100km s − Mpc − ) = 0 . σ = 0 .
75 .
Galaxy-galaxy lensing provides a statistical measure of theprofile of the tangential shear, γ t ( R ), averaged over a thinannulus at the projected radius R around the lens galax-ies. This quantity is related to the excess surface density(hereafter ESD) around the lens galaxy, ∆Σ, as∆Σ( R ) = γ t ( R )Σ crit = ¯Σ( < R ) − Σ( R ) , (1)where ¯Σ( < R ) is the average surface mass density within R ,and Σ( R ) is the azimuthally averaged surface density at R .Note that, according to this relation, ∆Σ( R ) is independentof a uniform background. In the above equation,Σ crit = c πG D s D l D ls (1 + z l ) (2)is the critical surface density in comoving coordinates, with D s and D l the angular distances of the lens and source, D ls the angular distance between the source and the lens, and z l the redshift of the lens.By definition, the surface density, Σ( R ), is related tothe projection of the galaxy-matter cross-correlation func-tion, ξ g , m ( r ), along the line-of-sight. In the distant observerapproximationΣ( R ) = ¯ ρ Z ∞−∞ h ξ g , m ( p R + χ ) i dχ , (3) c (cid:13) , 1– ?? odeling Galaxy-Galaxy Weak Lensing with SDSS Groups where ¯ ρ is the mean density of the universe and χ is theline-of-sight distance from the lens.The cross-correlation between galaxies and dark mattercan, in general, be divided into a 1-halo term and a 2-haloterm. The 1-halo term measures the cross-correlation be-tween galaxies and dark matter particles in their own hosthaloes, while the 2-halo term measures the cross-correlationbetween galaxies and dark matter particles in other haloes.In the present work, we are interested in the lensing signalson scales R h − Mpc where the observational measure-ments are the most accurate. As we will show in §
4, onsuch scales the signal is mainly dominated by the 1-haloterm. Nevertheless, our model also takes the contribution ofthe 2-halo term into account. More importantly, since cen-tral galaxies (those residing at the center of a dark matterhalo) and satellite galaxies (those orbiting around a cen-tral galaxy) contribute very different lensing signals (e.g.Natarajan, Kneib & Smail 2002; Yang et al. 2006; Limousinet al. 2007), it is important to model the contributions fromcentral and satellite galaxies separately.As an illustration, in Fig. 1 we show the ESDs expectedfrom a single galaxy in a host halo of mass 10 h − M ⊙ .The solid line represents the lensing signal expected for thecentral galaxy of the halo. While the dotted and dashedlines show the lensing signal of a satellite galaxy residingin a sub-halo of 10 h − M ⊙ with a projected halo-centricdistance r p = 0 . h − Mpc and r p = 0 . h − Mpc from thecenter of the host halo, respectively. In the calculation, thedark matter mass distribution in the host halo is assumed tofollow the Navarro, Frank & White (1997) profile and thatin the sub-haloes is assumed to follow the Hayashi et al.(2003) model. These models are described in detail in § R ) is thenestimated by counting the number of dark matter particlesin a annulus with radius R centred on the selected galaxies.Fig. 1 shows clearly that the lensing signals of the centraland the satellite are quite different. The ESD of the cen-tral galaxy follows the mass distribution of the host halo,decreasing monotonically with R . The ESD of a satellite,on the other hand, consists of two parts: one from the sub-halo associated the satellite, which contributes to the innerpart, and the other from the host halo, which dominatesat larger R . This simple model demonstrates clearly that, inorder to model the galaxy-galaxy lensing signal produced bya population of lens galaxies, one needs to model carefullythe distribution of matter around both host haloes and sub-haloes. To do this, we need not only to identify the haloes inwhich each lens galaxy resides, but also to model the massand density profile of each host halo and subhalo. In addi-tion we also need to model the distribution of dark matterrelative to galaxies. In the following section, we describe ourmodeling with the use of observed galaxy groups. Our analysis is based on the SDSS galaxy group catalogueconstructed by Y07. The groups are selected with the adap-
Figure 1.
The ESD expected for a single galaxy. Here the hosthalo mass is assumed to be 10 h − M ⊙ . The solid line representsthe lensing signal for the central galaxy in such a halo. The dottedline represents the lensing signal of a satellite galaxy residing ina sub-halo of mass 10 h − M ⊙ which has a projected distance r p = 0 . h − Mpc from the center of the host halo. The dashed lineis the same as the dotted line, except that the subhalo’s projecteddistance is 0 . h − Mpc from the center of the host halo. tive halo-based group finder developed by Yang et al. (2005),from the New York University Value Added Galaxy Cata-log (NYU-VAGC; Blanton et al. 2005) which is based on theSDSS Data Release 4 (Adelman-McCarthy et al. 2006). Onlygalaxies with redshifts in the range 0 . z .
2, and withredshift completeness C > .
7, are used in the group identi-fication. The magnitudes and colors of all galaxies are basedon the standard SDSS Petrosian technique (Petrosian 1976;Strauss et al. 2002), and have been corrected for galacticextinction (Schlegel, Finkbeiner & Davis 1998). All mag-nitudes have been K-corrected and evolution-corrected to z = 0 . f c > . f c > .
8, where f c is defined as the ratio between thenumber of true members that are selected as the membersof the group and the number of the total true members ofthe group. c (cid:13) , 1– ?? Ran Li et. al
Figure 2.
The halo mass M S (estimated using stellar mass), versus M ∗ (lower panels) and L (upper panels) of the galaxies in the haloes.The left panels are for central galaxies and the right panels are for satellite galaxies. An important aspect of the group catalog construction isthe determination of the halo mass, M vir , of each group. InY07, two estimators are adopted. The first, M L , is estimatedusing the ranking of the characteristic luminosity of a group,which is the total luminosity of all member galaxies in thegroup with M r − h − . L . ). The second, M S , is estimated using the ranking ofthe characteristic stellar mass, M stellar which is defined to bethe total stellar mass of group members with M r − h − .
5. For each galaxy the stellar mass is estimated from its absolute magnitude and color using the fitting formulagiven by Bell et al. (2003).The basic assumption of the ranking method is thatthere is a one-to-one relation between M stellar (or L . ) andthe group mass. Using the dark matter halo mass functionpredicted by a model of structure formation, one can assigna halo mass to each group according to its M stellar - ranking(or L . - ranking). In this paper, we use the mass functionobtained by Warren et al. (2006). Note that this one-to-onemapping is applicable only when the group sample is com-plete. In Y07, three complete samples are constructed in c (cid:13) , 1– ?? odeling Galaxy-Galaxy Weak Lensing with SDSS Groups Figure 3.
The distribution of the host halo masses for the central and satellite galaxies in different luminosity bins, as indicated by the r -band absolute-magnitude range in each panel. three redshift ranges. Only groups in the complete samplesare used in the ranking. The mass of other groups are esti-mated by a linear interpolation based on the M stellar - M vir relation (or the L . - M vir ) obtained from the completesample. Detailed tests using mock galaxy redshift sampleshave shown that the 1- σ error of the estimated halo massis ∼ . M L and M S , agree remarkably well with each other, with ascatter that decreases from about 0.1 dex at the low-massend to about 0.05 dex at the high-mass end. Since the cor-relation between M stellar and halo mass is somewhat tighter than that between L . and halo mass, we adopt M S as ourfiducial halo mass throughout. As we demonstrate in § M L instead yields results that are fairly similar.Fig. 2 shows the relation between the host halo mass, M S , and the galaxy stellar mass M ∗ (the lower two panels)or the galaxy luminosity L (the upper two panels). Resultsare shown separately for central galaxies (left panels) andsatellite galaxies (right panels). As one can see, the stellarmass (luminosity) of central galaxies is quite tightly cor-related with their host halo masses. However, for satellitegalaxies of a given stellar mass (or luminosity), their host c (cid:13) , 1– ?? Ran Li et. al halo mass covers a very large range, reflecting the fact thatmany low-mass galaxies are satellites in massive haloes. Thedistributions of host halo masses, M S , for central or satel-lite galaxies in different luminosity bins are shown in Fig.3. On average, brighter central galaxies reside in more mas-sive haloes. For faint galaxies, the halo-mass distribution isbroader, again because many faint galaxies are satellites inmassive systems.In the group catalogue, the mass assignment describedabove is used only for groups where the brightest galaxyis brighter than M r − h = − .
5. This is because themass ranking used in the group catalog is based on the totalstellar mass (or total luminosity) of the member galaxiesthat are brighter than M r − h = − .
5. The groupswith no galaxies brighter than this magnitude thus have noassigned rank. As described in Y07, the reason for choosingthis maginitude is a compromise between having a completesample in a relatively large volume and having more groupsthat are represented by a number of member galaxies. Forgroups in which all member galaxies have M r − h > − .
5, a different method has to be adopted. In modeling theluminosity function and stellar mass function of the centralgalaxies based on the same SDSS group catalogue as usedhere, Yang et al. (2008) obtain an average relation betweenthe luminosity (or stellar mass) of the central galaxy andthe halo mass down to M r − h ∼ −
17. We adopt thisrelation to assign halo masses to all groups (including thosecontaining only one isolated galaxy) represented by centralswith M r − h > − .
5. For convenience, the halo massesobtained in this way are also referred to as M S (based onthe stellar mass of central galaxies) and M L (based on the r -band luminosity of the central galaxies), respectively. With the group catalogue described above, we can modelthe dark matter distribution by convolving the halo distri-bution with the density profiles of individual haloes. In ourmodeling of the density profiles, the host halo of a group isassumed to be centered on the central galaxy. There are twoways to define a central galaxy: one is to define the central ina group to be the galaxy with the highest stellar mass, andthe other is to define the central to be the brightest member.For most groups these two definitions give the same results,but there are very few cases (less than ∼ ∼
10% (van den Bosch, Tormen & Giocoli2005). Some of the subhaloes are associated with ‘satellitegalaxies’ in a halo. In our modeling of the galaxy-galaxyweak lensing, we only take into account subhaloes associ-ated with satellite galaxies, treating other subhaloes as partof the host halo. Giocoli, Tormen & van den Bosch (2008)provide a fitting function of the average mass function forsubhaloes at the time of their accretion into the parent haloof a given mass. Using this mass function, we first sample a set of masses for each group mass. We then set the massoriginally associated with a satellite galaxy according to thestellar mass ranking of the satellites in the group. Here weimplicitly assume that the initial subhalo mass function isthe same as the mass function of the subhalos that hostsatellites. This assumption is not proved by any observa-tions, and we have to live with it since a more realistic modelis not currently available. Fortunately, subhalos only con-tribute a small fraction to the total lensing signal on smallscales. The uncertainty here will not have a significant im-pact on any of our conclusions. To obtain the final mass inthe subhalo at the present time, the evolution of the sub-haloes needs to be taken into account. In other words, weneed to know the fraction of the mass that is stripped andhow the structure of a subhalo changes after the stripping.Here it is convenient to introduce a parameter f m which isthe retained mass fraction of the subhalo. Gao et al. (2004)studied the radial dependence of the retained mass fraction f m from a large sample of subhaloes in a large cosmologicalsimulation. In their work, f m is considered as a function of r s /r vir , h , where r s is the distance of the subhalo from thecenter of the host halo and r vir , h is the virial radius of thehost halo. The simulation of Gao et al. gives f m = 0 . r s /r vir , h ) / . (4)We will adopt this in our modeling of the masses associatedwith subhaloes. However, in the group catalogue, only theprojected distance, r p , from the group center is available.The 3D-distance, r s , is obtained by randomly sampling theNFW profile of the host halo with the given projected radius r p . Thus, the mass assigned to a subhalo is determined bythe following three factors: (1) the stellar mass of the satel-lite galaxy; (2) the host halo mass; (3) the distance betweenthe satellite and the center of the host. Here the host halomass comes into our calculation in two ways. It not onlydetermines the subhalo mass function, but also affects theparameter f m in Eqs.4. Note that the accretion history ofthe host halo may also affect the value of f m . We have toneglect such effect because it is unclear how to model theaccretion histories for individual groups.For host haloes, we use the following NFW profile(Navarro, Frank & White 1997) to model the mass distribu-tion: ρ ( r ) = δ ¯ ρ ( r/r c )(1 + r/r c ) . (5)where ¯ ρ is the mean density of the universe, r c is a scaleradius, related to virial radius r vir by the concentration, c = r vir /r c , and δ is a characteristic over-density related to theaverage over-density of a virialized halo, ∆ vir , by δ = ∆ vir c ln(1 + c ) − c/ (1 + c ) . (6)We adopt the value of ∆ vir given by the spherical collapsemodel (see Nakamura & Suto 1997; Henry 2000). Numeri-cal simulations show that halo concentrations are correlatedwith halo mass, and we use the relations given by Macci`oet al. (2007), converted to our definition of halo mass. Notethat here we use r c , instead of the conventional notation r s ,to denote the scale radius of the NFW profile, as r s has beenused to denote the distance of a subhalo from the center ofits host. c (cid:13) , 1– ?? odeling Galaxy-Galaxy Weak Lensing with SDSS Groups For sub-haloes, we model their density profiles using theresults obtained by Hayashi et al. (2003), who found that thedensity profiles of stripped sub-haloes can be approximatedas ρ s ( r ) = f t r/r t, eff ) ρ ( r ) , (7)where f t is a dimensionless factor describing the reductionin the central density, and r t, eff is a cut-off radius imposedby the tidal force of the host halo. For f t = 1 and r t, eff ≫ r c , ρ s ( r ) reduces to the standard NFW profile ρ ( r ). Here ρ ( r )is calculated using the mass of the subhalo at the time ofits accretion into the host halo. Both f t and r t, eff depend onthe mass fraction of the sub-halo that remains bound, f m .Based on N -body simulations, Hayashi et al. obtained thefollowing fitting formulae relating f t and r t, eff to f m :log( r t, eff /r c ) = 1 .
02 + 1 .
38 log f m + 0 . f m ) ; (8)log( f t ) = − . .
35 log f m +0 . f m ) +0 . f m ) . (9)It should be pointed out, though, that there are substantialuncertainties in modeling the mass distribution around in-dividual satellite galaxies. In particular, many of the resultsabout subhaloes are obtained from N -body simulations, andit is unclear how significant the effect of including baryonicmatter is. Fortunately, the total mass associated with satel-lite galaxies is small (see e.g. Weinberg et al. 2008). Fur-thermore, the contribution of the subhaloes associated withthe satellite galaxies to the galaxy-galaxy lensing signal isconfined to small scales. We therefore expect that these un-certainties will not change our results significantly.With the mass distributions described above, we usea Monte-Carlo method to sample each of the profiles witha random set of mass particles. Note that the halo massassigned to a group in the SDSS Group Catalog is M ,which is the mass enclosed in the radius, r , defined suchthat M = 4 πr (180¯ ρ ) /
3. We therefore sample the par-ticle distribution within r . After all the particles in eachhalo are sampled, we project the positions of all the par-ticles to a plane and calculate ∆Σ( R ) by stacking galax-ies in each of the luminosity bin. Since the mass distribu-tion is isotropic, an arbitrary direction can be chosen forthe stacking. Thus, the projection effect is naturally in-cluded in our calculation. Each of the particles has a mass of10 h − M ⊙ . Our test using particles of lower masses showsthat the mass resolution adopted here is sufficient for ourpurpose. Using 2 times more particles leads to a differenceof about 5% at R ∼ . h − Mpc, and almost no differenceat R > . h − Mpc.
Before presenting our model predictions, we first describethe observational results that we will use for comparison.The observational results to be used were obtained by Man-delbaum et al. (2006), who analyzed the galaxy-galaxy lens-ing effects using galaxies in a sample constructed from theSDSS DR4 spectroscopic sample. Their sample of lensing
Table 1.
The properties of galaxy samples. In each case, theabsolute-magnitude range, the mean redshift, the mean luminos-ity, and the fraction of late-types are listed. Note that L ∗ =1 . × h − L ⊙ Sample M r h z i h L/L ∗ i f late L1 − < M r < −
17 0.031 0.075 0.81L2 − < M r < −
18 0.048 0.191 0.70L3 − < M r < −
19 0.074 0.465 0.54L4 − < M r < −
20 0.111 1.13 0.35L5f − . < M r < −
21 0.145 2.09 0. 22L5b − . < M r < −
22 0.150 3.22 0.12L6f − < M r < − . galaxies is similar to the galaxy sample used in Y07 to con-struct the group catalogue used here. The only difference isthat Mandelbaum et al’s sample includes all galaxies withredshifts in the range 0 . < z < .
35, while the galaxiesin Y07’s group catalogue are in 0 . z .
2. Since, asto be described below, we are interested in the lensing sig-nals around galaxies of given luminosity and morphologicaltype, this difference in redshift range is not expected to havea significant impact on our results. For the faint luminositybins, our galaxy samples should be almost identical to thatof Mandelbaum et al. (2006), because all faint galaxies areat z . ∼ h − M ⊙ , and the change in the halo mass functionaround this mass is small between z = 0 . .
35. Follow-ing Mandelbaum et al. (2006), we split the galaxy sampleinto 7 subsamples according to galaxy luminosity. Table 1shows the properties of these subsamples: the luminosityrange covered by each subsample, the mean redshift, themean luminosity, and the fraction of late-type galaxies. Asexpected the mean redshifts of brightest bins are differentfrom the corresponding redshifts in Mandelbaum et al.Also following Mandelbaum et al. (2006), we split eachgalaxy subsample into two according to galaxy morphology.The separation is made according to the parameter frac devgenerated by the PHOTO pipeline. The value of frac devis obtained by fitting the galaxy profile, in a given band, toa model profile given by frac dev × F deV + (1 − frac dev) × F exp , where F deV and F exp are the de Vaucouleurs and ex-ponential profiles, respectively. As in Mandelbaum et al., weuse the average of frac dev in the g , r and i bands. Galaxieswith frac dev > . frac dev < . c (cid:13) , 1– ?? Ran Li et. al
Figure 4.
Comparison of the lensing signal predicted by the fiducial model with the observational results. Here the ESD is plottedas the function of the transverse distance R for lensing galaxies in different luminosity bins. Data points with error bars are theobservational results of Mandelbaum et al. , while the lines are the model predictions. The dotted, dashed and dot-dashed lines representthe contributions of the ‘1-halo term’ of central galaxies, the ‘1-halo term’ of satellite galaxies, and the ‘2-halo term’ (of both centralsand satellites), respectively. The solid lines show the predicted total ESD. The r -band magnitude range for each case can be found inTable 1. details). Thus, we assume the observational results are unbi-ased, and compare them directly with our model predictions.The observation data used here was kindly provided byR. Mandelbaum. The data used in Fig. 6 has been publishedin Mandelbaum et al. (2006) where lensing signal is calcu-lated for early and late type galaxies separately. R. Mandel-baum also provided the lensing data combining early andlate type galaxies for us to make the comparisons presentedin all other figures. Note that here we only show the com- parison of the lensing signal for galaxies divided accordingto luminosity.The errorbars on the observational points are 1 σ statis-tical error. The systematic error of the galaxy-galaxy lensinghas been discussed in detail in Mandelbaum et al. (2005).The test has been carried out for three source samples: r < r >
21, and high redshift LRGs. The overall sys-tematic error is found to be comparable to or slightly largerthan the statistical error shown here. c (cid:13) , 1– ?? odeling Galaxy-Galaxy Weak Lensing with SDSS Groups Figure 5.
The contribution to the ESD plotted separately for dark matter haloes of different masses. In each panel, the dotted lineshows the contribution from haloes with M S > h − M ⊙ . The dashed line shows the contribution from haloes with 10 h − M ⊙ M S < h − M ⊙ , and the dot-dashed line shows the contribution from haloes with M S < h − M ⊙ . The solid line shows the totallensing signal predicted by the fiducial model. For comparison, the observational data are included as data points with error-bars. In Fig. 4 we show the lensing signal around galaxies in differ-ent luminosity bins obtained from our fiducial model, whichhas model parameters as described in the last section andassumes the WMAP3 cosmology. Here the ESD is plottedas a function of the projected distance R from galaxies.The solid line shows the averaged ESD of all galaxies inthe corresponding luminosity bin. The amplitude of the pre-dicted ESD increases with galaxy luminosity, reflecting thefact that brighter galaxies on average reside in more mas-sive haloes, as shown in Figs. 2 and 3. These results are tobe compared with the data points which show the obser- vational results obtained by Mandelbaum et al. (2006) forthe same luminosity bins. Overall, our fiducial model repro-duces the observational data reasonably well, especially forbright galaxy bins where the observational results are themost reliable. The reduced χ is 3 . χ of 0 .
9. Given that we do not adjust any model parame-ters, the χ indicates a good agreement. For the three low-luminosity bins, the predicted ESD is lower than the corre-sponding observational result. For L1 and L2, the observa-tional data are very uncertain. For L3, if we take the ob-servational data points at face-value, the discrepancy with c (cid:13) , 1– ?? 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 lategalaxies. The data points with error-bars show the observational results. The model predictions of the ESD using stellar mass as halomass indicator are shown as the solid lines. For comparison, the dashed lines show the corresponding model predictions using M L as thehalo masses. the model prediction is significant. As described in § M r − h < − .
5, their halo masses are not obtainedfrom the ranking of M stellar , but from the average stellarmass-halo mass relation of central galaxies that is requiredto match the observed stellar mass function of central galax-ies. While all the galaxies in the bright luminosity bins havetheir host halo mass assigned by ranking method, the frac-tion of the galaxies in halos that have masses assigned ac-cording to the mass-halo mass relation is about 30% in L3 and about 70% in L2 and L1. It is possible that this rela-tion underestimates the halo mass. In order to see the effectcaused by such uncertainties, we have used a set of param-eters from Yang et al. (2008) that are still allowed by theobserved stellar mass function but give larger halo massesto the hosts of faint central galaxies. This increases the pre-dicted ESD for L3 by ∼ R ∼ . . h − Mpc are actually slightly higher than c (cid:13) , 1– ?? odeling Galaxy-Galaxy Weak Lensing with SDSS Groups Figure 7.
The dependence of the predicted ∆Σ( R ) on various model parameters. For comparison, the observational data are includedas data points with error-bars. The left column shows the dependence on halo concentration: the halo concentrations in CL (dotted line)and CH (dashed line) are assumed to be 1/2 and 2 times those in the fiducial model (solid line), respectively. The middle column ofshows the effects of halo center offset. The dotted line and the dashed line show the results of models Dev1 and Dev2 model, respectively,while the solid line is the fiducial model. The right column shows the effect of assuming triaxial halo density profile. The dotted lineshows the result of model TRI, while the solid line again shows the fiducial model. Both the observation and model predicted ∆Σ( R )are normalized by the fiducial prediction. for the brighter sample L4, while in our model the ESD forL3 is always lower than that for L4. There is an effect thatmay help to reduce the discrepancy between our model pre-diction and the observational results for low-mass galaxies.Since the group catalogue is only complete down to certainhalo mass limit at different redshift (see Y07), additionalassumptions have to be made in order to model the dis- tribution of the haloes below the mass limit. In the modeldescribed above, we have assumed that the haloes below themass limit have a random distribution, so that they do notcontribute to the ESD. However, in reality these low-masshaloes are correlated with the more massive ones. As a re-sult, our assumption will underestimate the 2-halo term ofthe ESD. c (cid:13) , 1– ?? Ran Li et. al
Figure 8.
The model prediction of ESD assuming different cosmological models. The solid and dotted lines show the fiducial model andmodel using WMAP1 parameters, respectively . The observational data are plotted as data points with error-bars.
In order to understand how the predicted ESD is pro-duced, we also show separately the contributions from dif-ferent sources. The dotted lines show the ESD contributedby the 1-halo term of central galaxies. For all the luminos-ity bins, this term dominates the ESD at small R . For thebrightest two bins, this term dominates the ESD over theentire range of R studied. This reflects the fact that almostall the galaxies in these two bins are central galaxies and thehaloes in which they reside are more extended. The dashedlines show the ESD contributed by the 1-halo term of satel-lite galaxies. This term first decreases with R and then in-creases to a peak value before declining at large R . Thisowes to the fact that, in the inner part, the lensing signalproduced by satellite galaxies is dominated by the subhaloes associated with them, while at larger R the lensing signalproduced by satellite galaxies is dominated by their host ha-los. The value of R at which the ESD reaches the minimumcorresponds roughly to the average halo-centric distance ofthe subhaloes in the luminosity bin. Note that the contribu-tion of satellites to the total ESD is only important at large R in the low-luminosity bins. This reflects the fact that asignificant fraction of the low-luminosity satellites reside inmassive haloes. Note that although the 1-halo satellite termdominates the lensing signal at R > . h − Mpc for faint lu-minosity bins, the discrepancy between the observation andour prediction cannot be simply solved by boosting up thesatellite contribution. The reason is that the increase of the1-halo satellite term requires the increase of the host halo c (cid:13) , 1– ?? odeling Galaxy-Galaxy Weak Lensing with SDSS Groups mass of the satellites, which will make the 1-halo centralterm of the bright bins increase as well, causing significantdiscrepancy for the bright bins. Finally, the dash-dotted linesrepresent the contribution of the 2-halo term. As expected,this term is relevant only on relatively large scales. In ourmodel this term never dominates at R h − Mpc. How-ever, as discussed before this term may be underestimatedhere. In C08, the 2-halo term is found to be comparable tothe 1-halo satellite term even at R ∼ . h − Mpc. Unfortu-nately the 2-halo term in C08 may be overestimated becausehalo-exclusion effect is not properly included.The large fluctuations seen in L6f are due to the smallnumber of galaxies in this luminosity bin.Since the halo mass of each lensing galaxy is known inour model, we can also examine the contributions to thetotal ESD in terms of the halo mass. Fig. 5 shows the re-sults where the host haloes are split into three bins of M S : M S > h − M ⊙ (dotted lines); 10 h − M ⊙ M S < h − M ⊙ (dashed lines); and M S < h − M ⊙ (dashed-dotted lines). As one can see, the lensing signals in brighterbins are dominated by more massive haloes. Very massivehaloes with M S > h − M ⊙ are not the dominant contrib-utor, even for the brightest luminosity bin considered here,because the total number of galaxies hosted by such haloesare relatively small. For the low-luminosity bins, relativelymassive haloes dominate the ESD at large R , because a sig-nificant fraction of the low-luminosity satellites are hostedby massive haloes (see Figs. 2 and 3). In Fig. 6 we present the results separately for early-type andlate-type galaxies. For a given luminosity bin, the predictedESD has a higher amplitude for early-type galaxies, clearlydue to the fact that early-type galaxies are more likely toreside in massive haloes (see e.g. van den Bosch, Yang &Mo 2003). For the faint samples, L1 and L2, the behaviorof the predicted ESD for early-type galaxies resembles thatof satellite galaxies in these luminosity bins, while the pre-dicted ESD for the late-type galaxies looks like of the centralgalaxies in the corresponding luminosity bins. This, again,reflects the fact that faint early-type galaxies are mostlysatellites in massive haloes, while the faint late-type popula-tion is dominated by the central galaxies in low-mass haloes.The dashes lines in Fig. 6 show the results obtainedusing M L as halo mass, rather than M S (see § M S . However, forthe late-type samples, the ESDs obtained using M L are sig-nificantly higher than those obtained using M S , especiallyfor the brighter samples. This is mainly due to the fact thatlate-type galaxies contain significant amounts of young stars,so that their stellar mass-to-light ratios are relatively low.Consequently, they are assigned a larger halo mass based ontheir luminosity than based on their stellar mass.The model predictions are compared with the observa-tional results of Mandelbaum et al. (2006). Here again, themodel prediction based on M S matches the observationaldata for the four bright samples. For the three faint bins, themodel predictions are again lower than the observational re-sults. As shown in Fig. 6, it seems that our model predictionagrees better with the observation for the late type galaxies. For example, for L3 our model prediction matches the ob-servation reasonably well for the late-type subsample (withthe reduced χ equal to 1 . χ equal to5 . R ∼ . h − Mpc. Unfortunately, it is still unclear if therequired halo mass increase is feasible in current models ofgalaxy formation.
The ESD signal predicted by the model outlined in § § § spher-ical dark matter halo. In this subsection, we test how ourresults are affected by these assumptions.The concentration parameter is a measure of theamount of dark matter in the central regions of the haloes.Accordingly, different models for the concentration-halomass relations are expected to result in different predictionsfor the ESD signal (at least on small scales). The fiducialmodel described in § N -body simulations. Finally, attempts to measure halo con-centrations observationally have thus far given conflictingresults (e.g. van den Bosch & Swaters 2001; Comerford &Natarajan 2007, and referenecs therein). To examine howthe lensing predictions depend on changes in halo concenta-tion we consider two models: model CH, in which the con- c (cid:13) , 1– ?? Ran Li et. al centrations are 2 times as high as in our fiducial model, andmodel CL, in which the concentrations are 2 times smaller.The left column of Fig. 7 shows the predictions of mod-els CH (dashed lines) and CL (dotted lines) compared toour fiducial model (solid lines). Results are only shown forfour luminosity bins, as indicated. Note that the model withhigher (lower) halo concentrations predicts ESDs that arehigher (lower). The effect is stronger on scales where the1-halo central term dominates (see Fig 4). Accordingly, inthe case of the brightest sample (L6f), models CL and CHdiffer from the fiducial model on scales up to R ≃ h − Mpc,while in sample L4 the differences are only appreciable outto R ≃ . h − Mpc. We conclude that the predicted ESDdepends quite strongly on the assumed halo concentrations,indicating that galaxy-galaxy lensing has the potential toconstrain the density profiles of dark matter haloes (seeMandelbaum et al. 2008). In this paper, we have assumedthat the concentration of a halo depends only on its mass,we have ignored the possible halo age-dependence of theconcentration-mass relation (e.g. Wechsler et al. 2002; Zhaoet al. 2003; Lu et al. 2006). If for a given mass older haloshave higher concentration, and if the formation of a galaxy ina halo depends strongly on the formation history of the halo,then the age-dependence of the halo concentration must betaken into account. Unfortunately, it is unclear how to con-nect the halo age (hence halo concentration) with the prop-erties of galaxies. For a given halo mass, the dispersion inthe concentration is about 0.12 dex (e.g. Jing 2000), whichcorresponds to a change of about 20% in the predicted ESDon R ∼ . h − Mpc.In our fiducial model, central galaxies are assumed toreside at the center of their dark matter haloes. However, asshown in van den Bosch et al. (2005), in haloes with masses M h > h − M ⊙ there is evidence to suggest that centralgalaxies are offset from their halo centers by ∼ M S > h − M ⊙ are offsetfrom their halo centers by an amount that is drawn froma Gaussian distribution with zero mean. We use two dif-ferent values for the dispersion of the distribution: 3% ofthe virial radius (model ‘Dev1’) and 6% of the virial radius(model ‘Dev2’). The corresponding lensing predictions areshown in the middle column of Fig. 7 as the dotted (Dev1)and dashed (Dev2) lines. Note that the offsets only affectthe lensing signal for the brightest sample (L6f), where alarger offset results in a stronger suppression of the ESD onsmall scales ( R < . h − Mpc). For fainter samples, no (sig-nificant) differences with respect to the fiducial model areapparent, which owes to the fact that fainter centrals typi-cally reside in haloes with M h < h − M ⊙ which do nothave an offset (at least in our models).As a final test, we examine the impact of halo shapeson the galaxy-galaxy lensing signal. In our fiducial model,dark matter haloes are assumed to be spherically symmetric.However, N -body simulations show that, in general, they aretriaxial rather than spherical. Jing & Suto (1998) proposeda fitting formula for triaxial dark matter haloes, which hasbeen applied to both strong and weak lensing analyses (e.g. Oguri, Lee & Suto 2003; Oguri & Keeton 2004; Tang & Fan2005). In order to examine the impact of our assumptionof halo sphericity, we consider an alternative model (model‘TRI’), in which we assume that the dark matter densitydistribution is given by ρ TRI ( R ), where R specifies an ellip-soidal surface: R = „ x a + y b + z c « c . (10)Here a b c are the three principal semi-axes of theellipsoid. We set ρ TRI ( R ) = ρ ( R ′ ), where ρ ( R ′ ) is the NFWprofile, so that the total mass within a sphere of radius R ′ in ρ ( R ′ ) is equal to the mass within the elliptical shell at R .For the axis ratios we adopt the distribution function givenby Jing & Suto (2002): p ( a/c ) = 1 √ π × . „ M vir M NL « . z )] . × exp − h ( a/c )( M vir /M NL ) . z )] . − . i . (11)and p ( a/b | a/c ) = 32(1 − max( a/c, . × " − „ a/b − − max( a/c, . − max( a/b, . « , (12)where M NL is the characteristic mass scale, on which therms of the top-hat smoothed over-density is equal to 1 .
68. Inpractice, we proceed as follows: For each dark matter halo wefirst draw the axis ratio a/c and a/b using Eqs. (11) and (12),respectively. Next we draw a random 3D orientation of theprincipal axes, and project the dark matter particles alongthe (fiducial) line-of-sight. Next, for each halo, we determinethe major axis of the projected distribution which we alignwith the major axis of the central galaxy. This assumptionis motivated by observational claims that the major axis ofa central galaxy tends to aligned with that of its host halo(e.g. Yang et al. 2006; Faltenbacher et al. 2008; Wang et al.2008). As shown in the right-hand panels of Fig. 7, changingfrom spherical haloes (solid lines) to triaxial haloes (dottedlines) has almost no impact on the predicted ESDs. Thisshould not come as an entire surprise, since the ESDs areazimuthally averaged over many haloes, which have randomorientations on the sky. Note that here we assume that thehalo is perfectly aligned with the central galaxy. However,the observational results mentioned above actually suggesta misalignment. We have tried a model in which the orienta-tion of the host halo is uncorrelated with that of the centralgalaxy. The change in the results is very small comparedwith the model assuming perfect alignment.
Another very important assumption in our model predic-tion is the cosmological model in the calculations of the halomass function and the geometrical properties of spacetime.The redshifts of galaxies considered here are restricted to c (cid:13) , 1– ?? odeling Galaxy-Galaxy Weak Lensing with SDSS Groups z .
2, and the impact of changing cosmological parame-ters on the spacetime geometry is quite small in our analysis.On the other hand, changing cosmological parameters canchange the halo masses assigned to individual groups, whichmay have significant impact on the expected lensing signal.In our fiducial model, we adopt the cosmology parametersobtained from the WMAP 3-year data, with Ω m = 0 . Λ = 0 . n = 0 . σ = 0 .
75 (Spergel et al. 2007).As comparison, we will show some results obtained assum-ing another set of cosmological parameters with Ω m = 0 . Λ = 0 . n = 1 .
0, and σ = 0 .
9, which is strongly sup-ported by the first year data release of the WMAP mission(see Spergel et al. 2003) and has been considered in manyprevious studies. In what follows we will refer to this secondset of parameters as the WMAP1 cosmology. Note that thecosmological parameters given by the recent WMAP 5-yeardata (Komatsu et al. 2008) are in between those of WMAP1and WMAP3.Fig. 8 compares the ESD predicted by the fiducial modelusing the WMAP3 cosmology (solid lines) and that pre-dicted by the WMAP1 cosmology (dotted lines). As one cansee, the ESD predicted by WMAP1 is significantly higherthan that predicted by WMAP3, especially for bright galax-ies. Most of this increase is due to changes in the halo massfunction, which causes (massive) groups to be assigned alarger halo mass. The changes in the halo concentrations andthe spacetime geometry play only a minor role. A compari-son with the SDSS data clearly favors the WMAP3 cosmol-ogy over the WMAP1 cosmology, especially for the brighterluminosity bins. The reduced χ for the WMAP1 cosmologyis 21 .
3, much larger than 3 . σ , or our halo mass assignment is in serious error. Inthe following subsection we show that the uncertainties inour halo mass assignments are unlikely to change our resultssignificantly. We therefore conclude that the galaxy-galaxylensing data prefer a ΛCDM model with a relatively low σ .If we use WMAP5 parameters, the model prediction is inbetween WMAP1 and WMAP3, which is still too high tomatched the observed ESD of bright galaxies. In our model, the masses of groups are assigned accordingto the stellar-mass ranking and the halo mass function pre-dicted by the adopted cosmology. The underlying assump-tion is that the mass function of the host haloes of groupsis the same as that predicted by the cosmological model.However, even if the cosmological model adopted is a goodapproximation to the real universe, the observed halo den-sity may be different from the model prediction because ofcosmic variance introduced by the finite observational vol-ume. The effect of such variance is expected to be mostimportant for massive haloes, because the number densityof such systems is small. If, for example, the number den-sity of massive haloes in the observational sample is, dueto cosmic variance, smaller than the model prediction, themass assignment with the use of the theoretical halo massfunction would assign a higher halo mass to groups. Conse- quently, the ESD of bright galaxies, which are biased towardmassive haloes, would be overestimated. Here we test theimportance of such effects by considering the uncertaintiesdue to Poisson fluctuations. We use the halo mass functionpredicted with the WMAP3 cosmology to generate a set ofrandom halo samples, each of which contains the same num-ber of groups as the observational sample. The halo massesin each of these samples is then ranked in descending orderand the halo mass of a given rank is then assigned to thegroup with the same rank in stellar mass. We find that thescatter in the ESD obtained in this way is negligibly small,even for the brightest sample. The reason is that, even forthe brightest sample, the ESD is dominated by haloes withintermediate masses, 10 h − M ⊙ < M S < h − M ⊙ (seeFig. 5), and the total number of such groups is quite large.Another uncertainty in the mass assignment may arisefrom fiber collisions. In the SDSS survey, no two fibers on thesame SDSS plate can be closer than 55 arcsec. Although thisfiber collision constraint is partially alleviated by the factthat neighboring plates have overlap regions, ∼ c (cid:13) , 1– ?? Ran Li et. al our conclusions about the comparison between the observa-tional data and the model predictions are robust against theuncertainties due to fiber collisions.The scatter in the relation between halo mass and stel-lar mass (or luminosity) may also produce some uncertain-ties in our model prediction. As shown in Mandelbaum &Seljak (2007), the scatter is in partial degeneracy with cos-mology model. In our investigation, we have fixed cosmolog-ical parameters and allowed no dispersion in the halo mass-total stellar mass relation. If, for example, we assume a lognormal distribution with a dispersion of 0.3 index in halomass for a given stellar mass, the predicted ESD would be afew percent larger than that predicted by the fiducial model.
In this paper we model the galaxy-galaxy lensing signal ex-pected for SDSS galaxies, using the galaxy groups selectedfrom the SDSS to represent the dark matter haloes withinwhich the galaxies reside. We use the properties of the darkhalo population, such as mass function, density profiles andshapes, expected from the current ΛCDM cosmogony tomodel the dark matter distribution in each of the groupsidentified in the SDSS volume. The use of the real galaxygroups allows us to predict the galaxy-galaxy lensing signalsseparately for galaxies of different luminosity, morphologi-cal types, and in different environments (e.g. central versussatellite galaxies). We check the robustness of our model pre-dictions by changing the assumptions about the dark matterdistribution in individual groups (such as the shape, densityprofile, and center offset of dark matter haloes), as well asthe cosmological model used to predict the properties of thehalo population. We compare our model predictions with theobservational data of Mandelbaum et al. (2006) for similarsamples of lens galaxies. Although there is some discrepancyfor lens galaxies in the low-luminosity bins, the overall obser-vational results can be well understood in the current ΛCDMcosmogony. In particular, the observed results can be wellreproduced in a ΛCDM model with parameters based onthe WMAP3 data, but a ΛCDM model with a significantlyhigher σ , such as the one based on the WMAP1 data, sig-nificantly over-predicts the galaxy-galaxy lensing signal. Ourresults also suggest that, once a correct model of structureformation is adopted, the halo masses assigned to galaxygroups based on ranking their stellar masses with the halomass function, are statistically reliable. The results obtainedimply that galaxy-galaxy lensing is a powerful tool to con-strain both the mass distribution associated with galaxiesand cosmological models. In the future, when deep imag-ing surveys provide more sources with high image qualityin the SDSS sky coverage, the galaxy-galaxy lensing signalsproduced by the SDSS galaxies can be estimated to muchhigher accuracy. The same analysis as presented here is ex-pected to provide stringent constraints on the properties ofthe dark matter haloes associated with galaxies and galaxysystems, as well as on cosmological parameters. ACKNOWLEDGMENTS
We thank Rachel Mandelbaum for providing the SDSS lens-ing data in electronic format. Part of the computation wascarried out on the SGI Altix 330 system at the Depart-ment of Astronomy, Peking University. Li Ran is supportedby the National Scholarship from China Scholarship Coun-cil. XY is supported by the
One Hundred Talents projectof the Chinese Academy of Sciences and grants from NSFC(Nos. 10533030, 10673023). HJM would like to acknowledgethe support of NSF AST-0607535, NASA AISR-126270 andNSF IIS-0611948. Zuhui Fan would like to acknowledge thesupports from NSFC under grants. 10373001, 10533010, and10773001, and 973 Program (No. 2007CB815401).
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