Galaxy Clustering & Galaxy-Galaxy Lensing: A Promising Union to Constrain Cosmological Parameters
Marcello Cacciato, Frank C. van den Bosch, Surhud More, Ran Li, H.J. Mo, Xiaohu Yang
aa r X i v : . [ a s t r o - ph ] J u l Mon. Not. R. Astron. Soc. , 1–19 (2008) Printed 30 October 2018 (MN L A TEX style file v2.2)
Galaxy Clustering & Galaxy-Galaxy Lensing:A Promising Union to Constrain Cosmological Parameters
Marcello Cacciato ⋆ , Frank C. van den Bosch , Surhud More , Ran Li , ,H.J. Mo , Xiaohu Yang Max-Planck-Institute for Astronomy, K¨onigstuhl 17, D-69117 Heidelberg, Germany Department of Astronomy, University of Massachussetts, Amherst, MA 01003-9305, USA Department of Astronomy, Peking University, Beijing 100871, China Shanghai Astronomical Observatory, Nandan Road 80, Shanghai 200030, China
ABSTRACT
Galaxy clustering and galaxy-galaxy lensing probe the connection between galaxiesand their dark matter haloes in complementary ways. Since the clustering of darkmatter haloes depends on cosmology, the halo occupation statistics inferred from theobserved clustering properties of galaxies are degenerate with the adopted cosmology.Consequently, different cosmologies imply different mass-to-light ratios for dark matterhaloes. Galaxy-galaxy lensing, which yields direct constraints on the actual mass-to-light ratios, can therefore be used to break this degeneracy, and thus to constraincosmological parameters. In this paper we establish the link between galaxy lumi-nosity and dark matter halo mass using the conditional luminosity function (CLF),Φ( L | M )d L , which gives the number of galaxies with luminosities in the range L ± d L/ M . We constrain the CLF parameters using the galaxyluminosity function and the luminosity dependence of the correlation lengths of galax-ies. The resulting CLF models are used to predict the galaxy-galaxy lensing signal.For a cosmology that agrees with constraints from the cosmic microwave background,i.e. (Ω m , σ ) = (0 . , . m , σ ) = (0 . , . Key words: galaxies: halos — large-scale structure of Universe — dark matter —cosmological parameters — gravitational lensing — methods: statistical
With the advent of large galaxy redshift surveys, it has be-come possible to obtain accurate measurements of the clus-tering of galaxies as a function of their properties, such as lu-minosity, morphology and color (e.g. Guzzo et al. 2000; Nor-berg et al. 2001, 2002; Zehavi et al. 2005; Wang et al. 2007). ⋆ International Max-Planck Research School FellowE-mail: [email protected]
Since galaxies are believed to form and reside in dark mat-ter haloes, the clustering strength of a given population ofgalaxies can be compared to that of dark matter haloes aspredicted by numerical simulations or the extended Press-Schechter formalism. Such a comparison reveals a wealth ofinformation about the so-called galaxy-dark matter connec-tion (e.g. Jing, Mo & B¨orner 1998; Peacock & Smith 2000;Berlind & Weinberg 2002; Yang, Mo & van den Bosch 2003;van den Bosch, Yang & Mo 2003; van den Bosch et al. 2007).Unfortunately, this method of constraining the link be-tween galaxies and dark matter haloes using galaxy clus- c (cid:13) Cacciato et al. tering has one important shortcoming: the halo occupationstatistics inferred from the observed clustering propertiesdepend on the cosmological parameters adopted. More pre-cisely, models based on different cosmologies can fit the clus-tering data equally well by simply relying on different halooccupation statistics or, equivalently, different mass-to-lightratios. In order to break this degeneracy between cosmol-ogy and halo occupation statistics independent constraintson the mass-to-light ratios are required (e.g. van den Bosch,Mo & Yang 2003; Tinker et al. 2005). One method thatcan provide these constraints is galaxy-galaxy lensing (here-after g-g lensing), which probes the mass distributions (andhence the halo masses) around galaxies. This implies thatthe combination of clustering and lensing in principle holdsthe potential to put constraints on cosmological parameters(Seljak et al. 2005; Yoo et al. 2006).The first attempt to detect g-g lensing was made byTyson et al. (1984), but because of the relatively poor qual-ity of their data they were unable to detect a statistically sig-nificant signal. With the advent of wider and deeper surveysbecoming available, however, g-g lensing has now been de-tected with very high significance, and as function of variousproperties of the lensing galaxies (e.g. Griffiths et al. 1996;Hudson et al. 1998; McKay et al. 2001; Guzik & Seljak 2002;Hoekstra et al. 2003, 2004; Sheldon et al. 2004, 2007a,b;Mandelbaum et al. 2006; Heymans et al. 2006; Johnstonet al. 2007; Parker et al. 2007; Mandelbaum, Seljak & Hi-rata 2008). Unfortunately, a proper interpretation of thesedata in terms of the link between galaxies and dark mat-ter haloes has been hampered by the fact that the lensingsignal can typically only be detected when stacking the sig-nal of many lenses. Since not all lenses reside in haloes ofthe same mass, the resulting signal is a non-trivial aver-age of the lensing signal due to haloes of different masses.Most studies to date have assumed that the relation betweenthe luminosity of a lens galaxy and the mass of its halo isgiven by a simple power-law relation with zero scatter (seeLimousin et al. 2007 for a detailed overview). However, ithas become clear, recently, that the scatter in this relationbetween light and mass can be very substantial (More etal. 2008b, and references therein). As shown by Tasitsiomiet al. (2004), this scatter has a very significant impact onthe actual lensing signal, and thus has to be accounted forin the analysis. In addition, central galaxies (those residingat the center of a dark matter halo) and satellite galaxies(those orbiting around a central galaxy) contribute very dif-ferent lensing signals, even when they reside in haloes ofthe same mass (e.g. Natarajan, Kneib & Smail 2002; Yanget al. 2006; Limousin et al. 2007). This has to be properlyaccounted for (see e.g. Guzik & Seljak 2002), and requiresknowledge of both the satellite fractions and of the spatialnumber density distribution of satellite galaxies within theirdark matter haloes.Over the years, numerous techniques have been de-veloped to interpret galaxy-galaxy lensing measurements(Schneider & Rix 1997; Natarajan & Kneib 1997; Brain-erd & Wright 2002; Guzik & Seljak 2001). Several authorshave also used numerical simulations to investigate the linkbetween g-g lensing and the galaxy-dark matter connnection(e.g., Tasitsiomi et al. 2004; Limousin et al. 2005; Natarajan,De Lucia & Springel 2007; Hayashi & White 2007). It hasbecome clear from these studies that g-g lensing in principle contains a wealth of information regarding the mass distri-butions around galaxies; in addition to simply probing halomasses, g-g lensing also holds the potential to measure theshapes, concentrations and radii of dark matter haloes, andthe first observational results along these lines have alreadybeen obtained (Natarajan et al. 2002; Hoekstra et al. 2004;Mandelbaum et al. 2006; Limousin et al. 2007; Mandelbaumet al. 2008).In this paper we use an analytical model, similar to thatdeveloped by Seljak (2000) and Guzik & Seljak (2001), topredict the g-g lensing signal as a function of the luminosityof the lenses starting from a model for the halo occupationstatistics that is constrained to fit the abundances and clus-tering properties of the lens galaxies. A comparison of thesepredictions with the data thus allows us to test the mass-to-light ratios inferred from the halo occupation model, andultimately to constrain cosmological parameters. The modelassumes that haloes have NFW (Navarro, Frenk & White1997) density distributions, and that satellite galaxies fol-low a radial number density distribution that is unbiasedwith respect to the dark matter. The occupation statisticsare described via the conditional luminosity function (CLF;see Yang et al. 2003), which specifies the average number ofgalaxies of given luminosity that reside in a halo of givenmass. This CLF is ideally suited to model g-g lensing, as itallows one to properly account for the scatter in the relationbetween luminosity and halo mass, and to split the galaxypopulation in centrals and satellites. We demonstrate howthese different galaxy populations contribute to the lensingsignal in different luminosity bins, and show that uncertain-ties related to the expected concentrations of dark matterhaloes and the radial number density distributions of satel-lite galaxies do not have a significant impact on our results.Assuming a flat ΛCDM cosmology with parameters sup-ported by the third year data release of the Wilkinson Mi-crowave Anisotropy Probe (WMAP, see Spergel et al. 2007),we obtain a CLF that accurately fits the abundances andclustering properties of SDSS galaxies. Using our analyticalmodel, we show that this same CLF also accurately matchesthe g-g lensing data obtained from the SDSS by Seljak etal. (2005) and Mandelbaum et al. (2006) without any addi-tional tuning of the model parameters . However, if we repeatthe same exercise for a cosmology with a matter density andpower-spectrum normalization that are slightly ( ∼
20 per-cent) higher, the model that fits the clustering data canno longer simultaneously fit the g-g lensing data. This con-firms that a joint analysis of clustering and g-g lensing is anextremely promising method to constrain cosmological pa-rameters. In a companion paper (Li et al. 2008), we use theSDSS galaxy group catalogue of Yang et al. (2007) to pre-dict the g-g lensing signal, which we compare to data fromthe SDSS. Although, Li et al. obtain their halo occupationstatistics from a galaxy group catalogue, rather than fromthe galaxy clustering properties, they obtain very similarresults.The present paper is organized as follows. We reviewthe necessary formalism of g-g lensing in §
2, with a detaileddescription of the model used to interpret the g-g lensingsignal. The CLF, used to describe the connection betweengalaxies and dark matter haloes, is introduced in §
3. Theproperties of the predicted g-g lensing signal are illustratedin § c (cid:13) , 1–19 lustering and Galaxy-Galaxy Lensing dictions and SDSS data. A detailed analysis of the assump-tions entering the model is presented in §
5. Conclusions arepresented in § Galaxy-galaxy lensing measures the tangential shear distor-tions, γ t , in the shapes of background galaxies (hereaftersources) induced by the mass distribution around foregroundgalaxies (hereafter lenses). Since the tangential shear dis-tortions due to a typical lens galaxy (and its associateddark matter halo) are extremely small, and since back-ground sources have non-zero intrinsic ellipticities, measur-ing γ t with sufficient signal-to-noise requires large numbersof background galaxies. Except for extremely deep surveysbehind clusters of galaxies, which have a large surface area,the number density of background sources is insufficient for areliable measurement of γ t around individual lenses. In prac-tice, this problem can be circumvented by stacking manylenses according to some observable property. For exam-ple, Mandelbaum et al. (2006) measured γ t as a function ofthe transverse comoving distance R by stacking thousandsof lenses in a given luminosity bin [ L , L ]. The resultingshear γ t ( R | L , L ) holds information regarding the charac-teristic mass of the haloes that host galaxies with luminos-ity L L L , and hence can be used to constrain thegalaxy-dark matter connection.The tangential shear as a function of the projected ra-dius R around the lenses is related to the excess surfacedensity (ESD) profile, ∆Σ( R ), according to∆Σ( R ) = Σ( < R ) − Σ( R ) = γ t ( R )Σ crit . (1)where Σ( R ) is the projected surface density and Σ( < R ) isits average inside R ,¯Σ( < R ) = 2 R Z R Σ( R ′ ) R ′ dR ′ , (2)(Miralda-Escud´e 1991; Sheldon et al. 2004). The so-calledcritical surface density, Σ crit , is a geometrical parametergiven by Σ crit = c πG ω S ω L ω LS (1 + z L ) , (3)with ω S , ω L and ω LS the comoving distances to the source,the lens and between the two, respectively, and with z L theredshift of the lens.The projected surface density is related to the galaxy-dark matter cross correlation, ξ g , dm ( r ), according toΣ( R ) = ρ Z ω s [1 + ξ g , dm ( r )] d ω , (4)where ρ is the average density of matter in the Universe andthe integral is along the line of sight with ω the comovingdistance from the observer. The three-dimensional comovingdistance r is related to ω through r = ω + ω − ω L ω cos θ (see Fig. 1 for an illustration of the geometry). Since ξ g , dm ( r )goes to zero in the limit r → ∞ , and since in practice θ issmall, we can approximate Eq. (4) usingΣ( R ) = 2 ρ Z ∞ R [1 + ξ g , dm ( r )] r d r √ r − R , (5) Figure 1.
Illustration of the geometry between source, lens andobserver which is the expression we adopt throughout.The main goal of this paper is to test the halo occu-pation statistics inferred from galaxy clustering data withg-g lensing data. As is evident from the above equations,the lensing signal ∆Σ( R ) is completely specified by thegalaxy dark matter cross correlation, which, as we demon-strate below, can be computed from a given halo occupationmodel. For computational convenience, we will be working inFourier space, where the related quantity is the galaxy-darkmatter cross power spectrum P g , dm ( k ) = 4 π Z ∞ ξ g , dm ( r ) sin( kr ) kr r d r . (6)In order to compute this power spectrum, we follow Seljak(2000) and Guzik & Seljak (2001), and adopt the halo model,according to which all dark matter is partitioned over darkmatter haloes (see also Mandelbaum et al. 2005a). As usualin the halo model, it is convenient to split P g , dm ( k ) into twoterms; a 1-halo term, which describes the cross correlationbetween galaxies and the dark matter particles that residein the same halo, and a 2-halo term, where each galaxy iscross correlated with the dark matter in all haloes exceptfor the one that hosts the galaxy in question. The compu-tation of these two terms has to account for two importantcomplications. First of all, because of the stacking procedureused in order to achieve sufficient signal-to-noise, the ESDcontains signal from haloes with different masses. A properestimate of P g , dm ( k ), therefore, requires the full probabilitydistribution that a galaxy with the stacking property used(in this case luminosity) resides in a dark matter halo ofmass M . Secondly, central galaxies (those residing at thecenter of a dark matter halo) and satellite galaxies (thoseorbiting around a central galaxy) contribute very differentlensing signals, even when they reside in haloes of the samemass (e.g., Yang et al. 2006). This has to be properly ac-counted for, and requires knowledge of both the satellitefractions and of the spatial number density distribution ofsatellite galaxies within their dark matter haloes. Based onthese considerations, we split P g , dm ( k ) in four terms: P g , dm ( k ) = f c h P , cg , dm ( k ) + P , cg , dm ( k ) i (7)+ f s h P , sg , dm ( k ) + P , sg , dm ( k ) i , where ‘c’ and ‘s’ stand for ‘central’ and ‘satellite’, respec-tively. The reason for explicitely writing the central andsatellite fractions ( f c and f s = 1 − f c , respectively) in theabove equation will become apparent below, in which wedescribe each of these four terms in turn. c (cid:13) , 1–19 Cacciato et al.
The one-halo central term of the power spectrum describesthe dark matter distribution inside haloes hosting centralgalaxies. For a single, central lensing galaxy, it simply re-flects the Fourier transform of the overdensity of the darkmatter halo in which the lens resides: P , cg , dm ( k ) = Mρ u dm ( k | M ) , (8)where u dm ( k | M ) is the normalized Fourier transform of themass density profile, ρ ( r | M ): u dm ( k | M ) = 4 π Z r ρ ( r | M ) M sin( kr ) kr r d r . (9)with r the radius of the halo (see § L , L ], we have that P , cg , dm ( k ) = 1 ρ Z ∞ P c ( M | L , L ) M u dm ( k | M ) d M , (10)where P c ( M | L , L ) is the probability that a central galaxywith luminosity L L L resides in a halo of mass M . This probability function reflects the halo occupationstatistics, and, using Bayes’ theorem, can be written as P c ( M | L , L ) d M = h N c i M ( L , L ) n ( M ) n c ( L , L ) d M . (11)Here h N c i M ( L , L ) is the average number of central galax-ies with luminosities in the range [ L , L ] that reside in ahalo of mass M , n ( M ) is the halo mass function and n c ( L , L ) = Z ∞ h N c i M ( L , L ) n ( M ) d M , (12)is the comoving number density of central galaxies in thegiven luminosity range.Combining (10) and (11), the first term of the galaxy-dark matter power spectrum can be written as f c P , cg , dm ( k ) = 1¯ n tot ¯ ρ Z h N c i M ( L , L ) M u dm ( k | M ) n ( M ) d M (13)where n tot = n c ( L , L ) /f c is the total number density ofall galaxies (centrals plus satellites) with luminosities in therange [ L , L ]. Note that, for brevity, we don’t explicitelywrite the luminosity dependence of f c and n tot , but it isunderstood that f c = f c ( L , L ) and n tot = n tot ( L , L ).The 1-halo satellite term is similar to the 1-halo cen-tral term, except for the fact that satellite galaxies do notreside at the center of their dark matter halo, but followa number density distribution n s ( r | M ). Consequently, the1-halo lensing signal due to satellite galaxies involves a con-volution of n s ( r | M ) with the mass density profile ρ ( r | M )of the host halo in which they reside. Using that in Fourierspace a convolution corresponds to a simple multiplication,we obtain: P , sg , dm ( k ) = 1 ρ Z ∞ P s ( M | L , L ) u s ( k | M ) M u dm ( k | M ) d M , (14)with u s ( k | M ) = 4 π Z r n s ( r | M ) h N s i M ( L , L ) sin( kr ) kr r d r (15) the Fourier transform of n s ( r | M ) normalized by h N s i M ( L , L ) which is the average number of satel-lites with L L L that reside in a halo of mass M .We assume that there is no luminosity segregation amongstthe satellites, so that they all follow the same radial profile,independent of their luminosity. We write the probabilitythat a satellite galaxy with L L L resides in a haloof mass M as P s ( M | L , L ) d M = h N s i M ( L , L ) n ( M ) n s ( L , L ) d M , (16)with n s ( L , L ) = Z ∞ h N s i M ( L , L ) n ( M ) d M , (17)the comoving number density of satellite galaxies with lumi-nosities in the range [ L , L ]. The 1-halo satellite term canthus be written as f s P , sg , dm ( k ) = 1¯ n tot ¯ ρ Z h N s i M ( L , L ) u s ( k | M ) M u dm ( k | M ) n ( M ) d M . (18)where we have used that n tot = n s ( L , L ) /f s . Note that f s = f s ( L , L ). The 2-halo term of the power spectrum describes the corre-lation between galaxies and dark matter particles belongingto separate haloes. Within the halo model, this means crosscorrelating each galaxy with all the dark matter haloes otherthan the one in which the galaxy in question resides. Usingthe fact that dark matter haloes are a biased tracer of thedark matter mass distribution, the contribution to the 2-haloterm due to central galaxies can be written as P , cg , dm ( k ) = P NLdm ( k ) ρ Z ∞ P c ( M | L , L ) b ( M ) d M Z ∞ M ′ u dm ( k | M ′ ) b ( M ′ ) n ( M ′ ) d M ′ . (19)where P NLdm ( k ) and b ( M ) are the non-linear power spectrumof the dark matter and the halo bias function, respectively.The first integral reflects the contribution of the centralgalaxies, while the second integral describes the dark matterdensity field partitioned over haloes. Using (11) we obtain f c P , cg , dm ( k ) = P NLdm ( k ) n tot ρ Z ∞ h N c i M ( L , L ) b ( M ) n ( M ) d M Z ∞ M ′ u dm ( k | M ′ ) b ( M ′ ) n ( M ′ ) d M ′ . (20)Similarly, the satellite part of the 2-halo term is given by f s P , sg , dm ( k ) = P NLdm ( k ) n tot ρ Z ∞ M ′ u dm ( k | M ′ ) b ( M ′ ) n ( M ′ ) d M ′ Z ∞ h N s i M ( L , L ) u s ( k | M ) b ( M ) n ( M ) d M . (21)where the second integral now accounts for the number den-sity distribution of satellite galaxies in haloes of mass M .Note that equations (20) and (21) ignore halo exclusion ,i.e. the fact that, in the halo model, haloes can not overlap. c (cid:13) , 1–19 lustering and Galaxy-Galaxy Lensing In the Appendix, we present an approximate method to takehalo exclusion into account. Far from being a detailed treat-ment, the suggested procedure accounts only for the mostrelevant effect, i.e. the exclusion of dark matter particlesresiding in the same host halo of central galaxies (see Ap-pendix for further details). Unless stated otherwise, all theresults shown throughout the paper are obtained applyinghalo exclusion as modelled in the Appendix.In addition, a technical, as well as conceptual, issuearises in calculating the 2-halo terms introduced in equa-tions (20) and (21). Let us rewrite these two equations inthe following compact form: P , cg , dm ( k ) = P NLdm ( k ) I N c I M ( k ) P , sg , dm ( k ) = P NLdm ( k ) I N s ( k ) I M ( k ) , (22)where I N c , I N s ( k ) and I M ( k ) are I N c = Z ∞ h N c i M ( L , L ) n c b ( M ) n ( M ) d M , I N s ( k ) = Z ∞ h N s i M ( L , L ) n s u s ( k | M ) b ( M ) n ( M ) d M , I M ( k ) = Z ∞ Mρ u dm ( k | M ) b ( M ) n ( M ) d M . (23)The evaluation of these integrals is somewhat tedious nu-merically, as it requires knowledge of the halo mass func-tion and the halo bias function over the entire mass range[0 , ∞ ). Since these have only been tested against nu-merical simulations over a limited range of halo masses(10 h − M ⊙ < ∼ M < ∼ h − M ⊙ ), it is also unclear how ac-curate they are. In practice, though, these problems can becircumvented as follows. First of all, because of the exponen-tial cut-off in the halo mass function, it is sufficiently accu-rate to perform the integrations of Eq. (23) only up to M =10 h − M ⊙ . Secondly, I N c and I N s ( k ) contain the halo oc-cupation statistics, h N c i M ( L , L ) and h N s i M ( L , L ), re-spectively, which, for all luminosities of interest in this pa-per, are equal to zero for M < ∼ h − M ⊙ . Therefore, I N c and I N s ( k ) can be computed accurately by only integratingover the mass range [10 − ] h − M ⊙ . Unfortunately, theintegrand of I M ( k ) does not become negligibly small belowa given halo mass. However, in this case we can use the ap-proach introduced by Yoo et al. (2006): we write I M ( k ) asthe sum of two terms, I M ( k ) = I M ( k ) + I M ( k ), where: I M ( k ) = Z M min M ¯ ρ u dm ( k | M ) b ( M ) n ( M ) d M , I M ( k ) = Z ∞ M min M ¯ ρ u dm ( k | M ) b ( M ) n ( M ) d M . (24)Following Yoo et al. (2006), we use the fact that u dm ( k | M ) =1 over the relevant range of k as long as M is sufficientlysmall. This allows us to write I M ( k ) ≃ Z M min M ¯ ρ b ( M ) n ( M ) d M = 1 − Z ∞ M min M ¯ ρ b ( M ) n ( M ) d M . (25)where the last equality follows from the fact that thedistribution of matter is by definition unbiased with re-spect to itself. Detailed tests have shown that this pro-cedure yields results that are sufficiently accurate as long
Table 1.
Cosmological ParametersΩ m Ω Λ Ω b h n σ WMAP3 0 .
238 0 .
762 0 .
041 0 .
734 0 .
951 0 . . . .
040 0 . . . as M min < ∼ h − M ⊙ . Throughout we adopt M min =10 h − M ⊙ . The computation of the galaxy-dark matter cross correlation(or its power spectrum) as outlined in the previous subsec-tions, requires the following ingredients: • The halo mass function, n ( M ), specifying the comovingnumber density of dark matter haloes of mass M . • The halo bias function, b ( M ), which describes howhaloes of mass M are biased with respect to the overall darkmatter distribution. • The non-linear power spectrum of the dark matter dis-tribution, P NLdm ( k ). • The mass density distribution of dark matter haloes, ρ ( r | M ). • The number density distribution of satellite galaxies indark matter haloes, n s ( r | M ). • The halo occupation statistics for central and satellitegalaxies, as parameterized by h N c i M and h N s i M , respec-tively.All these ingredients depend on cosmology. In this pa-per we consider two flat ΛCDM cosmologies. The first has amatter density Ω m = 0 . b =0 . h = H / (100 km s − Mpc − ) =0 . n = 0 .
951 and a normalization σ = 0 . m = 0 .
3, Ω b = 0 . h = 0 . n = 1 . σ = 0 .
9. With strong support from the first year data re-lease of the WMAP mission (see Spergel et al. 2003), thischoice of parameters has been considered in many previousstudies. In what follows we will refer to this second set ofparameters as the WMAP1 cosmology. For clarity, the pa-rameters of both cosmologies are listed in Table 1.We define dark matter haloes as spheres with an averageoverdensity of 180, with a mass given by M = 4 π ρ ) r . (26)Here r is the radius of the halo. We assume that darkmatter haloes follow the NFW (Navarro, Frenk & White1997) density distribution ρ ( r ) = δ ρ ( r/r ∗ )(1 + r/r ∗ ) , (27)where r ∗ is a characteristic radius and δ is a dimensionlessamplitude which can be expressed in terms of the halo con- c (cid:13) , 1–19 Cacciato et al. centration parameter c dm ≡ r /r ∗ as δ = 1803 c ln(1 + c dm ) − c dm / (1 + c dm ) . (28)Numerical simulations show that c dm is correlated with halomass, and we use the relations given by Macci`o et al. (2007),converted to our definition of halo mass.For the halo mass function, n ( M ), and halo bias func-tion, b ( M ), we use the functional forms suggested by War-ren et al. (2006) and Tinker et al. (2005), respectively, whichhave been shown to be in good agreement with numericalsimulations. The linear power spectrum of density perturba-tions is computed using the transfer function of Eisenstein& Hu (1998), which properly accounts for the baryons, whilethe evolved, non-linear power spectrum of the dark matter, P NLdm ( k ), is computed using the fitting formula of Smith etal. (2003).For the number density distribution of the satellitegalaxies, we assume a generalized NFW profile (e.g., vanden Bosch et al. 2004): n s ( r | M ) ∝ „ r R r ∗ « − α „ r R r ∗ « α − , (29)which scales as n s ∝ r − α and n s ∝ r − at small and largeradii, respectively. Similar to the dark matter mass distribu-tion, n s ( r | M ) has an effective scale radius R r ∗ , and can beparameterized via the concentration parameter c s = c dm / R .Observations of the number density distribution of satel-lite galaxies in clusters and groups seem to suggest that n s ( r | M ) is in good agreement with an NFW profile, forwhich α = 1 (e.g., Beers & Tonry 1986; Carlberg, Yee &Ellingson 1997a; van der Marel et al. 2000; Lin, Mohr &Stanford 2004; van den Bosch et al. 2005a). Several studieshave suggested, however, that the satellite galaxies are lesscentrally concentrated than the dark matter, correspond-ing to R > α = R = 1, forwhich u s ( k | M ) = u dm ( k | M ) (i.e., satellite galaxies followthe same number density distribution as the dark matterparticles). In § α and R .The final ingredient is a model for the halo occupationstatistics. In their attempt to model the g-g lensing signalobtained from the SDSS, Seljak et al. (2005) and Mandel-baum et al. (2006) made the oversimplified assumption ofa deterministic relation between central galaxy luminosityand host halo mass. In particular, they used h N c i M ( L , L ) = M = f M ( L , L )0 otherwise (30)where f M ( L , L ) is the ‘characteristic’ mass of a halo thathosts a central galaxy with L L L . However, a realis-tic relation between central galaxy luminosity and host halomass will have some scatter. As demonstrated by Tasitsiomiet al. (2004), this scatter can have an important impact onthe g-g lensing signal (see also § h N s i M ( L , L ) ∝ M if M > f M ( L , L ) M otherwise (31) In this paper we improve upon the analysis by Seljak etal. (2005) and Mandelbaum et al. (2006) by using a more re-alistic model for the halo occupation statistics. Furthermore,rather than fitting the model to the lensing data, we con-strain the occupation statistics using clustering data fromthe SDSS combined with a large galaxy group catalogue.Subsequently we use that model to predict the g-g lensingsignal which we compare to g-g lensing data obtained fromthe SDSS.As a final remark, we emphasise that different quanti-ties, e.g. n ( M ), b ( M ), and P NLdm ( k ), depend on redshift, z ,even though we have not made this explicit in the equations. In order to specify the halo occupation statistics, we usethe CLF, Φ( L | M )d L , which specifies the average numberof galaxies with luminosities in the range L ± d L/ M (Yang, Mo & van den Bosch2003; van den Bosch, Yang, Mo 2003). Following Cooray &Milosavljevi´c (2005) and Cooray (2006), we write the CLFas Φ( L | M ) = Φ c ( L | M ) + Φ s ( L | M ) , (32)where Φ c ( L | M ) and Φ s ( L | M ) represent central and satellitegalaxies, respectively. The occupation numbers required forthe computation of the galaxy-dark matter cross correlationthen simply follow from h N x i M ( L , L ) = Z L L Φ x ( L | M )d L . (33)where ‘x’ refers to either ‘c’ (centrals) or ‘s’ (satellites). Mo-tivated by the results of Yang, Mo & van den Bosch (2008;hereafter YMB08), who analyzed the CLF obtained fromthe SDSS galaxy group catalogue of Yang et al. (2007), weassume the contribution from the central galaxies to be alog-normal:Φ c ( L | M ) = 1 √ π ln(10) σ c L exp » − (log L − log L c ) σ – . (34)Note that σ c is the scatter in log L (of central galaxies) at afixed halo mass. Moreover, log L c is, by definition, the expec-tation value for the (10-based) logarithm of the luminosityof the central galaxy, i.e.,log L c = Z ∞ Φ c ( L | M ) log L d L . (35)For the contribution from the satellite galaxies we adopt amodified Schechter function:Φ s ( L | M ) = φ ∗ s L ∗ s „ LL ∗ s « α s exp " − „ LL ∗ s « . (36)which decreases faster than a Schechter function at thebright end. Note that L c , σ c , φ ∗ s , α s and L ∗ s are all functionsof the halo mass M . In the parametrization of these massdependencies, we again are guided by the results of YMB08.In particular, for the luminosity of the central galaxies we c (cid:13) , 1–19 lustering and Galaxy-Galaxy Lensing Table 2.
Correlation lengths.Sample . M r − h < z > r V1 ( − . , − .
5] 0 .
173 7 . ± . − . , − .
0] 0 .
135 6 . ± . − . , − .
5] 0 .
109 5 . ± . − . , − .
0] 0 .
089 5 . ± . − . , − .
0] 0 .
058 4 . ± . − . , − .
0] 0 .
038 4 . ± . h − Mpc), obtained by fitting a power-law to the projected cor-relation function over the radial range [0 . , . h − Mpc. SeeWang et al. (2007) for details. adopt L c ( M ) = L ( M/M ) γ [1 + ( M/M )] γ − γ , (37)so that L c ∝ M γ for M ≪ M and L c ∝ M γ for M ≫ M . Here M is a characteristic mass scale, and L = 2 γ − γ L c ( M ) is a normalization. Using the SDSSgalaxy group catalogue, YMB08 found that to good approx-imation L ∗ s ( M ) = 0 . L c ( M ) (38)and we adopt this parameterization throughout. For thefaint-end slope and normalization of Φ s ( L | M ) we adopt α s ( M ) = − . a „ − π arctan[ a log( M/M )] « , (39)and log[ φ ∗ s ( M )] = b + b (log M ) + b (log M ) , (40)with M = M/ (10 h − M ⊙ ). This adds a total of six freeparameters: a , a , b , b , b and the characteristic halo mass M . Neither of these functional forms has a physical motiva-tion; they merely were found to adequately describe the re-sults obtained by YMB08. Finally, for simplicity, and to limitthe number of free parameters, we assume that σ c ( M ) = σ c is a constant. As shown in More et al. (2008b), this assump-tion is supported by the kinematics of satellite galaxies inthe SDSS. Thus, altogether the CLF has a total of elevenfree parameters.Note that, with the parametrization of the CLF intro-duced above, the halo occupation statistics can be rewrittenas: h N c i M ( L , L ) = Z L L Φ c ( L | M )d L = 12 h erf( x ) − erf( x ) i (41) h N s i M ( L , L ) = Z L L Φ s ( L | M )d L = φ ∗ s ( Γ " α s , „ L L ∗ s « − Γ " α s , „ L L ∗ s « , (42) where erf( x i ) is the error function calculated at x i =log( L i /L c ) / ( √ σ c ) with i = 1 , L ), and the galaxy-galaxy correla-tion lengths as a function of luminosity, r ( L ). Here weuse the luminosity function (hereafter LF) of Blanton etal. (2003a) uniformly sampled at 41 magnitudes covering therange − . . M r − h − .
4. Here . M r indicatesthe r -band magnitude K+E corrected to z = 0 . L c ( M ), α s ( M )and φ ∗ s ( M ) obtained by YMB08.For a given set of model parameters, we compute theLF using Φ( L ) = Z ∞ Φ( L | M ) n ( M ) d M . (43)The galaxy-galaxy correlation function for galaxies with lu-minosities in the interval [ L , L ] is computed using ξ gg ( r ) = b ( L , L ) ζ ( r ) ξ NLdm ( r ) . (44)Here ξ NLdm ( r ) is the non-linear correlation function of the darkmatter, which is the Fourier transform of P NLdm ( k ), ζ ( r ) = [1 + 1 . ξ NLdm ( r )] . [1 + 0 . ξ NLdm ( r )] . , (45)is the radial scale dependence of the bias as obtained by Tin-ker et al. (2005), and b gal ( L , L ) is the bias of the galaxies,which is related to the CLF according to b gal ( L , L ) = R ∞ h N i M b ( M ) n ( M ) d M R ∞ h N i M n ( M ) d M , (46)with h N i M = Z L L Φ( L | M )d L = h N c i M ( L , L ) + h N s i M ( L , L ) . (47)the average number of galaxies with luminosities in the range[ L , L ] that reside in a halo of mass M .To determine the likelihood function of our free pa-rameters we follow van den Bosch et al. (2007) and usethe Monte-Carlo Markov Chain (hereafter MCMC) tech-nique. The goodness-of-fit of each model is judged using χ = χ + χ r + χ with χ = X i =1 " Φ( L i ) − ˆΦ( L i )∆ ˆΦ( L i ) , (48) χ r = X i =1 " ξ gg ( r ,i ) − ξ gg ( r ,i ) , (49) c (cid:13) , 1–19 Cacciato et al.
Figure 2.
Upper row, left and central panels . The luminosity function of galaxies and the luminosity dependence of the galaxy correlationlength are plotted. Data come from the analysis of Blanton et al. (2003a) and Wang et al. 2007. The blue contours indicate the 68 and 95percent confidence level obtained from the MCMC. The agreement is extremely accurate for the luminosity function whereas is satisfyingfor the correlation length.
Lower row, three panels . The additional information coming from the group catalogue of YMB08 is plottedtogether with the corresponding 68 and 95 percent confidence level derived with the MCMC. In particular, the halo mass dependence ofthe central galaxy luminosity, the satellite conditional luminosity function normalization φ ∗ s and the the exponent α s are shown in theleft, central and middle panel, respectively. Upper row, right panel . The 68 and 95 percent confidence levels of the satellite fraction, f s ,obtained from the CLF (see eq. [51]). Table 3.
Best-fit CLF parameters obtained from SDSS clustering analysisCosmology log L log M γ γ a a log M b b b σ c χ (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)WMAP3 9.935 11.07 3.273 0.255 0.501 2.106 14.28 -0.766 1.008 -0.094 0.143 1.42WMAP1 9.915 11.16 3.336 0.248 0.484 2.888 14.54 -0.854 0.906 -0.062 0.140 1.70The best-fit CLF parameters obtained from the MCMC analysis for the WMAP3 and WMAP1 cosmologies and the value of thecorresponding reduced χ . Masses and luminosities are in h − M ⊙ and h − L ⊙ , respectively. and χ = X i =1 " log L c ( M i ) − log ˆ L c ( M i )∆ log ˆ L c ( M i ) + X i =1 » α s ( M i ) − ˆ α s ( M i )∆ˆ α s ( M i ) – + X i =1 " φ ∗ s ( M i ) − ˆ φ ∗ s ( M i )∆ ˆ φ ∗ s ( M i ) . (50)Here ˆ . indicates an observed quantity and the subscripts‘Φ’, ‘ r ’ and ‘GC’ refer to the luminosity function, thegalaxy-galaxy correlation length and the group catalogue, respectively. Note that, by definition, ˆ ξ gg ( r ,i ) = 1. Table 4lists the best-fit parameters obtained with the MCMC tech-nique for both the WMAP1 and WMAP3 cosmologies, aswell as the corresponding value of χ = χ /N dof . Here N dof = 74 −
11 = 63 is the number of degrees of freedom.
Fig. 2 shows the results obtained for the WMAP3 cosmology.In each panel the blue contours indicate the 68 and 95 per-cent confidence levels obtained from the MCMC. The upperleft-hand panel shows that the CLF model accurately fitsthe galaxy LF of Blanton et al. (2003a). The fit to the cor- c (cid:13) , 1–19 lustering and Galaxy-Galaxy Lensing Figure 3.
Same as Fig. 2 but for the WMAP1 cosmology. relation lengths as function of luminosity, shown in the up-per middle panel, is less accurate, although data and modeltypically agree at the 1 σ level. The lower panels of Fig. 2show the 68 and 95 percent confidence levels on L c ( M ), φ ∗ s ( M ) and α s ( M ), compared with the results obtained byYMB08 from the SDSS group catalogue of Y07. Since thesedata have been used as additional constraints, it should notcome as a big surprise that the CLF is in good agreementwith these data. We emphasise, though, that it is not triv-ial that a single halo occupation model can be found thatcan simultaneously fit the LF, the luminosity dependenceof the galaxy-galaxy correlation functions, and the resultsobtained from a galaxy group catalogue.Finally, the upper right-hand panel of Fig. 2 shows thesatellite fraction, f s ( L ) = 1Φ( L ) Z ∞ Φ s ( L | M ) n ( M ) d M . (51)as function of luminosity. This is found to decrease from ∼ . ± .
03 at . M r − h = −
17 to virtually zero at . M r − h = −
23. The fact that the satellite fractiondecreases with increasing luminosity is in qualitative agree-ment with previous studies (Tinker et al. 2006; Mandelbaumet al. 2006; van den Bosch et al. 2007).We have repeated the same exercise for the WMAP1cosmology. As evident from Fig. 3, for this cosmology wecan obtain a CLF that fits the data almost equally well (thereduced χ is only slightly higher than for the WMAP3 cos-mology; see Table 3). Note that the group data (shown inthe lower panels) differ from that in Fig. 2, even though thegroup catalogue is the same. This owes to the fact that the halo mass assignments of the groups are cosmology depen-dent (see Y07 for details). The satellite fractions inferredfor this cosmology, shown in the upper right-hand panelof Fig. 3, are similar, though slightly higher, than for theWMAP3 cosmology, in excellent agreement with van denBosch et al. (2007).The fact that both cosmologies allow an (almost)equally good fit to these data, despite the relatively largedifferences in halo mass function and halo bias, illustratesthat the abundance and clustering properties of galaxies al-low a fair amount of freedom in cosmological parameters.However, as demonstrated in van den Bosch, Mo & Yang(2003), the best-fit CLFs for different cosmologies predictdifferent mass-to-light ratios as function of halo mass. Thisis evident from Fig. 4, which shows the mass-to-light ratios M/ h L . i M as function of halo mass inferred from our CLFMCMCs for the WMAP1 and WMAP3 cosmologies. Here h L . i M is the average, total luminosity of all galaxies with . M r − h − . M ,which follows from the CLF according to h L . i M = Z ∞ L min Φ( L | M ) L d L , (52)with L min the luminosity corresponding to a magnitude . M r − h = − .
5. Clearly, the mass-to-light ratiosinferred for the WMAP1 cosmology are significantly higherthan for the WMAP3 cosmology (see also van den Boschet al. 2007, where a similar result was obtained using datafrom the 2dFGRS). Hence, the abundance and clusteringproperties of galaxies can be used to constrain cosmologicalparameters, as long as one has independent constraints on c (cid:13) , 1–19 Cacciato et al.
Figure 4.
The 68 and 95 percent confidence levels for the mass-to-light ratios, M/ h L . i M , obtained from the CLF MCMCs forthe WMAP3 and WMAP1 cosmologies. the mass-to-light ratios of dark matter haloes. This is ex-actly what is provided by g-g lensing. In the next section,we therefore use the CLF models presented here to predictthe g-g lensing signal, which we compare to SDSS data. In order to compute the ESD, ∆Σ, as a function of the co-moving separation on the sky, R , we proceed as follows. Westart by calculating the four different terms of the galaxy-dark matter cross power spectrum defined in § § ξ µ, xg , dm ( r ) = 12 π Z f x P µ, xg , dm ( k ) sin( kr ) kr k d k , (53)where ‘ µ ’ stands for 1h or 2h, and ‘x’ refers to either ‘c’(centrals) or ‘s’ (satellites). These are used to compute thecorresponding four terms of the surface density, Σ µ, x ( R ),Σ µ, x ( R ) = 2 ρ Z ∞ R ξ µ, xg , dm ( r ) r d r √ r − R . (54)Note that we are allowed to neglect the contribution comingfrom the constant background density, ¯ ρ , (cf. equation [5])because it will cancel in defining the ESD (this is known ingravitational lensing theory as the mass-sheet degeneracy).The final ESD then simply follows from∆Σ( R ) = ∆Σ , c ( R ) + ∆Σ , s ( R )+ ∆Σ , c ( R ) + ∆Σ , s ( R ) . (55) in which the relative weight of each term is already includedvia the central and satellite fractions in the definitions ofthe corresponding power spectra .Before comparing the g-g lensing predictions from ourCLF models to actual data, we first demonstrate how thefour different terms contribute to the total ESD. The left-hand panel of Fig. 5 shows the ∆Σ( R ) obtained from ourbest-fit CLF model for the WMAP3 cosmology for three dif-ferent luminosity bins, as indicated . Note that the fainterluminosity bins reveal a more ‘structured’ excess surfacedensity profile, with a pronouced ‘bump’ at R ∼ h − Mpc,which is absent in the ∆Σ( R ) of the brighter galaxies. Thereason for this is evident from the middle and right-handpanels of Fig. 5, which show the contributions to ∆Σ( R )from the four different terms for the faint ( − > . M r − h > −
17) and bright ( − > . M r − h > − . h − Mpc < ∼ R < ∼ h − Mpc, and is responsible forthe pronounced bump on intermediate scales. In the caseof the bright bin, however, the 1-halo central term domi-nates all the way out to R ∼ h − Mpc. This owes to thefact that bright centrals reside in more massive haloes, whichare larger and cause a stronger lensing signal, and due to thefact that the satellite fraction of brighter galaxies is smaller.The fact that the 1-halo satellite term peaks at intermediatescales, rather than at R = 0, owes to the fact that ∆Σ , s ( R )reflects a convolution of the host halo mass density pro-file with the number density distribution of satellite galax-ies. On large scales ( R > ∼ h − Mpc), which roughly reflectstwo times the virial radius of the most massive dark mat-ter haloes, the ESD is dominated by the 2-halo terms. Notethat the faint galaxies, with − > . M r − h > − − > . M r − h > −
22, indicating that they have sim-ilar values for their bias. This owes to the fact that manyof the faint galaxies are satellites which reside in massivehaloes. Note also that the 2-halo central term reveals afairly abrupt truncation at small radii, which owes to halo-exclusion (see Appendix). This truncation also leaves a sig-nature in the total lensing signal, which is more pronouncedfor the fainter lenses. We caution, however, that the sharp-ness of this feature is partially an artefact due to our ap-proximate implementation of halo-exclusion. Nevertheless,it is clear from Fig. 5 that the excess surface densities ob-tained from g-g lensing measurements contain a wealth ofinformation regarding the galaxy-dark matter connection.
The g-g lensing data used here is described in Seljak etal. (2005) and Mandelbaum et al. (2006) and has beenkindly provided to us by R. Mandelbaum. The catalogue of Note that our notation differs slightly from that in Mandelbaumet al. (2006) Here, for simplicity, we have used the halo mass function andhalo bias function computed at z = 0. In § (cid:13) , 1–19 lustering and Galaxy-Galaxy Lensing Figure 5.
The predicted ESD up to large scales ( R ∼ h − Mpc) for three luminosity bins, as indicated. The solid lines refer to thetotal signal as predicted according to our model. The dotted lines refer to the 1-halo central term, whereas the dashed lines refer to the1-halo satellite term. Note that they dominate the signal on different scales (see text). The long dashed lines refer to the 2-halo centralterm. It rises steeply at relatively large scales due to our halo exclusion treatment (see Appendix). The 2-halo satellite term is indicatedwith the dotted-dashed line.
Table 4.
Luminosity bins of the SDSS g-g lensing dataID . M r − h h z i (1) (2) (3)L1 ( − . , − .
0] 0.032L2 ( − . , − .
0] 0.047L3 ( − . , − .
0] 0.071L4 ( − . , − .
0] 0.10L5f ( − . , − .
0] 0.14L5b ( − . , − .
5] 0.17L6f ( − . , − .
0] 0.20Luminosity bins of the lenses. Column (1) lists the ID of each lu-minosity bin, following the notation of Mandelbaum et al. (2006).Column (2) indicates the magnitude range of each luminosity bin(all magnitudes are K+E corrected to z = 0 . lenses consists of 351 ,
507 galaxies with magnitudes − > . M r − h > −
23 and redshifts 0 . < z < .
35 takenfrom the main galaxy catalogue of the SDSS Data Release4 (Adelman-McCarthy et al. 2006). This sample is split in7 luminosity bins (see Table 4), and for each of these lumi-nosity bins the excess surface density profiles, ∆Σ( R ), havebeen determined from measurements of the shapes of morethan 30 million galaxies in the SDSS imaging data down toan apparent r -band magnitude of r = 21 .
8. The resultingdata are shown as solid dots with errorbars in Fig. 6. Werefer the reader to Mandelbaum et al. (2005b, 2006) for adetailed description of the data and of the methods used todetermine the ESD profiles.
Using the methodology outlined in § § § R ), at the mean redshift of the sample, i.e., we usethe halo mass function, n ( M ), the halo bias function, b ( M ),and the non-linear power spectrum, P NLdm ( k ) that correspondto the mean redshift listed in the third column of Table 4.We have verified, though, that computing ∆Σ( R ) simply at z = 0 instead has a negligible impact on the results.The results are shown in Fig. 6, where the solid dotswith errorbars correspond to the SDSS data and the solidlines are the predictions of our best-fit CLF model (whoseparameters are listed in Table 3). Note that this model fitsthe data remarkably well, which is quantified by the factthat the reduced χ is 3 .
1. We emphasize that there are nofree parameters here: the ESD has been computed using theCLF that has been constrained using the LF and the clus-tering data. The good agreement between model and lensingdata thus provides independent support for the halo occu-pation statistics described by our WMAP3 CLF model, inparticular for the mass-to-light ratios and satellite fractions,which have an important impact on the lensing signal.The different curves in each of the panels in Fig. 6show the contribution to the lensing signal due to the fourseparate terms: ∆Σ , c (dotted lines), ∆Σ , s (short-dashedlines), ∆Σ , c (long-dashed lines), and ∆Σ , s (dot-dashedlines). In agreement with the examples shown in Fig. 5, the1-halo central term becomes increasingly more dominant formore luminous lenses. In fact, in the brightest luminositybin (L6f) it dominates over the entire radial range probed.In the low-luminosity bins, most of the observed lensing sig-nal at R > ∼ h − kpc is dominated by the 1-halo satelliteterm. The fact that our model accurately fits the data, thussupports the satellite fractions inferred from our CLF model,and shown in the upper right-hand panel of Fig. 2.Both Seljak et al. (2005) and Mandelbaum et al. (2006)did not account for the contributions of the 2-halo terms intheir analyses of the galaxy-galaxy lensing signal. Our model c (cid:13) , 1–19 Cacciato et al.
Figure 6.
The excess surface density ∆Σ as a function of the comoving transverse separation R is plotted for different bins in luminosityof the lens galaxy (see Table 4). The solid line represents the total signal as predicted by the model, data points and error bars come fromSeljak et al. (2005), see text. The different contributions to the signal are also plotted. The dotted line represents the 1-halo central termwhich obviously dominates at the smallest scales in all cases. Note that this term dominates on larger and larger scales when brightergalaxies are considered, reflecting the idea that brighter galaxies live on average in more massive haloes. The dashed line representsthe 1-halo satellite term which is dominant only for faint galaxies and only on intermediate scales (0.1-1 h − Mpc). The 2-halo centralis plotted with a long dashed line and it becomes relevant on large scales (
R > h − Mpc). Note that the strong truncation of thisterm at small scales is due to our implementation of halo exclusion (see Appendix). The 2-halo satellite term (dotted-dashed line) neverdominates but it can contribute up to 20% of the total signal. indicates that, although the 2-halo terms never dominatethe total signal, they can contribute as much as 50 percentat large radii ( R ≃ h − Mpc). We thus conclude that the2-halo terms cannot simply be ignored.
As shown in § c (cid:13) , 1–19 lustering and Galaxy-Galaxy Lensing Figure 7.
The predictions for the lensing signal, ∆Σ( R ), are shown for two different sets of cosmological parameters (WMAP1 andWMAP3, see text). The green (blue) shaded area corresponds to the 95% confidence level of the WMAP1 (WMAP3) model. Note that,although the cosmological parameters of these two cosmologies only differ by < ∼
20 percent (see Table 1), the ESD predictions are verydifferent, and can easily be discriminated. ferent. Since the galaxy-galaxy lensing signal is very sensi-tive to these mass-to-light ratios, it is to be expected thatour WMAP3 and WMAP1 CLFs will predict significantlydifferent ESD profiles, thus allowing us to discriminate be-tween these two cosmologies.Fig. 7 shows the 95 percent confidence levels for ∆Σ( R )obtained from our CLF MCMCs for both the WMAP3(blue) and WMAP1 (green) cosmologies. Indeed, as antici-pated, for the WMAP1 cosmology we obtain excess surfacedensities that are significantly higher than for the WMAP3 cosmology, in accord with the higher mass-to-light ratios (cf.Fig. 4). A comparison with the SDSS data clearly favors theWMAP3 cosmology over the WMAP1 cosmology. In fact, forthe latter our best-fit CLF model yields a reduced χ of 29 . χ = 3 . m and σ differ only by ∼ ∼ n ). Yet,we can very significantly favor one cosmology over the other.This indicates that the combination of clustering data and c (cid:13) , 1–19 Cacciato et al.
Figure 8.
The average number of central galaxies as a functionof halo mass obtained from our best-fit CLF for the WMAP3cosmology. This is equivalent to the probability P c ( M | L , L )that a central galaxy with L L L is hosted by a haloof mass M . Results are shown for four different luminosity bins,as indicated. Note that brighter centrals reside, on average, inmore massive haloes. In addition, the width of P c ( M | L , L ) alsoincreases with luminosity. g-g lensing data can be used to put tight constraints on cos-mological parameters. A detailed analysis along these linesis deferred to a forthcoming paper (Cacciato et al. , in prepa-ration). Although our computation of the g-g lensing signal does notinvolve any free parameters (these are already constrainedby the clustering data), a number of assumptions are made.In particular, haloes are assumed to be spherical and tofollow a NFW density distribution, central galaxies are as-sumed to reside exactly at the center of their dark matterhaloes, and satellite galaxies are assumed to follow a ra-dial number density distribution that has the same shapeas the dark matter mass distribution. In addition, we madeassumptions regarding the functional form of the CLF. Al-though most of these simplifications are reasonable, andhave support from independent studies, they may have anon-negligible impact on the predictions of the g-g lensingsignal. If this is the case, they will affect the reliability ofthe cosmological constraints inferred from the data. In thissection we therefore investigate how strongly our model pre-dictions depend on these oversimplified assumptions.Some of these dependencies were already investigatedin our companion paper (Li et al. 2008). In particular, wehave shown that the fact that realistic dark matter haloesare ellipsoidal, rather than spherical, can be safely ignored(i.e., its impact on the ESD profiles is completely negligible).On the other hand, if central galaxies are not located exactly at the center of their dark matter haloes, this may have anon-negligible impact on the lensing signal on small scales( R < ∼ . h − Mpc). Fortunately, for realistic amplitudes ofthis offset (van den Bosch et al. 2005b), the effect is fairlysmall and only restricted to the most luminous galaxies (seeLi et al. 2008 for details).Below we investigate three additional model dependen-cies: the scatter in the relation between light and mass, theconcentration of dark matter haloes, and the radial numberdensity distribution of satellite galaxies. To that extent wecompare our fiducial model (the best-fit CLF model for theWMAP3 cosmology presented above), to models in whichwe change only one parameter. L c − M relation An important improvement of our halo occupation modelover that used by Seljak et al. (2005) and Mandelbaum etal. (2006) is that we allow for scatter in the relation be-tween light and mass. In particular, we model the proba-bility function P c ( L | M ) = Φ c ( L | M ) as a log-normal with ascatter, σ c , that is assumed to be independent of halo mass.As demonstrated in More et al. (2008b), this assumption isconsistent with the kinematics of satellite galaxies, and it issupported by semi-analytical models for galaxy formation.Note, though, that this does not imply that the scatter in P c ( M | L ), which is the probability function which actuallyenters in the computation of the g-g lensing signal, is alsoconstant. In fact, it is not. This is illustrated in Fig. 8, whichshows P c ( M | L , L ) of our fiducial model for four luminos-ity bins. Two trends are evident: more luminous centralsreside, on average, in more massive haloes, and they havea larger scatter in halo masses. As discussed in More, vanden Bosch & Cacciato (2008), the fact that the scatter in P c ( M | L ) increases with luminosity simply owes to the factthat the slope of L c ( M ) becomes shallower with increasing M (see the lower right-hand panels of Figs. 2 and 3). As isevident from Fig. 8, this is a strong effect, with the scatter in P c ( M | L ) becoming extremely large at the bright end. Notethat this scatter is not dominated by the width of the lumi-nosity bin. Hence, even if one were able to use infinitesimallynarrow luminosity bins when stacking lenses, the scatter in P c ( M | L ) will still be very appreciable.As first shown by Tasitsiomi et al. (2004), scatter in therelation between light and mass can have a very significantimpact on the ESDs. This is demonstrated in the upper pan-els of Fig. 9, which show the impact on ∆Σ( R ) of changing σ c by 0 .
05 compared to our best-fit CLF value of σ c = 0 . σ c have a negli-gible impact on ∆Σ( R ) for the low luminosity bins. At thebright end, however, relatively small changes in σ c have avery significant impact on ∆Σ( R ). In particular, increas-ing the amount of scatter reduces the ESD. This behaviorowes to the shape of the halo mass function. Increasing thescatter adds both low mass and high mass haloes to the dis-tribution, and the overall change in the average halo massdepends on the slope of the halo mass function. Brightergalaxies live on average in more massive haloes where thehalo mass function is steeper. In particular, when the aver-age halo mass is located at the exponential tail of the halomass function, an increase in the scatter adds many more c (cid:13) , 1–19 lustering and Galaxy-Galaxy Lensing Figure 9.
The impact of various model parameters on ∆Σ( R ). Results are shown for three luminosity bins, as indicated at the top of eachcolumn. In each panel the solid line corresponds to our fiducial model (the best-fit CLF model for the WMAP3 cosmology presented inFig. 6), while the dotted and dashed lines correspond to models in which we have only changed one parameter or model ingredient. Upperpanels: the impact of changes in the parameter σ c , which describes the amount of scatter in Φ c ( L | M ) (see equation [34]). Second rowfrom the top: the impact of changes in the halo concentration, c dm ( M ). In particular, we compare three models for the mass dependenceof c dm : Macci`o et al. (2007; MAC), Bullock et al. (2001; BUL), and Eke et al. (2001; ENS). Third row from the top: the impact ofchanges in R = c dm /c s , which controls the concentration of the radial number density distribution of satellite galaxies relative to that ofthe dark matter. Lower panels:
The impact of changes in α , which specifies the central slope of the radial number density distributionof satellite galaxies. See text for a detailed discussion. low mass haloes than massive haloes, causing a drastic shiftin the average halo mass towards lower values. On the otherhand, fainter galaxies live in less massive haloes, where theslope of the halo mass function is much shallower. Conse- quently, a change in the scatter does not cause a significantchange in the average mass.Clearly, if the g-g lensing signal is used to constrain cos-mological parameters, it is important that one has accurateconstraints on σ c . From the clustering analysis presented in c (cid:13) , 1–19 Cacciato et al. § . ± .
01 (for both WMAP1 and WMAP3).This is in good agreement with previous studies: Cooray(2006), using a CLF to model the SDSS r -band LF, obtained σ c = 0 . +0 . − . . YMB08, using a SDSS galaxy group cata-logue, obtained σ c = 0 . ± .
03, and More et al. (2008b),using the kinematics of satellite galaxies in the SDSS, find σ c = 0 . ± .
04 (all errors are 68% confidence levels). Al-though it is reassuring that very different methods obtainvalues that are in such good agreement, it is clear that theremaining uncertainty may have a weak impact on our abil-ity to constrain cosmological parameters. Fortunately, thescatter only impacts the results at the bright end, so thatone can always check the results by removing data from thebrightest luminosity bins.
The g-g lensing signal on small scales reflects the projectedmass distribution of the haloes hosting the lensing galaxies.Therefore, the detailed shape of ∆Σ( R ) on small scales issensitive to the mass distribution of dark matter haloes. Inour model, we have assumed that dark matter haloes followNFW profiles, which are characterized by their concentra-tion parameters, c dm . Halo concentrations are known to de-pend on both halo mass and cosmology, and various analyti-cal models have been developed to describe these dependen-cies (Navarro et al. 1997; Bullock et al. 2001; Eke, Navarro &Steinmetz 2001; Neto et al. 2007; Macci`o et al. 2007, 2008).Unfortunately, these models make slightly different predic-tions for the mass dependence of c dm (mainly due to the factthat the numerical simulations used to calibrate the modelscovered different limited mass ranges). In Li et al. (2008),we have shown that changing c dm by a factor of two has avery large impact on the ESD profiles. However, this is muchlarger than the typical discrepancies between the differentmodels for c dm ( M ). The second row of panels in Fig. 9 shows∆Σ( R ) obtained for three of these models: the solid lines (la-belled MAC) corresponds to our fiducial model for which wehave used the c dm ( M ) relation of Macci`o et al. (2007). Thedotted lines (labelled BUL) and dashed lines (labelled ENS)correspond to the c dm ( M ) relations of Bullock et al. (2001)and Eke et al. (2001), respectively. The BUL model predictshalo concentrations that are about 15 percent higher thanfor the MAC model. The ENS model predicts a c dm ( M ) thatis somewhat shallower than the BUL and MAC models. Asis evident from Fig. 9, though, the results based on thesethree different models are very similar. We thus concludethat our results are robust to uncertainties in the relationbetween halo mass and halo concentration. In our modelling of the g-g lensing signal, we have assumedthat the number density distribution of satellite galaxies canbe described by a generalised NFW profile (eq. [29]), whichis parameterized by two free parameters: α and R . In themodels presented above, we have assumed that α = R = 1,so that the number density distribution of satellite galax-ies has exactly the same shape as the dark matter distri-bution. As discussed in § R = 1 . R = 0 . R = 2 . R = c dm /c s , so that R > R <
1) corresponds to satellite galaxies being less(more) centrally concentrated than the dark matter. Notethat changes in R have a negligible effect on ∆Σ( R ) for thebright luminosity bins. This simply owes to the fact that theESD of bright lenses is completely dominated by the 1-halocentral term (i.e., the satellite fraction of bright galaxies isvery small). For the fainter luminosity bins, however, an in-crease (decrease) in R causes a decrease (increase) in ∆Σ( R )on intermediate scales (0 . h − Mpc < ∼ R < ∼ h − Mpc), whichis the scale on which the 1-halo satellite term dominates. Theeffect, though, is fairly small (typically smaller than the er-rorbars on the data points).The last row of Fig. 9 shows the impact of changing thecentral slope, α , of n s ( r ). If the number density distribu-tion of satellite galaxies has a central core ( α = 0), ratherthan a NFW-like cusp ( α = 1), it has a similar impact onthe lensing signal as assuming a less centrally concentrated n s ( r ). In fact, the ESD profiles for ( α, R ) = (0 . , .
0) arevery similar to those for ( α, R ) = (1 . , . n s ( r ) (see also Yoo et al. 2006). Clearly, ourconclusion that the WMAP3 cosmology is strongly preferredover the WMAP1 cosmology is not affected by uncertaintiesin the radial distribution of satellite galaxies. Galaxy clustering and galaxy-galaxy lensing probe thegalaxy-dark matter connection in complementary ways.Since the clustering of dark matter haloes depends on cos-mology, the halo occupation statistics inferred from the ob-served clustering properties of galaxies are degenerate withthe adopted cosmology. Consequently, different cosmologiesimply different mass-to-light ratios for dark matter haloes.Galaxy-galaxy lensing, on the other hand, yields direct con-straints on the actual mass-to-light ratios of dark matterhaloes. Combined, clustering and lensing therefore offer theopportunity to constrain cosmological parameters.Although the advent of wide and deep surveys has re-sulted in clear detections of galaxy-galaxy lensing, a properinterpretation of these data in terms of the link betweengalaxies and dark matter haloes has been hampered by thefact that the lensing signal can only be detected when stack-ing the signal of many lenses. Since not all lenses residein haloes of the same mass, the resulting signal is a non-trivial average of the lensing signal due to haloes of differentmasses. In addition, central galaxies (those residing at thecenter of a dark matter halo) and satellite galaxies (thoseorbiting around a central galaxy) contribute very different c (cid:13) , 1–19 lustering and Galaxy-Galaxy Lensing lensing signals, even when they reside in haloes of the samemass (e.g., Yang et al. 2006). This has to be properly ac-counted for, and requires knowledge of both the satellitefractions and of the spatial number density distribution ofsatellite galaxies within their dark matter haloes.In this paper, we model galaxy-galaxy lensing with theCLF, Φ( L | M ), which describes the average number of galax-ies of luminosity L that reside in a halo of mass M . This CLFis ideally suited to model galaxy-galaxy lensing. In particu-lar, it is straightforward to account for the fact that thereis scatter in the relation between the luminosity of a centralgalaxy and the mass of its dark matter halo. This repre-sents an improvement with respect to previous attempts tomodel the g-g lensing signal obtained from the SDSS, whichtypically ignored this scatter (e.g. Seljak et al. 2005; Man-delbaum et al. 2006). However, in agreement with Tasitsiomiet al. (2004), we have demonstrated that the scatter in thisrelation has an important impact on the g-g lensing signaland cannot be ignored. We also improved upon previousstudies by modelling the 2-halo term (the contribution tothe lensing signal due to the mass distribution outside ofthe halo hosting the lens galaxy), including an approximatetreatment for halo exclusion.Following Cooray & Milosavljevi´c (2005), we split theCLF in two components: one for the central galaxies andone for the satellites. This facilitates a proper treatment oftheir respective contributions to the g-g lensing signal. Thefunctional forms for the two CLF components are motivatedby results obtained by Yang et al. (2008) from a large galaxygroup catalogue. For a given cosmology, the free parametersof the CLF are constrained using the luminosity function,the correlation lengths as function of luminosity, and someproperties extracted from the group catalogue. We have per-formed our analysis for two different ΛCDM cosmologies: theWMAP1 cosmology, which has Ω m = 0 . σ = 0 . m = 0 .
238 and σ = 0 . ACKNOWLEDGEMENTS
M.C. acknowledges R. Mandelbaum for providing the lens-ing data in electronic format and for her kind cooperation.M.C. also thanks Alexie Leauthaud for useful discussions,as well as Nikhil Padmanabhan and Joanne Cohn for usefulcomments during the visit at UC Berkeley/LBNL.
REFERENCES
Adelman-McCarthy J.K., et al., 2006, ApJS, 162, 38Beers T.C., Tonry J.L., 1986, ApJ, 300, 557Berlind A. A., Weinberg D.H., 2002, ApJ, 575, 587Blanton M.R., et al. 2003a, ApJ, 592, 819Blanton M.R., et al. 2003b, AJ, 125, 2348Brainerd T.G., Wright O.C., 2002, ASPC, 283, 177Bullock J. S., Kolatt T. S., Sigad Y., Somerville R. S.,Kravtsov A. V., Klypin A. A., Primack J. R., Dekel A.,2001 MNRAS, 321, 559Carlberg R.G., Yee H.K.C., Ellingson E., 1997, ApJ, 478,462Chen J., 2007, preprint (arXiv:0712.0003)Cooray A., Milosavljevi´c, 2005, ApJ, 627, L89Cooray A., 2006, MNRAS, 365, 842De Lucia G., Kaufmann G., Springel V., White S.D.M.,Lanzoni B., Stoehr F., Tormen G., Yoshida N., 2004, MN-RAS, 348, 333Eke V.R., Navarro J.F., Steinmetz M., 2001, ApJ, 554, 114Eisenstein D.J., Hu W., 1998, ApJ, 496, 605Griffiths R.E., Casertano S., Im M., Ratnatunga K.U.,1996, MNRAS, 281, 1159Guzik J., Seljak U., 2001, MNRAS, 321, 439Guzik J., Seljak U., 2002, MNRAS, 335, 311Guzzo L., et al. , 2000, A&A, 355, 1Hayashi E., White S.D.M., 2007, preprint(arXiv:0709.3933)Heymans C., et al. , 2006, MNRAS, 371, L60Hoekstra H., Franx M., Kujiken K., Carlberg R.G., YeeH.K.C., 2003 , MNRAS, 340, 609Hoekstra H., Yee H.K.C., Gladders M.D., 2004 , ApJ, 606,67Hudson M.J., Gwin S.D.J., Dahle H., Kaiser N., 1998, ApJ,503, 531Johnston D.E. Sheldon E.S., Tasitsiomi A., Frieman J.A.,Wechsler R.H., McKay T.A., 2007, ApJ, 656, 27 c (cid:13) , 1–19 Cacciato et al.
Jing Y. P., Mo H.J., B¨orner G., 1998, ApJ, 494, 1Li R., Mo H.J., Fan Z., Cacciato M., van den Bosch F.C.,Yang X., More S., 2008, in preparationLimousin M, Kneib J.P., Natarajan P., 2005, MNRAS, 356,309Limousin M, Kneib J.P., Bardeau S., Natarajan P., CzoskeO., Smail I., Ebeling H., Smith G.P., 2007, A&A, 461, 881Lin Y.-T., Mohr J.J., Stanford S.A., 2004, ApJ, 610, 745Macci`o A.V., Dutton A.A., van den Bosch F.C., Moore B.,Potter D., Stadel J., 2007, MNRAS, 378, 55Macci`o A.V., Dutton A.A., van den Bosch F.C., Moore B.,2008, preprint (arXiv:0804.1926)Magliocchetti M., Porciani C., MNRAS, 2003, 346, 186Mandelbaum R., Tasitsiomi A., Seljak U., Kravtsov A.V.,Wechsler R.H., 2005a, MNRAS, 362, 1451Mandelbaum R., Hirata C. M., Seljak U., Guzik J.,Padmanabhan N., Blake C., Blanton M.R., Lupton R.,Brinkmann J., 2005b, MNRAS, 361, 1287Mandelbaum R., Seljak U., Kauffmann G., Hirata C.M.,Brinkmann J., 2006, MNRAS, 368, 715Mandelbaum R., Seljak U., Hirata C.M., 2008, preprint(arXiv:0805.2552)McKay et al. , 2001, preprint (arXiv: astro-ph/0108013)Miralda-Escud´e J., 1991, ApJ, 370, 1Moore B., Ghigna S., Governato F., Lake G., Quinn T.,Stadel J., Tozzi P., 1999, ApJ, 524L, 19More S., van den Bosch F.C., Cacciato M., 2008a, preprint(arXiv:0807.4529)More S., van den Bosch F.C., Cacciato M., Mo H.J., YangX., Li R., 2008b, preprint (arXiv:0807.4532)Navarro J.F., Frenk C.S., White S.D.M., 1997, ApJ, 490,493Natarajan P., Kneib J.P., 1997, MNRAS, 287, 833Natarajan P., Kneib J.P., Smail I., 2002, ApJ, 580, L11Natarajan P., De lucia G., Springel V., 2006, MNRAS, 376,180Norberg P., et al. , 2001, MNRAS, 328, 64Norberg P., et al. , 2002, MNRAS, 332, 827Parker L..C., Hoekstra H., Hudson M.J., van WaerberkeL., Mellier Y., 2007, ApJ, 669, 21Peacock J.A., Smith R.E., 2000, MNRAS, 318, 1144Schneider P., Rix H.-W., 1997, ApJ, 474, 25Seljak U., 2000, MNRAS, 318, 203Seljak U., et al., 2005, Phys Rev D, 71, 043511Sheldon E.S., et al., 2004, AJ, 127, 2544Sheldon E.S., et al., 2007a, preprint (arXiv:0709.1153)Sheldon E.S., et al., 2007b, preprint (arXiv:0709.1162)Smith R.E., et al., 2003, MNRAS, 341, 1311Spergel D.N., et al., 2003, ApJS, 148, 175Spergel D.N., et al. , 2007, ApJS, 170, 377Tasitsiomi A., Kravtsov A.V., Wechsler R.H., PrimackJ.R., 2004, ApJ, 614, 533Tinker J.L., Weinberg D.H., Zheng Z., Zehavi I., 2005, ApJ,631,41Tinker J.L., Weinberg D.H., Zheng Z., 2006, MNRAS, 368,85Tyson J.A., Valdes F., Jarvis J.F., Mills A.P., 1984, ApJ,281, L59van den Bosch F.C., Yang X., Mo H.J., 2003, MNRAS, 340,771van den Bosch F.C., Mo H.J., Yang X., 2003, MNRAS, 345,923 van den Bosch F.C., Norberg P., Mo H.J., Yang X., 2004,MNRAS, 352, 1302van den Bosch F.C., Yang X., Mo H.J., Norberg P., 2005a,MNRAS, 356, 1233van den Bosch F.C., Weinmann S.M., Yang X., Mo H.J.,Li C., Jing Y.P., 2005b, MNRAS, 361, 1203van den Bosch F.C., et al. 2007, MNRAS, 376, 841van der Marel R.P., Magorrian J., Carlberg R.G., YeeH.K.C., Ellingson E., 2000, AJ, 119, 2038Wang Y., Yang X., Mo H.J., van den Bosch, F.C., 2007,ApJ, 664, 608Warren M.S., Abazajan K., Holz D.E., Teodoro L., 2006,ApJ, 646, 881Yang X., Mo H.J., van den Bosch F.C., 2003, MNRAS, 339,1057Yang X., Mo H.J., van den Bosch F.C., Weinmann S.M.,Li C., Jing Y.P., 2005, MNRAS, 362, 711Yang X., Mo H.J., van den Bosch F.C., Jing Y.P., Wein-mann S.M., Meneghetti M., 2006, MNRAS, 373, 1159Yang X., Mo H.J., van den Bosch F.C., Pasquali A., Li C.,Barden M., 2007, ApJ, 671, 153 (Y07)Yang X., Mo H.J., van den Bosch F.C., 2008, ApJ, 676,248 (YMB08)Yoo J., Tinker J.L., Weinberg D. H., Zheng Z., Katz N.,Dav´e R., 2006, ApJ, 652, 26Zehavi I., et al., 2005, ApJ, 630, 1
APPENDIX A: HALO EXCLUSION
By definition, the 2-halo terms of the galaxy-dark mattercross correlation, ξ g , dm ( r ), only considers pairs of galaxiesand dark matter particles that reside in different haloes.Since two haloes can not overlap spatially, this implies thatthe 2-halo terms given by Eqs. (20) and (21) need to bemodified to take account of halo exclusion. The concept ofhalo exclusion is illustrated in Fig. A1 for the 2-halo centraland 2-halo satellite terms separately. Consider a sphericalhalo of mass M and radius r that hosts a central galaxy.It is clear that this central galaxy cannot contribute anysignal to the 2-halo term on scales smaller than r . Hence,if all central galaxies lived in haloes of a fixed mass M ,then 1 + ξ , cg , dm ( r ) = 0 for r < r . In reality, though, oneneeds to account for the fact that centrals occupy haloesthat span a range in halo masses, even if the centrals allhave the same luminosity. In the case of the satellite galaxiesthe situation is even more complicated. In particular, sincesatellite galaxies can reside at the outskirts of dark matterhaloes, the 2-halo satellite term can still have non-zero powerat r ≪ r . Thus, halo exclusion has less impact on the 2-halo satellite term than on the 2-halo central term.Although the concept of halo exclusion is quite simple,a proper implementation of it in the halo model is extremelytedious numerically. We therefore use only an approximatetreatment, which has the advantage that it is straightfor-ward to implement numerically. First of all, we ignore haloexclusion for the 2-halo satellite term. Since this term is al-ways smaller than the 2-halo central term, and since haloexclusion is less important for satellites than for centrals,this should not have a significant impact on the results. Forthe 2-halo central term we proceed as follows: for each lu-minosity bin, [ L , L ], we simply set 1 + ξ , cg , dm ( r ) = 0 for c (cid:13) , 1–19 lustering and Galaxy-Galaxy Lensing Figure A1.
Illustration of halo exclusion. The upper panel showstwo haloes, of masses M and M ′ , and corresponding radii r and r ′ , respectively. The halo of mass M hosts a central galaxy.Since two haloes cannot overlap, this central galaxy does not con-tribute any signal to the 2-halo central term of the galaxy-darkmatter cross correlation function on scales r < r . In the caseof the 2-halo satellite term, illustrated in the lower panel, there isstill a contribution even on very small scales ( r ≪ r ), simplybecause satellite galaxies can reside near the edge of the halo. r < r ( ¯ M ). Here ¯ M is the average halo mass of the centralgalaxies ¯ M = Z ∞ P c ( M | L , L ) M d M , (A1)where P c ( M | L , L ) is the probability that a central galaxywith luminosity L L L resides in a halo of mass M ,and is given by Eq. (11). The corresponding radius, r ( ¯ M ),follows from Eq. (26).Although this treatment of halo exclusion is clearlyoversimplified, we emphasize that previous attempts to in-clude halo exclusion in the halo model are also approxima-tions (e.g. Magliocchetti & Porciani 2003; Tinker et al. 2005;Yoo et al. 2006). In addition, as is evident from Fig. A2, haloexclusion only has a mild impact on the overall results. Theblack lines, labelled HE, show the ESDs obtained from ourfiducial model in which halo exclusion is implemented usingthe method outlined above. For comparison, the red lines,labelled NOHE, show the results in which we ignore halo ex-clusion altogether (i.e. in which the 2-halo terms are simplycomputed using Eqs. [20] and [21]). The dashed lines showthe corresponding 2-halo central terms, which are clearlysuppressed on small scales in the HE model. Since brightercentral galaxies are hosted by more massive (and thereforemore extended) haloes, the effect of halo exclusion is appar-ent out to larger radii for brighter galaxies. Note also thatthe truncation is fairly sharp; this, however, is partially anartefact due to our approximate treatment in which we haveonly considered the average halo mass ¯ M ( L , L ). In reality,the central galaxies live in haloes that span a range in halomasses, and thus a range in sizes. If this were to be takeninto account, the truncation would still occur at roughly thesame radius, but be less sharp.Although halo exclusion clearly has a strong impact onthe 2-halo central term, the impact on the total ESD is onlymodest. This mainly owes to the fact that the total signal on small scales is completely dominated by the 1-halo terms.Overall, halo exclusion only results in a small reduction ofthe total ESD on intermediate scales. Due to the arteficialsharpness of the break in the 2-halo central term, halo ex-clusion introduces a sharp feature in the total ESD at theradius corresponding to this break. Although the sharpnessof this feature is an artefact of our oversimplified treatmentof halo exclusion, it does not influence our overall results. Infact, including or excluding halo exclusion has only a smallimpact on the total χ -values of our models. For example, forthe WMAP3 cosmology, the reduced χ of our fiducial modelis 3 .
1, compared to 4 . c (cid:13) , 1–19 Cacciato et al.
Figure A2.
The ESD is shown for three luminosity bins. The black lines refer to the fiducial model (HE) and the red lines to themodel without halo exclusion (NOHE). The solid lines indicate the total signal, whereas the long dashed lines show the 2-halo centralterms (note that the we ignore halo exclusion for the 2-halo satellite term). Although the 2-halo central term is strongly affected by haloexclusion, the impact on the total ESD is only mild. Note that the sharpness of the dip in the black solid lines is (at least partially) anartefact of our oversimplified treatment of halo exclusion, as discussed in the text.. c (cid:13)000