Galaxy-Galaxy Lensing in EAGLE: comparison with data from 180 square degrees of the KiDS and GAMA surveys
Marco Velliscig, Marcello Cacciato, Henk Hoekstra, Joop Schaye, Catherine Heymans, Hendrik Hildebrandt, Jon Loveday, Peder Norberg, Cristóbal Sifón, Peter Schneider, Edo van Uitert, Massimo Viola, Sarah Brough, Thomas Erben, Benne W. Holwerda, Andrew M. Hopkins, Konrad Kuijken
MMon. Not. R. Astron. Soc. , 1–15 (2016) Printed 29 August 2018 (MN L A TEX style file v2.2)
Galaxy-Galaxy Lensing in EAGLE: comparison with data from 180square degrees of the KiDS and GAMA surveys
Marco Velliscig, (cid:63) Marcello Cacciato, Henk Hoekstra, Joop Schaye, Catherine Heymans, Hendrik Hildebrandt, Jon Loveday, Peder Norberg, Cristóbal Sifón, , Peter Schneider, Edo van Uitert, Massimo Viola, Sarah Brough, Thomas Erben, Benne W. Holwerda, Andrew M. Hopkins, Konrad Kuijken Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands Scottish Universities Physics Alliance, Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK Argelander-Institut für Astronomie, Auf dem Hügel 71, D-53121 Bonn, Germany Astronomy Centre, Department of Physics and Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, UK ICC and CEA, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA University College London, Gower Street, London WC1E 6BT, UK Australian Astronomical Observatory, PO Box 915, North Ryde, NSW 1670, Australia
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We present predictions for the galaxy-galaxy lensing profile from the EAGLE hydrodynamicalcosmological simulation at redshift z =0.18, in the spatial range 0 . < R / ( h − Mpc) <
2, andfor five logarithmically equi-spaced stellar mass bins in the range 10 . < log ( M star / M (cid:12) ) < .
8. We compare these excess surface density profiles to the observed signal from back-ground galaxies imaged by the Kilo Degree Survey around spectroscopically confirmed fore-ground galaxies from the GAMA survey. Exploiting the GAMA galaxy group catalogue, theprofiles of central and satellite galaxies are computed separately for groups with at least fivemembers to minimise contamination. EAGLE predictions are in broad agreement with theobserved profiles for both central and satellite galaxies, although the signal is underestimatedat R ≈ . − h − Mpc for the highest stellar mass bins. When central and satellite galaxiesare considered simultaneously, agreement is found only when the selection function of lensgalaxies is taken into account in detail. Specifically, in the case of GAMA galaxies, it is crucialto account for the variation of the fraction of satellite galaxies in bins of stellar mass inducedby the flux-limited nature of the survey. We report the inferred stellar-to-halo mass relationand we find good agreement with recent published results. We note how the precision of thegalaxy-galaxy lensing profiles in the simulation holds the potential to constrain fine-grainedaspects of the galaxy-dark matter connection.
Key words: cosmology: large-scale structure of the Universe, cosmology: theory, galaxies:haloes, galaxies: formation, Physical data and processes: gravitational lensing: weak, meth-ods: statistical;
The connection between observable galaxy properties and the un-derlying (mostly dark) matter density field is the result of galaxyformation and evolution in a cosmological context; as such, it isextensively studied from various complementary perspectives. Nu-merous methods are available to probe the mass of dark matter (cid:63)
E-mail: [email protected] haloes within the galaxy formation framework, such as galaxy clus-tering (see e.g. Jing et al. 1998; Peacock & Smith 2000; Zehaviet al. 2002; van den Bosch et al. 2003; Anderson et al. 2014),abundance matching (see e.g. Vale & Ostriker 2004; Moster et al.2013; Behroozi et al. 2013) and stacked satellite kinematics (seee.g. Zaritsky & White 1994; Prada et al. 2003; Conroy et al. 2005;More et al. 2011). These methods require, in various ways, priorknowledge of galaxy formation theory. They are, therefore, limitedin their capacity to produce a stellar mass versus halo mass relationthat can serve as a test for the galaxy formation framework itself. c (cid:13) a r X i v : . [ a s t r o - ph . GA ] D ec Marco Velliscig et al.
For single galaxies, direct methods for estimating the halomass are available (see for a recent review Courteau et al. 2014).The rotation curves of spiral galaxies or the velocity dispersionsof ellipticals can give estimates of the amount of matter associ-ated with a galaxy, albeit at relatively small scales. Furthermore,a galaxy can deflect the light of a background galaxy along theline of sight, possibly into multiple images, providing a measure-ment of the total projected mass within the Einstein radii of galaxies(Kochanek 1991; Bolton et al. 2008; Collett 2015, and referencestherein). The mass of a single group or cluster of galaxies can beestimated via the dynamics of its satellite galaxies (see e.g. Pradaet al. 2003; Conroy et al. 2005), using weak or strong lensing (seee.g. Hoekstra et al. 2015; Fort & Mellier 1994; Massey et al. 2010)or X-ray emission (Ettori et al. 2013, and references therein).For a population of galaxies, galaxy-galaxy weak lensing (seee.g. Brainerd et al. 1996; Wilson et al. 2001; Hoekstra et al. 2004;Mandelbaum et al. 2006; van Uitert et al. 2011; Velander et al.2014; Viola et al. 2015; van Uitert et al. 2015; Leauthaud et al.2015; Mandelbaum et al. 2016) offers the possibility to measure theaverage halo mass directly and therefore represents a viable alterna-tive to constrain the galaxy-dark matter connection and ultimatelytest galaxy formation models. Galaxy-galaxy lensing measures thedistortion and magnification of the light of faint background galax-ies (sources) caused by the deflection of light rays by interveningmatter along the line of sight (lenses). The effect is independentof the dynamical state of the lens, and the projected mass of thelens is measured without any assumption about the physical stateof the matter. The gravitational lensing signal due to a single galaxyis too weak to be detected (it is typically 10 to 100 times smallerthan the intrinsic ellipticity of galaxies) given the typical numberdensity of background sources in wide-field surveys. Therefore thegalaxy-galaxy lensing signal must be averaged over many lenses.From a more theoretical perspective, the link between haloesand galaxies can be studied with an ab-initio approach using SemiAnalytical models and hydrodynamical cosmological simulations.Simulations aim to directly model the physical processes that arethought to be important for the formation of galaxies, as well asthe energetic feedback from supernovae and AGN that is thoughtto regulate their growth (see Somerville & Davé 2015, for a recentreview). However, many of these processes are happening on scalesthat are unresolved by simulations and as such they must be treatedas ‘subgrid’ physics. To gain confidence in these physical recipes,it thus becomes crucial to compare predictions of these models tovarious observations. Arguably, a key test for such studies is to re-produce the observed abundances of galaxies as a function of theirstellar mass (galaxy stellar mass function; hereafter GSMF), as thisis interpreted as the achievement of a successful mapping betweenthe stellar mass and the halo mass. Intriguingly, reproducing a basicquantity such as the GSMF has proven to be extremely challeng-ing for models of galaxy formation. To overcome this limitation,one might reverse the logic and calibrate the unresolved physicalprocesses to reproduce the (present-day) GSMF. This approach,exploited at length in Semi Analytical models, has recently beenadopted in hydro-simulations as well (see e.g. the EAGLE and theBAHAMAS project, Schaye et al. 2015; Crain et al. 2015; Mc-Carthy et al. 2016)In this paper, we compute the predicted weak galaxy-galaxylensing (GGL) profiles of galaxies from the EAGLE hydrodynam-ical simulation sampled according to their stellar mass. These pre-dictions are compared with the observed signal measured usingbackground galaxies imaged by the Kilo Degree Survey (KiDS; deJong et al. 2015) around spectroscopically confirmed foreground galaxies from the Galaxy And Mass Assembly (GAMA) survey(Driver et al. 2011). We refer to this combined data set as KiD-SxGAMA. This comparison represents an independent test of thevalidity of the physical processes implemented in the EAGLE sim-ulation, as they were calibrated to reproduce the galaxy stellar massfunction as well as the observed distribution of galaxy sizes butnot the lensing profiles. As explained in the main body of the pa-per and in Appendix A, a comparison of the GGL profiles offersthe possibility to test fine-grained aspects of the galaxy-dark matterconnection.This paper is organized as follows. In Section 2 we briefly in-troduce the data sets and describe the methodology to obtain thegalaxy-galaxy lensing measurements. In Section 3 we describe theEAGLE simulation employed in this study, the algorithm used toproduce the group catalogue from simulations (§3.1) and the stepstaken to measure the galaxy-galaxy lensing signal in the simu-lations (§3.2). In Section 4 we report the results for the galaxy-galaxy lensing signal from simulations and the comparison withKiDSxGAMA data for central (§4.1) and satellites galaxies (§4.2).In §4.3 we compare the GGL profile for the whole galaxy popula-tion against the KiDSxGAMA observations. We discuss limitationsand possible future improvements of this study in Section 5, sum-marize our findings and conclude in Section 6. We fit the galaxy-galaxy lensing profiles from the EAGLE simulation with simpleanalytical models in Appendix A.Throughout the paper we assume a Λ CDM cosmologicalmodel defined by the following set of parameters { Ω m , Ω b , σ , n s , h ≡ H / h when plotting the galaxy-galaxy lensing profiles to ease the com-parison with other published results. The observational data presented in this paper are obtained fromtwo surveys: KiDS and GAMA. KiDS is an ESO optical imagingsurvey (de Jong et al. 2013) with the OmegaCAM wide-field im-ager on the VLT Survey Telescope. When completed, it will covera total area of 1500 square degrees in four bands ( u , g , r , i ). KiDSwas designed to have both good galaxy shape measurements andphotometric redshift estimates of (background) galaxies. Here weuse the latest KiDS-ESO data release which is described in Hilde-brandt et al. (2016). Details of the survey can be found in de Jonget al. (2015).KiDS overlaps with the GAMA spectroscopic survey (Driveret al. 2011) carried out using the AAOmega multi-object spectro-graph on the Anglo-Australian Telescope (AAT). GAMA equato-rial fields are 98% complete down to a r -band magnitude of 19.8,and cover approximately 180 sq. degrees of sky that fully overlapwith the KiDS footprint. The redshift distribution of GAMA galax-ies (median redshift z ≈ .
25) is ideal for measurements of thegalaxy-galaxy lensing signal using KiDS galaxies as backgroundsources (median redshift z ≈ . c (cid:13) , 1–15 alaxy-Galaxy Lensing in EAGLE M star M crit200 | cen M crit200 | sat M censub M satsub M satsub / M crit200 | sat d sat r dmhalf | cen r dmhalf | sat N gal M limitstar f sat * * * * * ** ** ** *(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)[10 . − .
6] 12 .
46 13 .
95 12 .
47 11 .
57 0 .
03 881 144 28 354 9 .
46 0 . . − .
9] 12 .
92 14 .
09 12 .
92 11 .
95 0 .
03 1081 239 44 150 9 .
91 0 . . − .
2] 13 .
13 14 .
14 13 .
15 12 .
46 0 .
11 1347 261 75 68 9 .
96 0 . . − .
5] 13 .
39 14 .
19 13 .
39 12 .
85 0 .
13 1718 318 108 22 10 .
33 0 . . − .
8] 13 .
69 14 .
24 13 .
69 13 .
61 0 .
30 2802 340 264 29 −− . Table 1.
Various quantities of interest extracted from the EAGLE simulation at z = .
18. From left to right of the columns list: (1) stellar mass range;(2) average halo mass, M crit200 , of haloes hosting central galaxies in each stellar mass bin; (3) same as (2) but for haloes hosting satellites in each stellarmass bin; (4) mean value of the subhalo mass for central galaxies, considering all the particles bound to the subhalo ∗∗∗ ; (5) same as (4) but for satellitegalaxies; (6) average ratio between the mass of the satellite subhalo, M sub , and the mass of its host halo M crit200 ; (7) average 3D distance between thesatellite galaxy and the centre of its host halo; (8) mean radius of central galaxies within which half of the mass in dark matter is enclosed; (9) same as(8) but for satellite galaxies; (10) total number of galaxies in the stellar mass bin; (11) Minimum stellar mass for which a galaxy is considered for thecomputation of the richness of its group in the EAGLE simulation. This value of M limitstar reproduces the satellite fraction in GAMA. Note that the valuefor the stellar mass bin [11 . − .
8] is ill-defined (see discussion in §3.3). (12) average satellite fraction in EAGLE expressed as the total number ofsatellites divided by the total number of galaxies in the mass bin. This value is equal to the satellite fraction in GAMA by construction.* log ( M / [ M (cid:12) ])** R / [kpc]*** Note taht column (2) and (4) have very similar values. This indicates that, in this sample, adopting a spherical overdensity threshold or a FoFalgorithm to define the halo yields to comparable halo masses. A detailed description of how the galaxy-galaxy lensing signalaround GAMA galaxies using KiDSxGAMA data is computed canbe found in Viola et al. (2015) and Dvornik et al. (in prep.). Here,we only summarize the important aspects that enter into the mea-surement.Shape measurements are based on the r -band exposures whichyield the highest image quality in KiDS. Images are processed withthe THELI pipeline (optimized for lensing applications, Erben et al.2005, 2009, 2013), and galaxy ellipticities are computed using the lens fit code (Miller et al. 2007; Kitching et al. 2008; Miller et al.2013). Shape measurements are calibrated against extensive imagesimulations (Fenech Conti et al. 2016). Biases from non-perfectPSF modelling, are quantified and found subdominant as detailedin Kuijken et al. (2015).For every lens-source pair, the measured ellipticity ( e , e ) ofthe source, as estimated by lens fit, is projected along the separationof the lens in a tangential ( e + ) and cross ( e × ) component as: (cid:32) e + e × (cid:33) = (cid:32) − cos(2 φ ) − sin(2 φ )sin(2 φ ) − cos(2 φ ) (cid:33) (cid:32) e e (cid:33) , (1)where φ is the angle between the x-axis and the lens-source sepa-ration vector. Every source lens pair is then weighted by the term:˜ w ls = w s (cid:104) Σ − (cid:105) (2)which is the product of the lens fit weight w s , computed accordingto the estimated reliability of the measured source ellipticity (Milleret al. 2007), and a term (cid:104) Σ − (cid:105) defined via (cid:104) Σ − (cid:105) ls = π Gc D l ( z l ) ∞ (cid:90) z l +∆ z D ls ( z l , z s ) D s ( z s ) n ( z s )d z s , (3)where D l is the angular diameter distance of the lens calculated us-ing the spectroscopic redshift z l , D s is the angular diameter distanceof the source, and we have used ∆ z = . n ( z s ) is the red-shift distribution of the background galaxy population, and D ls is the distance between the lens and the source. We emphasize herethat n ( z s ) is the global redshift distribution of the KiDS galaxies es-timated using the direct calibraton method described in Hildebrandtet al. (2016).The galaxy-galaxy lensing signal, also known as the excesssurface density, ESD, is computed in bins of projected distance R : ∆Σ ( R ) = γ t ( R ) (cid:104) Σ crit (cid:105) ls = (cid:32) (cid:80) ls ˜ w ls e + (cid:104) Σ crit (cid:105) ls (cid:80) ls ˜ w ls (cid:33) + K ( R ) (4)where (cid:104) Σ crit (cid:105) ls ≡ / (cid:104) Σ − (cid:105) ls . Here, the sum is over all lens-sourcepairs in the radial bin, and K ( R ) = (cid:80) ls β ls m s (cid:80) ls β ls (5)is the correction to the ESD profile that takes into account the mul-tiplicative bias m s , with β ls = D ls / D s . Typically, the value of m s isaround − .
012 which results in a 1 / (1 + K ( R )) correction of ∼ . σ ∆Σ = σ e + (cid:32) (cid:80) ls ˜ w (cid:104) Σ − (cid:105) ( (cid:80) ls ˜ w ls ) (cid:33) , (6)where σ e + is the variance of all source ellipticities combined. Wenote here that, from analytical and numerical estimates of the co-variance matrix, we find the covariance between radial bins negli-gible on the scales of interest here.Galaxy-galaxy lensing offers a indirect measure of the pro-jected mass density: ∆Σ ( R ) ≡ ¯ Σ ( < R ) − Σ ( R ) , (7)where ∆Σ is the difference between the surface density averagedwithin R , ¯ Σ ( < R ), and measured at R , Σ ( R ). In this work, we make use of the group catalogue of the GAMA sur-vey (G3Cv7, Robotham et al. 2011) and version 16 of the stellar c (cid:13)000
012 which results in a 1 / (1 + K ( R )) correction of ∼ . σ ∆Σ = σ e + (cid:32) (cid:80) ls ˜ w (cid:104) Σ − (cid:105) ( (cid:80) ls ˜ w ls ) (cid:33) , (6)where σ e + is the variance of all source ellipticities combined. Wenote here that, from analytical and numerical estimates of the co-variance matrix, we find the covariance between radial bins negli-gible on the scales of interest here.Galaxy-galaxy lensing offers a indirect measure of the pro-jected mass density: ∆Σ ( R ) ≡ ¯ Σ ( < R ) − Σ ( R ) , (7)where ∆Σ is the difference between the surface density averagedwithin R , ¯ Σ ( < R ), and measured at R , Σ ( R ). In this work, we make use of the group catalogue of the GAMA sur-vey (G3Cv7, Robotham et al. 2011) and version 16 of the stellar c (cid:13)000 , 1–15 Marco Velliscig et al. mass catalogue, which contains approximately 180 000 objects,divided into three separate 12 × b = . . The GAMA group catalogue has beentested against mock data and ensures reliable central-satellite dis-tinction against interlopers for groups with 5 or more members( N FoF ≥
5) above the completeness limit of GAMA of approxi-mately log ( M star / M (cid:12) ) = ∆Σ profileswould have a significantly lower signal-to-noise ratio. We opt forthe construction of a (nearly) volume-limited lens sample followingthe iterative methodology described in Lange et al. (2015). Fromthis sample, we select only galaxies that reside in groups with atleast 5 members.Fig. 1 shows the stellar mass-redshift plane for the GAMAgalaxies (grey points) in the area overlapping with KiDS. Blackand coloured points show which of those GAMA galaxies are inthe (nearly) volume limited sample and at the same time belong togroups with five or more members. Points are coloured accordingto the stellar mass bin they belong to (see column 1 of Table 1). We compare the observed ESD profile to the predictions fromthe hydrodynamical cosmological simulations from the EAGLEproject (Schaye et al. 2015; Crain et al. 2015) with a cubic vol-ume of 100 Mpc . EAGLE was run using a modified version of the N -Body Tree-PM smoothed particle hydrodynamics (SPH) code GADGET -3, which was last described in Springel (2005). The mainmodifications with respect to
GADGET -3 regard the formulation ofthe hydrodynamics, the time stepping and the subgrid physics. Darkmatter and baryons are represented by 2 × particles, with aninitial particle mass of m b = . × M (cid:12) and m dm = . × M (cid:12) We note that stellar masses of GAMA galaxies have been estimated inTaylor et al. (2011). In short, stellar population synthesis models fromBruzual & Charlot (2003) that assume a Chabrier (2003) Initial Mass Func-tion (IMF) are fit to the ugriz -photometry from SDSS. NIR photometryfrom VIKING is used when the rest-frame wavelength is less than 11 000Å. The EAGLE simulation catalogue used throughout this paper uses thesame value of the linking length (see §3.1).
Figure 1.
Stellar mass versus redshift of galaxies in the GAMA survey. Thefull sample is shown in grey. Coloured points refer to GAMA galaxies inthe (nearly) volume-limited sample (see §2.2) and in groups with at leastfive members. for baryons and dark matter, respectively. EAGLE was run usingthe set of cosmological values suggested by the initial results fromthe Planck mission { Ω m , Ω b , σ , n s , h } = {0.307, 0.04825, 0.8288,0.9611, 0.6777} (Table 9; Planck Collaboration et al. 2014).EAGLE includes element-by-element radiative cooling (for11 elements; Wiersma et al. 2009a), pressure and metallicity-dependent star formation (Schaye 2004; Schaye & Dalla Vecchia2008), with a Chabrier (2003) Initial Mass Function, stellar massloss (Wiersma et al. 2009b), thermal energy feedback from starformation (Dalla Vecchia & Schaye 2012), angular momentumdependent gas accretion onto supermassive black holes (Rosas-Guevara et al. 2015) and AGN feedback (Booth & Schaye 2009;Schaye et al. 2015). The subgrid feedback parameters were cali-brated to reproduce the present day observed galaxy stellar massfunction (GSMF) as well as the observed distribution of galaxysizes (Schaye et al. 2015). More information regarding the tech-nical implementation of hydrodynamical aspects as well as subgridphysics can be found in Schaye et al. (2015). Groups of connected particles are identified by applying the FoFalgorithm to the dark matter particles using a linking length of0 . SUBFIND (Springel et al. 2001;Dolag et al. 2009).
SUBFIND identifies local minima in the grav-itational potential using saddle points. All particles that are grav-itationally bound to a local minimum are grouped into a subhalo.Particles that are bound to a subhalo belong to that subhalo only. Wedefine the subhalo centre as the position of the particle for whichthe gravitational potential is minimum. The mass of a subhalo isthe sum of the masses of all the particles that belong to that sub-halo. The most massive subhalo is the central subhalo of a givenFoF group and all other subhaloes are satellites . c (cid:13) , 1–15 alaxy-Galaxy Lensing in EAGLE log [ M star / M fl ] s a t e lli t e f r a c t i o n log M limitfiducial − . M limitfiducial +0 . M limitfiducial − . M limitfiducial GAMA N GAMAFoF Figure 2.
Satellite fraction, f sat , in EAGLE (black curve) obtained with achoice of M limitstar that reproduces the GAMA satellite fraction (black trian-gles). Curves with different line styles and shades of grey show the satellitefraction with a choice of the M limitstar of respectively -1.5, -0.75 below and+0.25 dex above the reference values. The mass M crit200 and the radius r crit200 of the halo are assignedusing a spherical over-density algorithm centred on the minimumof the gravitational potential, such that r crit200 encompasses a regionwithin which the mean density is 200 times the critical density ofthe universe.The group finder of EAGLE links particles in real spacewhereas the GAMA group finder connects members in redshiftspace. This difference could be particularly important if a largefraction of interlopers were wrongly assigned to groups for GAMA.However, the GAMA group finder was tested against mock cata-logues and found to be robust against interlopers for groups withfive or more members (Robotham et al. 2011). We defer a moredetailed study of the impact of adopting exactly the same groupingalgorithm to a forthcoming publication by the KiDS collaboration. The galaxy-galaxy lensing signal from observations measures the ∆Σ profile (Sec. 2.1). Therefore, in order to compare to the obser-vations, we calculate the ∆Σ profiles from EAGLE. To do so, weproject all the particles within a sphere with radius 2 .
95 Mpc cen-tred on the location of the subhalo onto the x − y plane . We dividethe projected radial range into 150 bins equally spaced in log-space.At every projected radius R , we calculate the surface density within R , ¯ Σ ( < R ), as the sum of the mass of all the particles within the pro-jected radius R , M ( < R ), divided by the area A = π R . The surface Although this is strictly true only to the extent to which one knows thesource redshift distribution. We tested that the results do not differ significantly by choosing differentprojections or averaging over the three of them. density at R , Σ ( R ), is the mass enclosed in the annulus with innerradius ( R − δ R /
2) and outer radius ( R + δ R /
2) divided by the area2 π R δ R , where δ log R = log (2 . / R < . R ≈ .
43 Mpc instead of 2 .
95 Mpchas a negligible effect on the signal.Subhaloes are binned according to their stellar mass, calcu-lated as the sum over all stellar particles that belong to the subhalo.The ∆Σ in a given stellar mass bin is then calculated by averagingthe ∆Σ profiles of single subhaloes. The statistical errors are calcu-lated using bootstrapping: galaxies in each mass bin are re-sampled1000 times and the range of values that count for the 95% of the dis-tribution is taken as the 2-sigma error for the ESD profiles from thesimulation. In order to avoid selection bias, it is important that the sampleof galaxies that is selected in the simulations is a fair represen-tation of the galaxy sample in GAMA. The GAMA galaxy sam-ple (nearly volume-limited and with groups with 5 or more mem-bers) has a median redshift of z = .
16 and hence we compare thecorresponding galaxy-galaxy lensing signals with those obtainedfrom the snapshot of the EAGLE simulation closest in redshift, i.e. z =0.18. The slight discrepancy in redshift is likely unimportant asfrom z =0.25 to z =0 there is little evolution in the GSMF (Furlonget al. 2015). We verified that the effect of using EAGLE galaxies at z =0 is indeed negligible.A robust discrimination between satellites and central galax-ies is obtained by restricting our sample to galaxies that belong togroups with at least five members. To mimic this selection, we needto impose a minimum stellar mass from which we start countinggroup members in EAGLE. The choice of this M limitstar is somewhatarbitrary and could alter the ratio between the number of satelliteand central galaxies in a given stellar mass bin. By increasing thestellar mass limit, the number of central galaxies in groups that havefour or more satellites above the mass limit is reduced, whereasthe number of satellites of a given stellar mass remains mostly un-changed. Therefore, increasing the stellar mass limit has the neteffect of increasing the satellite fraction. We choose the value of M limitstar that results in the ratio of satellite to total galaxies found inGAMA for a given stellar mass bin. In the rest of the paper wealso show the effect of a different choice of M limitstar on the ∆Σ profileresults from EAGLE.Fig. 2 shows the satellite fraction in EAGLE for differentchoices of M limitstar . The black triangles show the satellite fractionin our galaxy sample from GAMA for galaxies in the same stellarmass bins. The black line represents the satellite fraction in EAGLEif we choose the value of M limitstar that reproduces the GAMA satel-lite fraction (see Table 1). With different linestyles and shades ofgrey we show the satellite fraction with a choice of the M limitstar ofrespectively -1.5, -0.75 below and +0.25 dex above the values thatreproduce the GAMA satellite fraction. For the fiducial choice of M limitstar the satellite fraction of GAMA is reproduced by construction, c (cid:13) , 1–15 Marco Velliscig et al. but we note that this would not necessarily be the case if the num-ber of galaxies in a stellar mass bin were too small to recover theexact satellite fraction. Decreasing (increasing) the value of M limitstar with respect to the fiducial value, has the net effect of decreasing(increasing) the satellite fraction. The fiducial values of M limitstar ineach stellar mass bin are (9 . , . , . , . , −− ), see also col-umn (11) of Table 1. We note that the value of M limitstar is ill-definedfor the most massive bin. In fact, the haloes that enter in this binsatisfy the richness cut for every value of M limitstar that is lower thanthe lower limit of the bin itself (log[ M star / M (cid:12) ] = . M limitstar are close to the completenesslimit at z=0.18 of the specific GAMA galaxy group sample adoptedthroughout the paper.Since the value of the satellite fraction, in our approximationof the GAMA selection function, is essential for the calculation ofthe combined signals from satellite and central galaxies, the choiceof M limitstar has a major effect on the comparison with observationswhen galaxies are not separated in centrals and satellites (see §4.3). In the following we present the results for the excess surface density ∆Σ computed from the simulations (for details see §3.2). We dividegalaxies into five stellar mass bins ranging from log ( M star / M (cid:12) ) = . ( M star / M (cid:12) ) = .
8. In the simulations we considerall stellar mass particles bound to a subhalo for the stellar massdetermination. We note that this choice may overestimate the stel-lar mass content since in observations stars in galaxy outskirts areoften not detectable. We address this caveat by correcting the stel-lar mass of GAMA galaxies by a multiplicative factor given by theratio between the galaxy’s measured flux in the r band and the in-tegral of its Sersic profile up to infinity (Taylor et al. 2011). In thisway we correct the stellar mass of galaxies by taking into accounttheir undetected flux. An alternative approach would be to consideronly stellar particles within a 30 kpc aperture for the stellar masscalculation in EAGLE (see the discussion in Schaye et al. 2015).Similarly, we would need to correct the observed stellar mass bythe multiplicative factor given by the ratio between the measuredflux (r band) of the galaxy and its integrated Sersic profile up to30 kpc. We tested this alternative approach, leading to very similarresults with the disadvantage of reducing the number of galaxiesavailable from the EAGLE simulations in the highest stellar massbins. We therefore opted for the former approach. The ESD in agiven stellar mass bin is computed by stacking the ∆Σ of all galax-ies in that mass bin.We compare each prediction from the simulation to the corre-sponding data from KiDSxGAMA. We first present results for cen-tral and satellite galaxies separately (see §4.1 and §4.2). We thenpresent the results for both galaxy types combined (§4.3). This sig-nal is the linear combination of the signal from satellite and centralgalaxies, where the relative importance of the two terms is modu-lated by the value of the satellite fraction (§4.3.1). Fig. 3 shows the ESD profile around central galaxies in the EAGLEsimulation as a function of the projected distance from the centreof the galaxy. For all mass bins ∆Σ is a decreasing function of theprojected radius. Fluctuations in the excess surface density profilescan arise due to the presence of matter associated to satellite galax-ies, but these are usually not massive enough to significantly alter R [ h Mpc ] [ h M p c ] Centrals M star / M <10.6 M star / M <10.9 M star / M <11.2 M star / M <11.5 M star / M <11.8 Figure 3.
Profiles of the excess surface density, ∆Σ , of matter around cen-tral galaxies up to projected separations of 2 h − Mpc from the centre ofthe galaxy. To mimic the GAMA selection function, only galaxies hostedby groups with five or more members with masses above the stellar masslimit listed in column 11 of Table 1 are taken into account for this anal-ysis. Central galaxies are divided into five stellar mass bins ranging fromlog ( M star / M (cid:12) ) = . ( M star / M (cid:12) ) = .
8. The vertical dashedline marks R = .
05 h − Mpc representative of the scales at which the innerpart of the dark matter halo dominates the signal. the azimuthally averaged excess surface density profile. Moreover,since the signal is averaged over many galaxies, any deviation dueto the presence of a relatively massive satellite would be averagedout in the stacking process.Table 1 reports values of the mean subhalo mass M censub for eachstellar mass bin. The ∆Σ ( R = .
05 h − Mpc) (the intersection be-tween the red dashed line in Fig. 3 and the ∆Σ profiles for differentstellar mass bins) and the mean mass M censub are monotonically in-creasing functions of the stellar mass. Both ∆Σ ( R = .
05 h − Mpc)and M censub are approximated reasonably well by single power lawfunctions of the stellar mass (not shown here), albeit with differentcoefficients. ∆Σ ( R = .
05 h − Mpc) shows a weaker dependence onstellar mass with respect to M sub which, in this stellar mass range,has a power law coefficient close to unity. Central galaxies withhigher ∆Σ amplitudes are hosted by more massive haloes. There-fore, as expected, the amplitude of the ∆Σ profile at small scales isa proxy for the typical mass of the subhaloes hosting central galax-ies in a given stellar mass bin. Fig. 4 shows the ∆Σ signal in EAGLE (red curves) whereas ∆Σ from the observations is indicated with black diamonds and ver-tical error bars. Curves with different shades of grey show the ESDprofiles in EAGLE with a different choice of the M limitstar (see §3.3).For stellar masses 10 . < log ( M star / M (cid:12) ) < .
6, the uncertain-ties in the data are large due to the limited number of low-massgalaxies that are centrals in rich groups ( N GAMAFoF ≥
5) and thereforeare not representative of the entire central galaxy population (Vi-ola et al. 2015). For stellar masses 10 . < log ( M star / M (cid:12) ) < . c (cid:13) , 1–15 alaxy-Galaxy Lensing in EAGLE ∆ Σ [ h M fl / p c ] . < log M star / M fl < . . < log M star / M fl < . . < log M star / M fl < . . < log M star / M fl < . R [ h − Mpc] . < log M star / M fl < . log M limitfiducial − . M limitfiducial + 0 . M limitfiducial − . M limitfiducial GAMA N GAMAFoF Centrals
Figure 4.
Excess surface density profiles from KiDSxGAMA (black diamonds) and in the EAGLE simulation (red curves) for central galaxies hosted bygroups with 5 or more members that each have stellar masses greater than M limitstar (listed in column 11 of Table 1) in order to mimic the GAMA selection ofgalaxies. Each panel contains a different bin in central galaxy stellar mass. Asymmetric error bars show the 2- σ error in each R bin. Curves with differentshades of grey show the ESD profiles in EAGLE with a choice of the M limitstar of respectively -1.5, -0.75 dex below and +0.25 dex above the values that reproducethe GAMA satellite fraction. agreement between data and predictions from the simulation and inwhat follows we discuss some features in more detail.The agreement between the ESD in EAGLE and KiDS sug-gests that central galaxies, with masses 10 . < log ( M star / M (cid:12) ) < . . < log ( M star / M (cid:12) ) < . ∆Σ seems tofavour a shallower excess surface density profile at radii larger than400 h − kpc. This might reflect a box-size effect, as more massive(more extended and less concentrated) haloes might be missing inthe small volume probed by the EAGLE simulation.The mean host halo masses predicted by EAGLE for galaxiesin the five stellar mass bins shown can be found in Table 1, column(2). We have computed analytical ∆Σ profiles corresponding tohaloes with Navarro et al. (1997, hereafter NFW) matter densityprofiles for the halo masses reported in column (2) of Table 1.These analytical profiles reproduce the overall normalisation of thesignal but poorly match the radial dependence of the numerical pro-files. In Appendix A, we discuss this test in detail, and we alsocomment on the cause of the limitations of simple analytical modelin accurately describing the ∆Σ profiles obtained from simulations.In the case of central galaxies the choice of M limitstar has a smalleffect on the ESD profile computed from the simulations as can be seen by comparing the grey lines in Fig. 4. To quantify this, weemploy the following statistics: χ = N points − (cid:88) i ( ∆Σ EAGLE i − ∆Σ data i ) σ EAGLE i + σ data i , (8)where N points is the number of stellar mass bins times the numberof data points per bin and i is an index running through all 60 datapoints. We obtain values χ = . M limitstar .We note that four points in each of the two most massive stellarmass bins lead to most of the deviations of χ from unity. Further-more, χ ranges from 1.4 to 1.8 as we change M limitstar from its fidu-cial value to the fiducial -1.5. We note that, throughout the paper,we are neglecting any off-diagonal terms of the covariance matrix.Although this might have a (supposedly small) effect on the abso-lute value of the χ , we are here mostly concerned with relativedifferences among models with different choices of a limiting stel-lar mass. In the context of this comparison, we consider the differ-ences reported above not worth further investigations.Higher values of M limitstar tend to produce higher amplitudes ofthe ESD profiles since higher mass subhaloes are being selected.The relative insensitivity on the exact choice of M limitstar suggest thatfor a comparison of ESD profiles of central galaxies only, the exactdetails of the galaxy selection are not crucial. We anticipate that thesame argument is not applicable when the ESD profiles of centraland satellite galaxies are analysed jointly since the choice of M limitstar determines the satellite fraction which in turn plays a major role inestablishing how the ESD profiles of central and satellite galaxiesare combined (see §4.3). c (cid:13) , 1–15 Marco Velliscig et al.
Unlike central galaxies, the ∆Σ profiles of the satellites galaxies are not necessarily expected to be simply decreasing functions of theseparation from the centre. For a single satellite galaxy the profileshould become negative at the projected separation from the centreof the host halo (Yang et al. 2006; Sifón et al. 2015). This effect isdue to the surface density at the centre of the host halo being largerthan the mean internal surface density, Σ ( R halocentre ) > ¯ Σ ( < R halocentre ).At larger separations than the separation to the host halo, the ∆Σ profile first increases due to the inclusion of the centre of thehost halo in the term ¯ Σ ( < R ), before decreasing again at still largerseparations. Stacking the ∆Σ of satellites in a given stellar mass binsmooths out the negative parts of the profiles since the separationsbetween satellites and their host halo vary. However, the increasein the signal at larger radii is preserved by the stacking.Fig. 5 shows the ∆Σ profile of satellite galaxies in the EAGLEsimulation. The amplitude of the ∆Σ profile at small separations( R = .
05 h − Mpc) is an increasing function of the stellar mass ofthe satellite. The same trend is shared by the average subhalo massfor satellite galaxies since satellites with higher stellar masses tendto be hosted by more massive dark matter subhaloes (see Table 1,column 5). As in the case of central galaxies, the similar depen-dence on the stellar mass suggests that the amplitude of ∆Σ at smallseparations can be considered a proxy for the mass of the subhalohosting the satellite galaxy.The radius at which the ∆Σ profile starts to be dominated bythe host halo mass (the satellite bump) increases with stellar mass.This effect is driven by the change in the average distance betweensatellites and their host haloes, which increases from ≈ ≈ R = . − Mpc), the ∆Σ profile startsto be dominated by the contribution of the halo hosting the satellitegalaxy. In this case ∆Σ shares a similar trend with stellar mass asthe mean host halo mass for satellite galaxies, M crit200 (see Table 1,column 3).The amplitude of the satellite bump is similar for all the stellarmass bins, which can be explained by the fact that the richness cuteffectively selects host haloes by mass. Indeed, most of the satel-lites with stellar mass 10 . < log ( M star / M (cid:12) ) < . . < log [ M crit200 / M (cid:12) ] < .
24. The prominenceof the satellite bump with respect to the overall normalisation de-creases with stellar mass, a trend that is explained by the fact thatthe ratio M satsub / M crit200 increases from 0.03 to 0.3 in the consideredmass range (see Table 1, column 6).The similar dependence of ∆Σ with halo mass at larger radiihighlights the fact that the amplitude of the satellite bump is tightlycorrelated to the host halo mass. In principle the amplitude of thesatellite bump should depend on the satellite’s subhalo mass as wellas on the host halo mass. In practice the satellite’s subhalo massis, except for the highest stellar mass bin, a small fraction of thehost halo mass and therefore it plays a minor role in setting theamplitude of the satellite bump. Fig. 6 shows the comparison between the observed ∆Σ profileof satellite galaxies (black squares) and the corresponding signalin the EAGLE simulation (blue curves) for the five stellar massbins. The ESD profiles computed for different choices of M limitstar areshown in grey. For 10 . < log ( M star / M (cid:12) ) < . R [ h Mpc ] [ h M p c ] Satellites M star / M <10.6 M star / M <10.9 M star / M <11.2 M star / M <11.5 M star / M <11.8 Figure 5.
As Fig. 3 but for satellite galaxies. To ease the comparison the re-sults for the central galaxies are reported with grey curves. The two verticallines mark R = .
05 h − Mpc and the R = . − Mpc. all broad agreement between simulation predictions and observa-tions.For log ( M star / M (cid:12) ) > . . < R < . − Mpc) scales is higher in the sim-ulations than in the observations although at low significance (lessthan two sigma).For stellar masses 10 . < log ( M star / M (cid:12) ) < . M limitstar has only a relatively minor effect on the ESD profileof satellite galaxies as computed from the simulation. In fact, the re-duced χ between the model and the data increases from its fiducialvalue of χ = . χ = . ( M limitstar ) = − . In this section we present the ESD calculated considering all galax-ies without distinguishing between centrals and satellites (studyingonly galaxies in rich groups). The ∆Σ profile of the whole popula-tion of galaxies of a given stellar mass is a linear combination ofthe profiles for satellites, ∆Σ sat , and centrals, ∆Σ cen : ∆Σ = f sat ∆Σ sat + (1 − f sat ) ∆Σ cen , (9)where f sat is the satellite fraction of galaxies in a given stellar massbin. The relative importance of each term is set by the value of f sat . Therefore the ∆Σ profile of the whole galaxy population lies inbetween those for satellite and central galaxies. c (cid:13) , 1–15 alaxy-Galaxy Lensing in EAGLE ∆ Σ [ h M fl / p c ] . < log M star / M fl < . . < log M star / M fl < . . < log M star / M fl < . . < log M star / M fl < . R [ h − Mpc] . < log M star / M fl < . log M limitfiducial − . M limitfiducial + 0 . M limitfiducial − . M limitfiducial GAMA N GAMAFoF Satellites
Figure 6.
Excess surface density profiles from KiDSxGAMA (black squares) and in the EAGLE simulation (blue curves) for satellite galaxies hosted bygroups with 5 or more members that each have stellar masses greater than M limitstar (listed in column (11) of Table 1) in order to mimic the GAMA selection ofgalaxies. Each panel contains a different bin in satellite galaxy stellar mass. As in Fig. 4 , asymmetric error bars show the 2- σ error in each R bin. Curves withdifferent shades of grey show the ESD profiles in EAGLE with a choice of the M limitstar of respectively -1.5, -0.75 dex below and +0.25 dex above the values thatreproduce the GAMA satellite fraction. Fig. 7 shows the comparison of the ∆Σ profiles obtained fromobservations (black triangles) and the EAGLE simulation (blackcurves). As in previous figures the ∆Σ profiles for different choicesof M limitstar are shown in grey.Most of the differences between ∆Σ in the simulation andobservations are in line with what we expect from our previ-ous results. Specifically, the amplitude of the satellite bump at11 . < log ( M star / M (cid:12) ) < . ∆Σ profile of satellite and central galaxies, as ∆Σ for all galaxies is a linear combination of the two (see Eq. 9).The degree of agreement between the EAGLE and GAMA re-sults is driven by the choice of a M limitstar that reproduces the satel-lite fraction of GAMA. In fact, for different choices of M limitstar ,the agreement between the simulation and observations worsensconsiderably. The χ between the model and the data increasesfrom its fiducial value of χ = . χ = ( M limitstar ) = − .
5. This dependence of χ on the choice of M limitstar is considerably stronger than when satellites and centrals areanalysed separately, suggesting that particular care has to be takenwhen selecting groups in EAGLE when satellites and centrals areanalysed jointly. On the other hand, this analysis shows that theGGL signal has the potential to test the mix of satellites and cen-trals predicted by simulations. In this section we discuss some of the limitations of our study andhighlight possible future improvements. The main issues are: • In the comparison between simulation and observations an im-portant role is played by stellar mass errors, both random and sys-tematic. We consider here the effect of a random error of ∼ . . Since the number density of galax-ies decreases with stellar mass, random errors always scatter morelow-mass galaxies to high masses than viceversa. The importanceof this effect depends on the steepness of the galaxy stellar massfunction (GSMF). For low masses, log ( M star / M (cid:12) ) < .
9, wherethe GSMF is reasonably flat, a comparable number of galaxies isscattered towards higher and lower masses. On the other hand, forhigher stellar masses where the GSMF is steeper, relatively morelow-mass galaxies are scattered towards higher masses. Thereforethe effect of random errors is expected to be stronger at highermasses (e.g. Furlong et al. 2015) . We verified that the uncertain- These errors can be significantly larger ( ∼ . (cid:13)000
9, wherethe GSMF is reasonably flat, a comparable number of galaxies isscattered towards higher and lower masses. On the other hand, forhigher stellar masses where the GSMF is steeper, relatively morelow-mass galaxies are scattered towards higher masses. Thereforethe effect of random errors is expected to be stronger at highermasses (e.g. Furlong et al. 2015) . We verified that the uncertain- These errors can be significantly larger ( ∼ . (cid:13)000 , 1–15 Marco Velliscig et al. ∆ Σ [ h M fl / p c ] . < log M star / M fl < . . < log M star / M fl < . . < log M star / M fl < . . < log M star / M fl < . R [ h − Mpc] . < log M star / M fl < . log M limitfiducial − . M limitfiducial + 0 . M limitfiducial − . M limitfiducial GAMA N GAMAFoF Centrals + Satellites
Figure 7.
Excess surface density profiles from KiDSxGAMA (black triangles) and in the EAGLE simulation (black curves) for all galaxies hosted by groupswith 5 or more members that each have stellar masses greater than M limitstar (listed in column (11) of Table 1) in order to mimic the GAMA selection of galaxies.Each panel refers to a different bin in galaxy stellar mass. As in Fig. 4 , asymmetric error bars show the 2- σ error in each R bin. Curves with different shadesof grey show the ESD profiles in EAGLE with a choice of the M limitstar of respectively -1.5, -0.75 dex below and +0.25 dex above the values that reproduce theGAMA satellite fraction. ties in the stellar mass determinations play a very minor role for allstellar mass bins. The effect of random errors on the ∆Σ profiles iswell within the errors on the simulation results. • The group finder of EAGLE identifies groups in real spacewhereas the GAMA group finder uses redshift space. This maycause differences in the ∆Σ profile, in particular if interlopers arewrongly assigned to groups, which would artificially increase therichness of the observed group. Therefore, the observed signalwould be artificially lowered by the contribution of less massivegroups hosting fewer than five members. To be fully consistent, thesame algorithm should be employed in both simulations and obser-vations. • The centring in observations is done according to the lightemitted by the galaxies – the centre of a group is defined as thelocation of the Brightest Group Galaxy – whereas in simulationswe adopt the position of the particle with the minimum gravita-tional potential as the centre. Schaller et al. (2015), have shownthat in EAGLE the majority of the galaxies ( > ∼ • In this work we mostly assume that the good agreement be-tween the simulation and observations stems from the ability of EAGLE to reproduce the observed GSMF. A comprehensive studyshould be made to test how sensitive this agreement is to the levelat which the GSMF is reproduced by the simulations. This can bestudied by employing the EAGLE models using different subgridparameters (Crain et al. 2015), although these simulations use vol-umes that are at least a factor of eight smaller than the main EAGLErun, which may be problematic.
In this work we compare the excess surface density profile ∆Σ ( R )obtained from the state-of-the-art hydrodynamical cosmologicalEAGLE simulation to the observed weak galaxy-galaxy lensingsignal using (source) galaxies with accurate shape measurementsfrom the KiDS survey around spectroscopically confirmed (lens)galaxies from the GAMA survey (referred throughout the paper asKiDSxGAMA). Results are presented for (lens) central and satel-lite galaxies in five logarithmically equi-spaced stellar mass bins inthe range 10 . < log ( M star / M (cid:12) ) < . r -band mag-nitude 19.8. This yields a relatively simple selection function. Wemimic this selection function by taking galaxies in the EAGLE sim-ulation with stellar masses above M limitstar (about 10 M (cid:12) , see Table1). The precise value of this limit in stellar mass is chosen in orderto reproduce the relative abundances of central and satellite galax-ies for the different stellar mass bins in GAMA. To minimize themis-identification of central and satellite galaxies, only groups with c (cid:13) , 1–15 alaxy-Galaxy Lensing in EAGLE at least five members are used in the data (Robotham et al. 2011).We apply the same ‘richness cut’ to the simulation.The ∆Σ profile of central galaxies (Fig. 3) in EAGLE isa decreasing function of the transverse separation with a scale-dependent logarithmic slope. We compare the ∆Σ signal of cen-tral galaxies in EAGLE with the observed signal in KiDSxGAMA.We find that both the normalization and the radial dependence ofthe signal from EAGLE are in broad agreement with the data forlog ( M star / M (cid:12) ) < . ∆Σ profile. For the highest stellar mass bins(11 . < log ( M star / M (cid:12) ) < .
8) EAGLE underestimates the sig-nal at large radii ( ∼ Mpc ) leads to a lack of massive clusters.For low stellar masses, the ∆Σ profile of satellite galaxiesis a non-monotonic function of the separation from the centre.This feature stems from the fact that the signal is dominated bytwo different components at different scales. The smallest scales( R < . − Mpc) are dominated by the subhalo attached to thesatellite galaxy. The largest scales (0 . < R < − Mpc ) aredominated by the contribution from the main host halo. For stellarmasses below log ( M star / M (cid:12) ) <
11, the EAGLE predictions andKiDS data are in agreement at all probed scales, suggesting thatsatellite galaxies in the simulations are hosted by subhaloes withthe correct mass and that they reside in host haloes with the cor-rect halo mass. The agreement is less satisfactory for galaxies withlog ( M star / M (cid:12) ) > . M limitstar is varied by almost two orders of magnitude, the difference be-tween the ∆Σ profiles is remarkably small. However, when the ESDprofiles of central and satellite galaxies are analysed jointly, theEAGLE predictions of the ESD profile are sensitive to the selec-tion function. We have calibrated the choice of the value of M limitstar to reproduce the GAMA satellite fraction in bins of stellar massas this quantity encapsulates the main effect. However, our anal-ysis makes apparent that, as the quality of the data improves, itwill become crucial to properly mimic selection effects to comparegalaxy-galaxy lensing observations and predictions from simula-tions, which will enable the satellite fraction to be tested directly. ACKNOWLEDGEMENTS
Author Contributions:
All authors contributed to the develop-ment and writing of this paper. The authorship list is given in threegroups: the lead authors (MVe, MC, HHo, JS), followed by two al-phabetical groups. The first alphabetical group includes those whoare key contributors to both the scientific analysis and the data prod-ucts. The second group covers those who have either made a signif-icant contribution to the data products, or to the scientific analysis.
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APPENDIX A: COMPARISON OF ANALYTICAL ANDNUMERICAL ∆Σ PROFILES
The measured galaxy-galaxy lensing signal is often interpreted inthe context of a Λ CDM framework where the baryon content ofa (lens) galaxy is embedded in a dark matter halo. The lensingeffect on the light rays emitted by background (source) galaxiesis therefore caused by the total matter density contrast along theline of sight. At the transverse separations of interest in this paper(0 . < R < h − Mpc) most of this matter contrast is actuallyassociated with the foreground (lens) galaxy. If one further limitsthe analysis to central galaxies, a simple yet effective model –oftenemployed in the literature– assumes that i) the contribution to thelensing signal from the stellar mass of a galaxy can be describedas a point-mass contribution ( ∆Σ star ( R ) ∝ R − ), ii) the contributionfrom (both cold and hot) gas can be ignored ( ∆Σ gas ( R ) ≈ ∆Σ halo ( R ), can bedescribed analytically assuming a spherical NFW matter densityprofile. The ESD profiles computed from the EAGLE simulationrepresent a benchmark against which the simple model outlinedabove can be tested. With this aim, we proceed with the followingtests. A1 ESD signal of the mean halo and stellar mass
For each stellar mass bin for which ∆Σ is computed, we knowwhich haloes contribute to the stack. We thus can compute themean halo mass for each stellar mass bin (see column 2 in Ta-ble 1). We compute ∆Σ halo ( R ) corresponding to this mass adoptingthe concentration-halo mass relation derived for (relaxed ) haloesof the EAGLE simulation (see Schaller et al. 2015). We also com-pute ∆Σ star ( R ) using the mean stellar mass in each bin. The resultsare shown in Figure A1 where the ESD profiles of the simula-tions (green points with error bars) have been rebinned to 10 radialpoints, and the different contributions are indicated with differentline styles and colours as indicated in the legend. We have repeated the exact test either using the median (stellar and halo)masses in the bins or using the entire distribution of (stellar and halo)masses and in both cases the results do not change significantly. We further comment on this in the next subsection.c (cid:13) , 1–15 alaxy-Galaxy Lensing in EAGLE ∆ Σ [ h M fl / p c ] . < log ( M star /M fl ) < . . < log ( M star /M fl ) < . . < log ( M star /M fl ) < . . < log ( M star /M fl ) < . Projected separation [ h − Mpc] . < log ( M star /M fl ) < . NFW ( M halo , c halo ) +PointMassNFW ( M halo , c halo ) Point MassNFW (Duffy et al. 08 concentration)EAGLE (rebinned)
Figure A1.
Excess surface density computed for central galaxies in EAGLE for different stellar mass bins (green circles with error bars). These profiles arethe rebinned versions of those plotted with a red line in Figure 4. Analytical predictions of the excess surface are plotted with different line styles (see legend).For the point mass term, we use the mean stellar mass in each bin. For the NFW term, we use the mean halo mass for each bin as reported in column 2 ofTable 1 in the main body of the paper and the corresponding halo concentration according to Schaller et al. (2015).
The analytical description of the ESD profiles is in fair agree-ment for all stellar mass bins only on scales larger than R ∼ . h − Mpc. On smaller scales the analytical description system-atically overestimates the results from the simulations. The agree-ment on relatively large scales suggests that the knowledge of themean total mass of the halo is indeed sufficient to describe thelensing signal at those scales. On smaller scale, however, the ESDprofile is clearly dependent on the actual matter density distribu-tion. The fact that the simulations are systematically below theanalytical predictions seems to indicate that the haloes that con-tribute to the signal are less centrally concentrated than what is as-sumed. Schaller et al. (2015) show that the dark matter haloes inthe EAGLE simulation have slightly different concentrations thanthose in the dark-matter only version of EAGLE. However, the dif-ference is not sufficient to explain the feature under inspection here.It is worth noting that the concentration-mass relation provided bySchaller et al. (2015) and adopted here for this test was derivedusing only relaxed haloes (for which a spherical NFW matter den-sity profile is an adequate description). Not all of the haloes thatenter the stack in each stellar mass bin are expected to be relaxedand this may be the cause of the difference between the analyticaland numerical ESD profiles. Duffy et al. (2008) reported indeed that, in the case of the OWLS simulations (Schaye et al. 2010), asample with only relaxed haloes yields on average higher concen-trations than a sample where also unrelaxed haloes are included.We show that using the halo concentration-mass relation in Duffyet al. (2008) for the full sample indeed leads to lower ∆Σ profiles onscales below ∼ . h − Mpc. This in turn yields a better agreementwith the profiles predicted from the EAGLE simulation (dot-dashedred line in Figure A1). Despite the improvement, significant differ-ences are still noticeable for the four lowest mass bins and on scales0 . < R / ( h − Mpc) < .
2. We defer a more quantitative analysisof this feature to further publications as it is beyond the scope ofthis paper.
A2 Fitting the numerical ∆Σ profiles The test described in § A1 shows that a simple analytical modelcannot reproduce the entire scale dependence of the ESD profilesobtained from the EAGLE simulation. The question then ariseswhether this severely hampers the possibility to retrieve halo prop-erties such as their masses and concentrations when such simplemodels are employed to fit the ESD profiles. To answer this ques- c (cid:13) , 1–15 Marco Velliscig et al. tion we define a model in which ∆Σ ( R ) = ∆Σ star ( R |(cid:104) M star (cid:105) ) + ∆Σ halo ( R | M halo , c halo ) . (A1)Here, (cid:104) M star (cid:105) is a free parameter that indicates the mean stellarmass in each stellar mass bin , M halo , and c halo are two free pa-rameters that completely specify the analytical ESD profile of ahalo with a NFW matter density profile. We treat the ESD pro-files from the EAGLE simulation as the data to be fit by thismodel. We fit (cid:104) M star (cid:105) , M halo , and c halo independently for each stel-lar mass bin. No priors are imposed on M halo , and c halo , whereaswe impose that (cid:104) M star (cid:105) is within the stellar mass bin limits. The fitis performed using a Markov Chain Monte Carlo (MCMC) tech-nique. Specifically, we employ the publicly available emcee code(Foreman-Mackey et al. 2013) and we check for convergence byensuring that the chain is much longer than the auto-correlationlength of each parameter. We find that the best-fit model yields a χ = . / (50 − = .
22, i.e. the simple model can adequatelydescribe the data, although more flexible models might yield evenbetter agreement. Figure A2 shows the ESD profiles from the sim-ulation (green points with error bars) and the median and the 68%credibility level (red curves and orange shaded regions) derivedfrom the MCMC run.The top panel of Figure A3 shows the constraints on themean stellar and halo masses obtained by fitting the ∆Σ of theEAGLE simulation with the simple analytical model describedabove. For comparison, we report the results by van Uitert et al.(2016) obtained simultaneously fitting the KiDSxGAMA galaxy-galaxy lensing profile and the GAMA stellar mass function. Wenote here that van Uitert et al. (2016) employed a sophisticated halomodel rather than a simple three-parameter (per bin) model likethe one adopted here. In the range, 10 . < log[ M star / M (cid:12) ] < . ∆Σ of theEAGLE simulation with the simple analytical model describedabove. As expected from the test in § A1 the posterior distribu-tions of the parameters M halo and c halo indicate that concentra-tions are systematically underestimated with respect to the fidu-cial concentration-halo mass relation (black points in Figure A3).The result we find confirms the notion that the halo concentrationsfound via a fitting of the ESD profiles have to be interpreted as ef-fective concentrations and are most likely to be lower than thosebased on fits of relaxed haloes in numerical simulations. This hasalready been noted in several observational works, e.g. in the con-text of fitting ESD profiles around GAMA galaxies using KiDSgalaxy images (see e.g. Viola et al. 2015; van Uitert et al. 2016). Acloser inspection of Figure A3 also shows that the retrieved meanhalo masses (circles with horizontal error bars) are unbiased withrespect to the actual mean halo mass in each stellar mass bin in allcases except for the bin 10 . < log [ M star / M (cid:12) ] < .
9. This is per-haps not surprising given that this is exactly the bin for which the We adopt the same stellar mass bins as in the main body of the paper. log[ M star /M fl ] l og [ M c r i t | ce n / M fl ] Van Uitert et al. 2016Analytical Fit to EAGLE ESD12.4 12.6 12.8 13.0 13.2 13.4 13.6 13.8 log[ M crit200 | cen /M fl ] h a l o c o n c e n t r a t i o n OWLS (relaxed)EAGLE (relaxed, z = 0 )OWLS (full)Analytical Fit Figure A3.
Top Panel.
Average halo mass (for haloes with N GAMAFoF ≥ ∆Σ profile (see detaill in § A2) and the resultfrom van Uitert et al. (2016) obtained simultaneously fitting galaxy-galaxylensing and the GAMA stellar mass function for the KiDSxGAMA galaxysample. Both blue and red horizontal bars indicate the width of the bin,whereas vertical bars indicate the 68% credibility interval for the inferredhalo mass. Bottom Panel.
Halo concentration-mass relation. Green circleswith error bars refer to the median and the 68% credibility interval obtainedfrom the MCMC used to fit the EAGLE ∆Σ profiles. The black line repre-sents the relation for relaxed haloes in the EAGLE simulation (see Schalleret al. 2015), where the black points indicate the mean halo masses of thefive stellar mass bins used in the analysis. For reference, the concentrationmass relations from Duffy et al. (2008) are also reported with red dashedand blue dotted lines, indicating the relation derived for relaxed-only andall haloes, respectively (see discussion in §A2). ESD profile from the EAGLE simulation differs the most from ananalytical ∆Σ profile that assumes a NFW matter density profile.Finally, we note that, at the smallest scales probed here ( R < . h − Mpc), the ∆Σ profile is sensitive to the point mass as-sumption employed to describe the contribution from the stellarcontent of the galaxy. The simulation disfavours a steep profile ∆Σ ( R ) ∝ R − and a better fit at those scales would require a moredetailed description of the stellar mass distribution in galaxies (seee.g. Kobayashi et al. 2015, for a similar discussion in the context ofthe galaxy-galaxy lensing quality in forthcoming surveys). c (cid:13) , 1–15 alaxy-Galaxy Lensing in EAGLE ∆ Σ [ h M fl / p c ] . < log ( M star /M fl ) < . . < log ( M star /M fl ) < . . < log ( M star /M fl ) < . . < log ( M star /M fl ) < . Projected separation [ h − Mpc] . < log ( M star /M fl ) < . Median . L . EAGLE (rebinned)
Figure A2.
Excess surface density computed for central galaxies in EAGLE for different stellar mass bins (green circles with error bars, these profiles arethe rebinned versions of those plotted with a red line in Figure 4.). Red curves (and orange shades) represent the median (and the 68% credibility intervals)derived from the MCMC employed to fit the data. A fair description of the data can be obtained for each bin except for that corresponding to galaxies in thestellar mass range 10 . < log( M star / M (cid:12) ) < . This paper has been typeset from a TEX/ L A TEX file prepared by theauthor. c (cid:13)000