The SL2S Galaxy-scale Lens Sample. V. Dark Matter Halos and Stellar IMF of Massive Early-type Galaxies out to Redshift 0.8
Alessandro Sonnenfeld, Tommaso Treu, Philip J. Marshall, Sherry H. Suyu, Raphael Gavazzi, Matthew W. Auger, Carlo Nipoti
aa r X i v : . [ a s t r o - ph . GA ] D ec Draft version December 22, 2014
Preprint typeset using L A TEX style emulateapj v. 5/2/11
THE SL2S GALAXY-SCALE LENS SAMPLE. V. DARK MATTER HALOS AND STELLAR IMF OF MASSIVEEARLY-TYPE GALAXIES OUT TO REDSHIFT 0.8
Alessandro Sonnenfeld ∗ , Tommaso Treu † , Philip J. Marshall , Sherry H. Suyu , Rapha¨el Gavazzi ,Matthew W. Auger , and Carlo Nipoti Draft version December 22, 2014
ABSTRACTWe investigate the cosmic evolution of the internal structure of massive early-type galaxies overhalf of the age of the Universe. We perform a joint lensing and stellar dynamics analysis of a sampleof 81 strong lenses from the SL2S and SLACS surveys and combine the results with a hierarchicalBayesian inference method to measure the distribution of dark matter mass and stellar IMF across thepopulation of massive early-type galaxies. Lensing selection effects are taken into account. We findthat the dark matter mass projected within the inner 5 kpc increases for increasing redshift, decreasesfor increasing stellar mass density, but is roughly constant along the evolutionary tracks of early-typegalaxies. The average dark matter slope is consistent with that of an NFW profile, but is not wellconstrained. The stellar IMF normalization is close to a Salpeter IMF at log M ∗ = 11 . Subject headings: galaxies: fundamental parameters — gravitational lensing — INTRODUCTION
Early-type galaxies (ETGs) constitute a family of ob-jects of remarkable regularity, captured by tight scalingrelations such as the fundamental plane (Dressler et al.1987; Djorgovski & Davis 1987) and the relations be-tween central black hole mass and galaxy proper-ties (Ferrarese & Merritt 2000; Gebhardt et al. 2000;Marconi & Hunt 2003; H¨aring & Rix 2004). Despitetremendous efforts, it is still unknown what the funda-mental source of this regularity is. Numerical simula-tions are now able to reproduce some of the key observ-ables of ETGs (Hopkins et al. 2009; Dubois et al. 2013;Remus et al. 2013; Feldmann & Mayer 2014), but theresolution and statistics that can be reached today arestill too low to allow for meaningful quantitative tests. Itis still very challenging to obtain realistic simulations ofthe baryonic component of ETGs, since this is affectedby a number of complex physical processes including starformation, feedback from supernovae and the effect of anactive galactic nucleus (AGN). Semi-analytical studies ofthe evolution of ETGs based on dissipationless mergersseem to be able to match the observed mean size-mass Physics Department, University of California, Santa Bar-bara, CA 93106, USA Department of Physics and Astronomy, University of Cali-fornia, Los Angeles, CA 90025-1547, USA Kavli Institute for Particle Astrophysics and Cosmology,P.O. Box 20450, MS29, Stanford, CA 94309, USA Institute of Astronomy and Astrophysics, Academia Sinica,P.O. Box 23-141, Taipei 10617, Taiwan Institut d’Astrophysique de Paris, UMR7095 CNRS - Uni-versit´e Pierre et Marie Curie, 98bis bd Arago, 75014 Paris,France Institute of Astronomy, University of Cambridge, MadingleyRd, Cambridge CB3 0HA, UK Department of Physics and Astronomy, Bologna University,viale Berti-Pichat 6/2, I-40127 Bologna, Italy * [email protected] † Packard Research Fellow relation, but there are difficulties in matching the ob-served scatter (Nipoti et al. 2012; Shankar et al. 2013).In addition, dissipationless mergers appear not to beable to reproduce the evolution (or lack thereof) in thetotal density profile of massive ETGs (Sonnenfeld et al.2014). Porter et al. (2014) suggest that dissipational ef-fects are critical for correctly predicting the normaliza-tion and scatter of the fundamental plane relation, butfurther tests are needed to check whether their modelmatches the entire set of observations of ETGs. Onthe observational side, most of the efforts in studies ofETGs have been focused on improving our current knowl-edge of the luminous component of these objects, namelythe stellar populations and their cosmic evolution (e.g.Fontana et al. 2004; Cimatti et al. 2006; Pozzetti et al.2010; Peng et al. 2010; Choi et al. 2014), while very lit-tle is known about the dark matter component. The un-derlying dark matter distribution is affected by baryonicphysics processes: adiabatic contraction of gas can leadto more concentrated dark matter halos (Gnedin et al.2011) whereas supernova feedback can remove dark mat-ter from the center of a galaxy (Pontzen & Governato2012). Observational constraints of dark matter haloscan be used to test some of the many models for the ef-fects of baryonic physics on the evolution of ETGs. Cur-rent observational constraints on the dark matter halosof ETGs are scarce and come mostly from the analy-sis of kinematical tracers data, either alone (see, e.g.,Cappellari et al. 2013a; Agnello et al. 2014, for recent re-sults) or in combination with strong gravitational lensing(see, e.g., Sonnenfeld et al. 2012; Newman et al. 2013;Barnab`e et al. 2013; Suyu et al. 2014, for recent results).The main advantage of strong lensing is that it allowsfor accurate and precise measurements of masses out tocosmological distances, making it possible to explore thetime dimension and address evolutionary questions (see,e.g., Treu 2010, for a recent review). Sonnenfeld et al.In this work, we use strong lensing and stellar velocitydispersion measurements for a set of ∼
80 lenses fromthe Strong Lensing Legacy Survey (SL2S) and the SloanACS Lens Survey (SLACS) to infer the properties of thepopulation of massive galaxies out to redshift ∼ .
8. Us-ing the same sample of lenses, Sonnenfeld et al. (2013b,hereafter Paper IV) measured the mean density slope γ ′ of the total density profile ρ ( r ) ∝ r − γ ′ across thepopulation of massive ETGs, finding that ETGs evolvewhile keeping approximately a constant density slope( dγ ′ /dz = − . ± . γ ′ is not of easy interpretation. It isnot clear how dark matter and baryons conspire to man-tain a constant density slope while the stellar compo-nent becomes less concentrated. Here we address thisquestion by fitting a two-component model, consistingof a stellar spheroid and a dark matter halo, to thesame data. Since dark matter is by definition massthat is not associated with the baryonic component ofa galaxy, in order to measure dark matter masses itis necessary to carefully account for all of the mass instars. Stellar and dark matter can be effectively disen-tangled only in systems with data of exceptional qual-ity; for typical strong lenses, dark matter halo propertiescan be inferred either by making assumptions about thestellar initial mass function (IMF) (Auger et al. 2010a),or by statistically combining information from manysystems (Rusin & Kochanek 2005; Jiang & Kochanek2007; Oguri et al. 2014), and despite recent progress(Cappellari et al. 2012; Conroy & van Dokkum 2012;Spiniello et al. 2014), the true IMF of ETGs is todaystill a subject of debate (Smith & Lucey 2013). In thiswork we study an ensemble of massive ETGs with thegoal of characterizing simultaneously their distribution ofdark matter halo and stellar IMF properties. We achieveit with a hierarchical Bayesian inference method: a ro-bust statistical tool that allows us to properly take intoaccount scatter in the population. We explicitly takeinto account the selection function of our lensing sur-veys, allowing us to learn about the general populationof galaxies rather than just characterizing the populationof strong lenses.This paper is organized as follows. In Section 2 wedescribe the sample of lenses used in our study. In Sec-tion 3 we describe the model adopted to describe thedensity profile of the lenses in our sample. In Section 4we introduce the statistical framework used for the anal-ysis of the population of ETGs. In Section 5 we explainhow the selection function of lensing surveys is taken intoaccount. In Section 6 we assume a Navarro Frenk andWhite (NFW Navarro et al. 1997) model for the darkmatter halo of all lenses and combine individual mea-surements to infer the properties of the population ofmassive ETGs. In Section 7 we generalize the analysisto the case of halos with free inner slope. After a discus-sion of our results in Section 8 we conclude in Section 9.Throughout this paper magnitudes are given in the ABsystem. We assume a concordance cosmology with mat-ter and dark energy density Ω m = 0 .
3, Ω Λ = 0 .
7, andHubble constant H =70 km s − Mpc − . THE SAMPLE
Similarly to our previous work in Paper IV, we wouldlike to study the mass distribution of a large sample of galaxies through strong lensing and stellar kinemat-ics and explore dependences of the mass structure onsize, stellar mass and redshift. In order to achievethis goal we need a sample of strong lenses with mea-surements of the lens and source redshifts, central stel-lar velocity dispersion and stellar population synthesis(SPS) stellar mass, over a significant range of redshifts.Similarly to Paper IV, we include in our analysis 25lenses from the SL2S survey and 53 lenses from theSLACS survey (Auger et al. 2010a). Lens models andSPS stellar masses of the 25 SL2S systems are taken fromSonnenfeld et al. (2013a) (hereafter Paper III), whileredshifts and velocity dispersions measurements are re-ported in Paper IV. With the intent of increasing thesize of the sample of SL2S lenses, we collected new spec-troscopic data for eight lenses and lens candidates withthe instrument X-Shooter on the Very Large Telescope(P.I. Gavazzi, program 092.B-0663) and with DEIMOSon the W.M. Keck telescope. These new spectroscopicobservations are summarized in Table 1. Three of theobjects targeted in these observations had already beenobserved (see Paper IV). These are systems for whichthe redshift of either the background lensed source orthe main deflector (in the case of SL2SJ021801-080247)was previously unknown. We observed them again withX-Shooter which, thanks to its extended wavelength cov-erage, increases greatly the chances of detecting emis-sion lines from the lensed sources, as we demonstratedin Paper IV. For spectra with a sufficiently high signal-to-noise ratio we measured the velocity dispersion of thelens galaxy, necessary for the joint lensing and stellardynamics analysis carried out in this paper. Velocitydispersion fits are performed with the same techniquedescribed in Paper IV. The measured velocity disper-sions are reported in Table 1. One-dimensional spectraof newly observed lenses are plotted in Figure 1, and 2dspectra around detected emission lines are shown in Fig-ure 2. Lens candidates with no previous spec-troscopic observations are also lacking published photo-metric measurements, lens models and stellar masses,all necessary ingredients for the analysis carried out inthis paper. We present here these pieces of information.These newly observed targets all have ground-based pho-tometric data from the instrument MEGA-Cam on theCanada-France Hawaii Telescope (CFHT) in u, g, r, i, z bands. Measurements of the photometric properties ofthe lens galaxies are performed with the same methodadopted in Paper III. This consists of fitting a de Vau-couleurs profile (de Vaucouleurs 1948) to the data in allfive bands simultaneously, while masking out portions ofthe image contaminated with flux from the (blue) lensedbackground source. The measured magnitudes of the lensgalaxies are then used to fit stellar population synthesis(SPS) models to infer their stellar masses, again follow-ing Paper III. The stellar masses thus derived dependon the assumed form of the stellar initial mass function(IMF). We infer two sets of SPS masses assuming eithera Chabrier (Chabrier 2003) or a Salpeter (Salpeter 1955)IMF. In addition to the five systems with no previousspectroscopic observations, we report stellar mass mea-surements for SL2SJ021801 − TABLE 1Spectroscopic observations.
Name obs. date Instrument slit width seeing exp. time z d z s σ S/N FWHM( ′′ ) ( ′′ ) ( ′′ ) (s) (km/s) (˚A − ) (km/s)SL2SJ020457 − ±
30 9 40SL2SJ020524 − ±
37 7 40SL2SJ021801 − ±
48 7 40SL2SJ022046 − · · · − · · · · · · − · · · · · · − ±
21 9 40SL2SJ085317 − · · · ±
20 14 160
Note . — Summary of spectroscopic observations and derived parameters.
SL2SJ020457-110309, z d = 0 . XSHOOTER
SL2SJ020524-093023, z d = 0 . XSHOOTER
SL2SJ021801-080247, z d = 0 . XSHOOTER
SL2SJ022708-065445, z d = 0 . XSHOOTER
Rest-frame wavelength ( ˚A)
SL2SJ023307-043838, z d = 0 . XSHOOTER
Rest-frame wavelength ( ˚A)
SL2SJ085317-020312, z d = 0 . DEIMOS
Fig. 1.—
1d spectra of new SL2S lenses and lens candidates (in black). Where available, we overplot the best fit spectrum obtained forthe velocity dispersion fitting (in red). Only the rest-frame wavelength region used in the fit is shown. Vertical gray bands are regions ofthe spectrum masked out of the fit and typically correspond to atmospheric absorption features. Each plot indicates the redshift of thegalaxy and the instrument used to acquire the data shown. ing SPS stellar masses, are reported in Table 2.Finally, we present lens models for systems with newspectroscopic observations that had not been analyzedin Paper III. The lens modeling technique that we adopthere is the same used in Paper III. We model the massdistribution of each lens as a singular isothermal ellip-soid and fit it by reconstructing the unlensed imageof the background source. We use the software
GLEE (Suyu & Halkola 2010) for this purpose. Each system isthen assigned a grade describing the likelihood of it beinga strong lens: grade A for definite lenses, B for proba-ble lenses, C for possible lenses and X for non-lenses.Images of the lens systems, together with images of themost likely source and image reconstruction are shown inFigure 3. The inferred lens model parameters and lensgrades are reported in Table 3. We provide here a brief summary of the outcome of the lens modeling of eachsystem. • SL2SJ020457-110309. The CFHT data reveal anearly-type galaxy with a bright blue image, tangen-tially elongated with respect to it. The blue objectis spectroscopically detected to be at higher red-shift than the main galaxy. The lens model howeverdoes not predict the presence of a counter-image.This is probably because the image of the back-ground source appears to be unusually straight, asopposed to the typical arc-like shape of stronglylensed images. While there is no doubt that theforeground galaxy is lensing the background source,our ground-based data does not allow us to deter-mine whether there is strong lensing or not, there- Sonnenfeld et al.
TABLE 2Lens photometric parameters.
Name R eff q PA u g r i z log M (Chab) ∗ log M (Salp) ∗ (arcsec) (degrees) ( M ⊙ ) ( M ⊙ )SL2SJ020457-110309 1 .
01 0 . − . .
81 21 .
93 20 .
78 19 .
82 19 .
27 11 . ± .
15 11 . ± . .
75 0 . − . .
69 22 .
01 20 .
55 19 .
50 19 .
06 11 . ± .
12 11 . ± . .
02 1 . − . .
05 22 .
07 21 .
32 20 .
33 19 .
64 11 . ± .
15 11 . ± . .
45 0 .
28 84 . .
55 22 .
49 21 .
18 20 .
19 19 .
76 10 . ± .
14 11 . ± . .
31 0 .
85 45 . .
44 21 .
98 20 .
63 19 .
41 19 .
03 11 . ± .
14 11 . ± . .
85 0 .
61 16 . .
45 22 .
81 21 .
39 20 .
12 19 .
67 11 . ± .
13 11 . ± . Note . — Best fit parameters for de Vaucouleurs models of the surface brightness profile of the lens galaxies, asobserved in CFHT data, after careful manual masking of the lensed images. Columns 2–4 correspond to the effectiveradius ( R eff ), the axis ratio of the elliptical isophotes ( q ), and the position angle measured east of north (PA). The typicaluncertainties are 30% on the effective radius, ∆ q ∼ .
03 for the axis ratio, a few degrees for the position angle, 0 . u -band magnitudes, 0 . g and r -band magnitudes and 0 . i and z bands. The last two columnsshow the stellar mass measured through stellar population synthesis modeling, assuming a Chabrier or a Salpeter IMF. A α A α A α A α Fig. 2.—
2d spectra of new SL2S lenses and lens candidatesaround detected emission lines from the lensed background source.Observer frame wavelength (in ˚A) is labeled on the horizontal axis. fore we assign a grade B to this system. • SL2SJ020524-093023. The visible lensed images inthis system consist of one arc. The lens model pre-dicts the presence of a counter-image, too faint tobe visible in CFHT data. • SL2SJ022708-065445. An extended blue arc isclearly visible West of a disky early-type galaxy.The reconstructed source appears to consist of twocomponents close to each other. In order to achievea satisfactory fit, we had to put a Gaussian prioron the position angle of the mass distribution, cen-tered on the PA of the light. • SL2SJ023307-043838. This double image systemallows us to robustly measure the Einstein radiusof the lens galaxy. • SL2SJ085317-020312. One extended arc is visible.We assign a Gaussian prior to the mass positionangle in order to obtain a reasonable fit.The systems SL2SJ021801-080247 and SL2SJ022046-094927, which were modeled in Paper III and labeledas grade B systems, are here upgraded to A lenses invirtue of the new spectroscopic data revealing that lensand arc are indeed at two different redshifts.The number of SL2S lenses with a complete set ofdata necessary for a lensing and dynamics analysis isnow 28, with the addition of systems SL2SJ020524-093023, SL2SJ021801-080247 and SL2SJ022046-094927to the sample analyzed in Paper IV. TWO COMPONENT MASS MODELS
The analysis presented in Paper IV is based on power-law model density profiles for the total (stellar and dark)mass. Though very instructive, studying the total den-sity profile leaves open questions on what the detailedstructure of the mass profile is. Different mass profilescould give rise to the same value of γ ′ when fitted witha power-law model. Massive ETGs have a slope closeto γ ′ ≈
2. Models in which the mass follows the lightcorrespond to steeper slopes ( γ ′ ≈ . γ ′ = 2 there must be a non-stellar (dark) componentwith a mean slope shallower than isothermal. We wantto disentangle the contribution of the dark component tothe mass distribution of our lenses from that of the stars.For this purpose, we consider mass models with two com-ponents: a stellar spheroid and a dark matter halo. Wemodel the stellar spheroid with a de Vaucouleurs pro-file with effective radius fixed to the observed one, anda uniform prior on the stellar mass-to-light ratio. Theark matter content and stellar IMF of massive early-type galaxies 5 TABLE 3Lens model parameters
Name R Ein q PA m s Grade Notes(arcsec) (degrees) (mag)SL2SJ020457-110309 0 . ± .
07 0 . ± .
19 47 . ± . .
61 BSL2SJ020524-093023 0 . ± .
09 0 . ± . − . ± . .
74 ASL2SJ022708-065445 0 . ± .
05 0 . ± . − . ± . .
51 A DiskySL2SJ023307-043838 1 . ± .
06 0 . ± .
04 50 . ± . .
40 ASL2SJ085317-020312 0 . ± .
12 0 . ± .
11 17 . ± . .
52 A
Note . — Peak value and 68% confidence interval of the posterior probability distributionof each lens parameter, marginalized over the other parameters. Columns 2–4 correspondto the Einstein radius ( R Ein ), the axis ratio of the elliptical isodensity contours ( q ), andthe position angle measured east of north (PA) of the SIE lens model. Column 5 showsthe magnitude of the de-lensed source in the g filter. The typical uncertainty on the sourcemagnitude is ∼ .
5. In Column 6 we report the grade of the lens system, describing thelikelihood of it being a strong lens. Column 7 lists notes on the lens morphology. dark matter halo is modeled with a generalized Navarro,Frenk & White (gNFW) profile (Zhao 1996): ρ DM ( r ) ∝ r γ DM (1 + r/r s ) − γ DM . (1)Both components are spherical. We fix the effective ra-dius of the stellar component to the observed one, andthe scale radius of the dark matter is fixed to r s = 10 R e ,which is a typical value seen in numerical simulations(e.g. Kravtsov 2013). The impact of this choice on ourinference will be discussed at the end of this Section.This mass model has three degrees of freedom, whichwe describe in terms of the stellar mass M ∗ , the pro-jected dark matter mass within a cylinder of 5 kpc ra-dius M DM5 , and the inner slope of the dark matter halo γ DM . We fit this model to the observed Einstein radiusand central velocity dispersion with the same procedureused in Paper IV. Model Einstein radii are calculatedgiven M ∗ , M DM5 and γ DM assuming spherical profiles,and model velocity dispersions are calculated throughthe spherical Jeans equation assuming isotropic orbits.The fit is done in a Bayesian framework, assuming a uni-form prior on log M ∗ , log M DM5 and γ DM , and restrict-ing the range of possible values for the latter quantity to0 . < γ DM < .
8. Note that this is very similar to the“free” model adopted by (Cappellari et al. 2012).As an example, we show in Figure 4 the posterior prob-ability distribution function of the model parameters forthe lens SL2SJ142059+563007. The model is largely un-constrained, since it consists of three free parameters thatare fit to only two pieces of data: the Einstein radius andthe central velocity dispersion. As expected and observedby previous authors (e.g. Treu & Koopmans 2002a,b),there is a strong degeneracy between the inner slope andnormalization of the dark matter component. The tiltof this degeneracy is determined in part by our choiceof parametrizing the dark matter halo in terms of themass enclosed within 5 kpc. This is not directly observ-able, while the mass enclosed within the Einstein radiusis better constrained by the data. For the lens in this ex-ample, the Einstein radius is larger than 5 kpc, thereforefor fixed dark matter mass within R Ein , the inferred massat 5 kpc will depend on the assumed value of the darkmatter slope. Nevertheless, 5 kpc is close in value to themedian Einstein radius of the lenses considered in this work and the choice of M DM5 to parametrize the darkmatter mass will prove useful later in this work, whenanalyzing the entire set of lenses statistically.For systems with data of exceptional quality, the de-generacy between dark matter mass and slope can bebroken without having to make additional assumptions(e.g. Sonnenfeld et al. 2012). In our work, we do notwish to constrain γ DM and M DM5 for individual systems,but we measure their population average values by sta-tistically combining measurements over a large numberof lenses. This will be the subject of Sections 4, 6 and7. Nevertheless, it is interesting to constrain the darkmatter content and the stellar mass of individual lenses.This can be done, provided we make a more restrictiveassumption on the shape of the dark matter halo. Wedo this by fixing the inner slope of the dark matter haloto γ DM = 1 and hence restrict ourselves to NFW densityprofiles for the rest of this section. The free parametersof the model are now the stellar mass M ∗ and the nor-malization of the dark matter halo, M DM5 . The model isvery similar to the one used by Treu et al. (2010). Theonly difference lies in the choice of the scale radius of theNFW component, r s . In Treu et al. (2010) this was fixedto 30 kpc, while here we fix it to ten times the effectiveradius of the stellar component.We fit this model to the lensing and stellar kinematicsdata of each one of the SL2S lenses, as well as lenses fromthe SLACS survey. Our two component model, with ade Vaucouleurs spheroid and an NFW dark matter halo,provides excellent fits to most of our lenses. The only ex-ceptions are a few SLACS lenses with very steep densityslope γ ′ > .
2, i.e. with relatively large velocity disper-sions for their Einstein radius (similar to PG1115+080Treu & Koopmans 2002b). In the context of our model, asteep density slope corresponds to a larger ratio betweenstellar and dark matter mass, since the NFW halo hasa much shallower density profile than a de Vaucouleursprofile at the scale relevant for our measurements, i.e. atthe effective radius. The steepest density profile we canconstruct with such a two component model is a galaxywith no dark matter. For these few SLACS lenses, evenif we assign the entire mass enclosed within the Einsteinradius to the spheroid, the model velocity dispersion isstill smaller than the measured one, although consistentwithin the uncertainty. A perfect match with the data Sonnenfeld et al.
Fig. 3.—
Lens models of five new SL2S galaxy-scale lens candidates. From left to right: CFHT image, input science image used for themodeling, image reconstruction, lensed source reconstruction, residual image normalized by the uncertainty on each pixel. would require γ DM >
2, excluded by our prior. The in-ference then favors small dark matter masses for thosesystems. Adopting a more flexible model for the stellardensity profile does not help in this case: Posacki et al.(2014) did a similar spheroid and halo decomposition tothe same SLACS lenses considered here using a multi-gaussian fit to the photometry, and still found very smalldark matter fractions for some of the objects.The derived model parameters for the SL2S lenses are reported in Table 4. The parameters consideredare the stellar mass, M ∗ , the projected dark mattermass enclosed within 5 kpc, M DM5 , the projected darkmatter mass enclosed within R e , M DMe , the fractionof dark matter mass projected within a cylinder of ra-dius R e , f DMe , and finally the
IMF mismatch parameter (Treu et al. 2010), defined as the ratio between the truestellar mass and its estimate based on stellar populationark matter content and stellar IMF of massive early-type galaxies 7
Fig. 4.—
Posterior probability distribution for a de Vau-couleurs + gNFW mass model of the gravitational lensSL2SJ142059+563007. The model parameters are the total stel-lar mass M ∗ , the inner slope of the dark matter halo γ DM and theprojected dark matter mass M DM5 enclosed within a cylinder of 5kpc radius. synthesis models assuming a Salpeter IMF, M (SPS) ∗ : α IMF ≡ M ∗ M (SPS) ∗ . (2)In this parametrization, a Chabrier IMF corresponds tolog α IMF ≈ − .
25. Individual measurements of the IMFmismatch parameter and the dark matter fraction areplotted as a function of redshift in Figures 5 and 6. Underthe above assumptions and with typical data quality, weare able to determine dark matter masses with a ∼ r s = 10 R e . Decreasing thevalue of the proportionality constant to r s = 5 R e resultsin dark matter masses smaller by ∼ .
10 dex and stel-lar masses larger by ∼ .
05 dex. We use these values asan estimate of the systematic uncertainty introduced byfixing the value of the dark matter scale radius. The sys-tematic uncertainty introduced by fixing the dark matterslope is only moderately larger, as can be deduced fromFigure 4.Most of the individual measurements of the IMF nor-malization are consistent with a Salpeter IMF. Thereare however a few outliers in the measurements shownin Figure 5. This is because the values of α IMF plot-ted in Figure 5 are obtained by marginalizing over thedark matter mass. The actual probability distributionsin the α IMF − M DM5 space are very elongated and extendcloser to the value log α IMF = 0 than the marginalizedposterior would suggest. The strong degeneracy betweenstellar and dark matter mass is taken fully into accountin the population analysis described in the next Section. HIERARCHICAL BAYESIAN INFERENCE
As shown above, the lensing and stellar kinematicsdata available for typical strong lenses are not sufficientto constrain both the slope and the normalization of thedark matter halo of individual objects. An important . . . . . z − . − . . . . l og α I M F SLACSSL2S
Fig. 5.—
IMF mismatch parameter α IMF = M ∗ /M (Salp) ∗ , re-ferred to a Salpeter IMF, as a function of redshift for galaxies ofthe SL2S, SLACS and LSD samples. Fig. 6.—
Fraction of mass in dark matter projected within acylinder of radius equal to the effective radius, as a function ofredshift. question is whether the IMF normalization or the darkmatter fraction evolve with time within the populationof ETGs. One possible way of addressing this questionis performing a linear fit for α IMF ( z ) and f DM e ( z ). How-ever, the analysis of Paper IV revealed that the densityslope γ ′ of ETGs is a function of mass and size as wellas redshift. This dependency of γ ′ on M ∗ and R e willpresumably be reflected on α IMF or f DM e . It is thenimportant to take all dependencies into account whenaddressing the time evolution of these two parameters.We want to characterize the population of early-typegalaxies which our strong lenses are drawn from. Thefocus is on the stellar mass, the IMF normalization, thedark matter mass within 5 kpc and the inner dark matterslope. Our lenses span a range of redshifts, stellar massesand sizes and we are interested to measure whether thereare structural variations with these quantities. In anal-ogy to the work of Paper IV, the strategy we adopt is ahierarchical Bayesian inference method. We assume thatthe values of the parameters describing individual galax-ies, ω i , are drawn from a parent distribution described Sonnenfeld et al. TABLE 4Stellar and dark matter masses of individual SL2S galaxies, assuming NFW halos.
Name z R e log M Salp ∗ log M LD ∗ log α IMF log M DM5 log M DMe f DMe (kpc) M ⊙ M ⊙ M ⊙ M ⊙ SL2SJ020524 − .
56 4 .
82 11 . ± .
12 11 . +0 . − . − . ± .
18 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ021247 − .
75 8 .
92 11 . ± .
17 11 . +0 . − . . ± .
19 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ021325 − .
72 17 .
67 11 . ± .
19 12 . +0 . − . . ± .
25 11 . +0 . − . . +0 . − . . +0 . − . SL2SJ021411 − .
61 6 .
29 11 . ± .
14 11 . +0 . − . − . ± .
17 11 . +0 . − . . +0 . − . . +0 . − . SL2SJ021737 − .
65 4 .
27 11 . ± .
16 11 . +0 . − . . ± .
19 11 . +0 . − . . +0 . − . . +0 . − . SL2SJ021801 − .
88 7 .
90 11 . ± .
14 11 . +0 . − . . ± .
46 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ021902 − .
39 3 .
01 11 . ± .
10 11 . +0 . − . . ± .
12 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ022046 − .
57 3 .
45 11 . ± .
11 11 . +0 . − . . ± .
14 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ022511 − .
24 8 .
59 11 . ± .
09 11 . +0 . − . − . ± .
15 11 . +0 . − . . +0 . − . . +0 . − . SL2SJ022610 − .
49 6 .
44 11 . ± .
11 11 . +0 . − . . ± .
16 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ023251 − .
35 4 .
78 11 . ± .
09 11 . +0 . − . . ± .
10 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ023307 − .
67 9 .
21 11 . ± .
13 11 . +0 . − . − . ± .
41 11 . +0 . − . . +0 . − . . +0 . − . SL2SJ084847 − .
68 3 .
21 11 . ± .
16 11 . +0 . − . − . ± .
28 11 . +0 . − . . +0 . − . . +0 . − . SL2SJ084909 − .
72 3 .
55 11 . ± .
13 11 . +0 . − . . ± .
15 11 . +0 . − . . +0 . − . . +0 . − . SL2SJ084959 − .
27 6 .
11 11 . ± .
09 11 . +0 . − . . ± .
10 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ085019 − .
34 1 .
35 11 . ± .
09 11 . +0 . − . . ± .
10 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ085540 − .
36 3 .
48 11 . ± .
10 11 . +0 . − . . ± .
13 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ090407 − .
61 16 .
81 11 . ± .
12 11 . +0 . − . − . ± .
48 11 . +0 . − . . +0 . − . . +0 . − . SL2SJ095921+020638 0 .
55 3 .
47 11 . ± .
11 11 . +0 . − . − . ± .
16 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ135949+553550 0 .
78 13 .
08 11 . ± .
15 11 . +0 . − . . ± .
22 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ140454+520024 0 .
46 11 .
78 12 . ± .
10 12 . +0 . − . . ± .
12 11 . +0 . − . . +0 . − . . +0 . − . SL2SJ140546+524311 0 .
53 4 .
58 11 . ± .
11 11 . +0 . − . . ± .
14 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ140650+522619 0 .
72 4 .
35 11 . ± .
15 11 . +0 . − . − . ± .
17 11 . +0 . − . . +0 . − . . +0 . − . SL2SJ141137+565119 0 .
32 3 .
04 11 . ± .
09 11 . +0 . − . − . ± .
16 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ142059+563007 0 .
48 7 .
86 11 . ± .
10 11 . +0 . − . − . ± .
16 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ220329+020518 0 .
40 3 .
86 11 . ± .
10 11 . +0 . − . − . ± .
24 11 . +0 . − . . +0 . − . . +0 . − . SL2SJ220506+014703 0 .
48 3 .
93 11 . ± .
10 11 . +0 . − . . ± .
13 10 . +0 . − . . +0 . − . . +0 . − . SL2SJ221326 − .
34 2 .
41 10 . ± .
10 10 . +0 . − . − . ± .
17 11 . +0 . − . . +0 . − . . +0 . − . SL2SJ222148+011542 0 .
33 5 .
27 11 . ± .
09 11 . +0 . − . − . ± .
19 10 . +0 . − . . +0 . − . . +0 . − . Note . — Redshifts, effective radii, stellar masses from SPS fitting (from Paper III) and from lensing and dynamics,projected dark matter masses within 5 kpc and within the effective radius, projected dark matter fractions within theeffective radius of SL2S lenses. by a set of hyper-parameters τ to be determined fromthe data d . From Bayes theorem,Pr( τ | d ) ∝ Pr( d | τ )Pr( τ ) . (3)In turn, the probability of observing the data d given thepopulation model τ can be written as the following prod-uct over the individual objects’ marginal distributions:Pr( d | τ ) = N Y i Z d ω i Pr( d i | ω i )Pr( ω i | τ ) . (4)The integrals run over all the parameters of individ-ual galaxies ω i . The first term in the product ofEquation 4 is the likelihood function for an individ-ual galaxy’s model parameters ω i given the data d i .The set of model parameters for an individual galaxy is ω i = ( M ∗ ,i , M DM5 ,i , γ DM ,i , α IMF ,i , z i , R e ,i ), where M DM5 is the dark matter mass within the effective radius. Thedata consist of the measured Einstein radius R Ein ,i , effec-tive radius, redshift, velocity dispersion within the aper-ture used for spectroscopic observations σ ap ,i and thestellar mass measured with the stellar population synthe-sis analysis, M (SPS) ∗ ,i . In order to speed up computations, redshifts and effective radii of all lenses are assumed to beknown exactly. This approximation does not introduceany significant uncertainty since the typical precision oneffective radii measurements is 10% (Sonnenfeld et al.2013a), which corresponds to a small uncertainty on thekey model parameters, and the uncertainty on redshiftsis δz = 0 . d i for lens i , given its parameters ω i can be factorized as follows:Pr( d i | ω i ) = Pr( R Ein ,i | ω i )Pr( σ ap ,i | ω i ) × Pr( M (SPS) ∗ ,i | ω i ) δ ( R (obs)eff ,i − R e ,i ) δ ( z (obs) i − z i )(5)This is possible because the observational uncertaintieson the measured Einstein radius, velocity dispersion andSPS stellar mass are independent of each other. For somelenses in the SL2S sample, more than one velocity disper-sion measurements is available (Sonnenfeld et al. 2013b).In those cases, the velocity dispersion term in Equation 5becomes a product over the multiple measurements.The second term in the integrand of (4) is the prob-ability for the galaxy’s individual stellar mass and halomass given the set of hyper-parameters τ . The hyper-parameters must describe the population of galaxies fromark matter content and stellar IMF of massive early-type galaxies 9which our strong lenses are drawn. Similarly to Paper IV,we assume that the structural properties of ETGs, in thiscase the dark matter mass and the IMF normalization,are a function of redshift, stellar mass and effective ra-dius. In the formalism of Kelly (2007), ξ i = { z i , M ∗ ,i , R e ,i } (6)are the independent variables, while η i = { M DM5 ,i , γ DM ,i , α IMF ,i } (7)are the dependent variables. It is useful to distinguishamong the hyper-parameters the ones that describe thedistribution in the independent variables, ψ , and thosedescribing the distribution of dependent variables, whichwe label as θ , following the notation of Kelly (2007). Thequantity Pr( ω i | τ ) has then the following form:Pr( ω i | τ ) = Pr( ξ i | ψ )Pr( η i | ξ i , θ ) , (8)where ω i = ξ i ∪ η i and τ = ψ ∪ θ . The probabilitydistribution of the independent variables describes howgalaxies in our sample are distributed in the { z, M ∗ , R e } space. It encodes both information on the distributionof galaxies in the Universe and the way lens candidatesare targeted in our lensing surveys, in terms of selectionsin stellar mass (or similarly, luminosity), redshift andsize. We assume that the distribution in the independentvariables can be written as the product of two Gaussiansin log M ∗ and log R e :Pr( ξ i | ψ ) =1 σ ∗ √ π exp (cid:20) − (log M ∗ ,i − µ ∗ ( ω i )) σ ∗ (cid:21) × σ R √ π exp (cid:20) − (log R e ,i − µ R ( ω i )) σ R (cid:21) . (9)The mean of these Gaussians is assumed to be differentfor lenses of different surveys: µ (SL2S) ∗ = ζ (SL2S) ∗ ( z i − .
5) + log µ (SL2S) ∗ , , (10) µ (SLACS) ∗ = ζ (SLACS) ∗ ( z i − .
2) + log µ (SLACS) ∗ , , (11) µ (SL2S) R = ζ (SL2S) R ( z i − .
5) + β (SL2S) R (log M ∗ − .
5) +log µ (SL2S) R, , (12) µ (SLACS) R = ζ (SLACS) R ( z i − .
2) + β (SLACS) R (log M ∗ − . µ (SLACS) R, . (13)We also assume different values of the dispersion σ ∗ , σ R for SL2S and SLACS lenses. Note that there’s no ex-plicit term for the distribution in z in Equation 9. Thisis equivalent to assuming a uniform distribution in red-shift. The more physically relevant quantity is the sec-ond term in Equation 8, which describes the propertiesof the dark matter halos and stellar IMF for galaxies ofgiven z , M ∗ and R e . The goal of this work is to un-derstand the properties of massive galaxies, irrespectiveof their lens nature. However, some galaxies are morelikely to be strong lenses than others, because the lens-ing probability depends in part on the density profile (Mandelbaum et al. 2009). Moreover, some strong lensesare more easily detectable than others, as discussedfor example by Arneson et al. (2012); Smith & Lucey(2013); Gavazzi et al. (2014). Then, in order to makeaccurate statements on the evolution on galaxies basedon strong lensing measurements, we must take into ac-count these selection effects. It is important to verifywhether the selection of lenses in the SLACS or SL2Ssurveys introduce a significant bias with respect to thegeneral population of ETGs, and to quantify it. Theterm Pr( η i | ξ i , θ ) should then include a term taking intoaccount the probability for a galaxy described by param-eters η i of being a strong lens detected in a survey. Wedescribe such probability with a set of hyper-parameters λ . The term relative to the dependent variables is thenalso assumed to be product of Gaussians, multiplied bya selection function term S ( η i | ξ i , λ ):Pr( η i | ξ i , θ , λ ) = S ( η i | ξ i , λ ) × σ DM √ π exp (cid:20) − (log M DM5 ,i − µ DM ( ξ i )) σ (cid:21) × σ γ √ π exp (cid:20) − ( γ DM ,i − µ γ ( ξ i )) σ γ (cid:21) × σ IMF √ π exp (cid:20) − (log α IMF ,i − µ IMF ( ξ i )) σ (cid:21) ×S ( η i | ξ i , λ ) . (14)The term S ( η i | ξ i , λ ), which will be discussed in Sec-tion 5, represents the lensing selection function. Thisterm multiplies the intrinsic distribution of galaxy pa-rameters, which we assumed to be described by a productof Gaussians, to give the distribution observed in stronglenses. Note that a similar decomposition could in prin-ciple be written for the distribution in the independentvariables, Pr( ξ i | ψ ). In practice, we are interested in re-covering the true distribution for the dependent variablesonly. The means of the Gaussians in Equation 14 arein general functions of galaxy redshift, stellar mass andeffective radius. In particular, we expect the dark mat-ter mass to grow with the stellar mass. We also expectthe ratio between stellar and dark mass and the darkmatter slope to vary with projected stellar mass density,Σ ∗ = M ∗ / (2 πR ), as the results of Paper IV highlightedhow the density profile of ETGs at fixed redshift dependsto first order on Σ ∗ , with systems with more compactstellar distributions having steeper density slopes. Wethen choose to parametrize the scaling relations of darkmatter halo and stellar IMF normalization in terms of M ∗ and Σ ∗ , as follows: µ DM = ζ DM ( z i − .
3) + β DM (log M ∗ ,i − .
5) + ξ DM log Σ ∗ / Σ + log M DM , (15) µ γ = γ (16) µ IMF = ζ IMF ( z i − .
3) + β IMF (log M ∗ − .
5) + ξ IMF log Σ ∗ / Σ + log α IMF , (17)Although it might seem more natural to assume a scal-ing in M ∗ and R e , which would isolate the dependence0 Sonnenfeld et al.on stellar mass to only one parameter, M ∗ and R e arehighly correlated because of the observed tight mass-sizerelation. As a result, dependences on M ∗ or R e are highlyinterchangeable and it is difficult to isolate the two withour data. A parameterization in terms of M ∗ and Σ ∗ mit-igates this effect. For the average dark matter slope wedo not allow for any scaling with any independent vari-able. This choice is driven by the little information avail-able from our data on the slope for an individual galaxy(see Figure 4). Allowing for too much freedom wouldresult in the average slope of the population of galaxiesbeing undetermined. As always, when the likelihood isnot very informative, it is important to choose very care-fully the model parameters and priors. To summarize,the set of hyper-parameters describing the distributionof independent variables is ψ = { ζ ∗ , µ ∗ , , σ ∗ , ζ R , β R , µ R, , σ R } , (18)with each parameter defined separately for the distri-bution of SL2S and SLACS lenses, while the hyper-parameters describing the dependent variables distribu-tion is θ = { ζ DM , β DM , ξ DM , M DM , , σ DM , γ , σ γ ,ζ IMF , β
IMF , ξ
IMF , α
IMF , , σ IMF , λ } . (19)Finally, we need to specify the form of the selection func-tion correction S ( η i | ξ i , λ ) in Equation 14. The followingsection is devoted to it. THE SELECTION FUNCTION
With the term “selection function” we define the map-ping between the global population of ETGs and the sub-set of the population sampled by our lens catalogs. Thegoal of this section is to characterize this selection func-tion in a both accurate and computationally tractableway. SL2S and SLACS, from which our lenses are cho-sen, are different lensing surveys and are in general sub-ject to different selection effects. Nevertheless, selectioneffects for the SL2S and SLACS surveys are qualitativelysimilar, and will be treated within the same framework.We can identify three main sources of selection. Thefirst one is the brightness of the lens. Both SLACS andSL2S samples were assembled by following-up massiveETGs, brighter than a threshold. For the subset ofSLACS galaxies we are considering, the lower limit tothe brightness was implicitly set by the requirement ofthe lens galaxy being targeted in the SDSS spectroscopicsurvey and having sufficient S/N to allow for a veloc-ity dispersion measurement (citation needed). For SL2S,only ETGs brighter than 21 . i -band were followed-up(Gavazzi et al. 2014). While the luminosity function ofETGs is well described by a one or two Schechter func-tions (Ilbert et al. 2013), selection in brightness resultsin a different luminosity function for strong lenses, witha cut at low luminosities. Luminosity is not directlyparametrized in the model described in Section 4, butit is tightly related to the stellar mass. This selectioneffect can then be captured by the parameters describ-ing the distribution in stellar mass in equations (10) and(11).The second selection effect is due to different lenseshaving different strong lensing cross-sections, X lens , i.e.different probability of producing systems of multiple im-ages of background sources. Mandelbaum et al. (2009) studied in detail how lensing cross-section depends onlens properties. As expected from general lensing the-ory, their main finding is that galaxy mass and densityprofile are the most important parameters determining X lens : more massive galaxies have larger lensing cross-section, and so do galaxies with a steeper density profile,at fixed mass. Quantitatively, the probability of a galaxydescribed by parameters ω i of being a strong lens is pro-portional to X lens ( ω i ). Therefore, the term S ( η i | ξ i , λ )in Equation 14, which is proportional to the probabil-ity of detecting a lens of parameters ω i given a selectionfunction described by λ , should also be proportional to X lens . The strong lensing cross-section of a lens with asmooth density profile monotonically decreasing with ra-dius is given by the area enclosed by the radial caustic,i.e. the points in the source plane mapped to points ofinfinite magnification in the radial direction. For simplic-ity, we calculate X lens ( ω i ) assuming spherical symmetry,as the area enclosed within the radial critical curve, un-like the tangential critical curve, is not very sensitiveto the ellipticity of the lens. Formally, the term X lens has units of solid angle. In practice, X lens is rescaledso that the probability Pr( η i | ξ i , θ , λ ) defined in Equa-tion 14 integrated over η i is normalized to unity. Thethird selection effect that we consider is the differentdetectability of strong lenses of different properties inthe two surveys considered, i.e. the probability, givena strong lens, of detecting it in a given lensing survey.The detection probability in the SL2S was studied byGavazzi et al. (2014), while that in SLACS-like surveyswas studied by Arneson et al. (2012). The most obvi-ous parameter determining the detection probability isthe brightness of the lensed background source: brighterarcs are easier to detect for both the SL2S and SLACSsurveys. In addition to the source brightness, anotherimportant parameter determining the detection proba-bility in both SL2S and SLACS is the Einstein radius.Gavazzi et al. (2014) have shown how the selection func-tion for SL2S lenses is mostly a function of R Ein , witha peak in the range 1 ′′ < R Ein < ′′ . SL2S lenses areselected photometrically by looking for blue arcs aroundred galaxy in ground based observations (Gavazzi et al.2012). This technique works best for lenses with Einsteinradius larger than ∼ ′′ , since arcs with R Ein smallerthan that are difficult to resolve in ground based pho-tometry. The upper limit is due to the fact that lenseswith radius smaller than 3 ′′ were preferentially targetedin the lens-finding algorithm, to favor galaxy-scale lensesover group-scale ones. For SLACS, lens candidates wereselected by looking for emission lines from lensed star-forming galaxies, and then confirmed by HST imaging.Lenses with too small Einstein radii are more difficultto confirm with this method, because of confusion be-tween the source and the deflector light. At the op-posite end, lenses with too large Einstein radii can es-cape the selection because the lensed features contributelittle to the flux deposited within the 1 . ′′ . . . . M ∗ . . . l og M D M . ” . ” . ” . ” . ” R Ein and σ lens map − . − . − . − . . . . . . l og σ l e n s Fig. 7.—
Solid lines: levels of constant angular Einstein radius(in arcsec) as a function of log M ∗ and log M DM5 for a lens atredshift z d = 0 .
3, with source redshift z s = 1 .
5, effective radius R e = 5kpc and γ DM = 1. Intensity map: logarithm of the stronglensing cross section, X lens in arcsec . imating the detection probability with a Gaussian func-tion in R Ein , which multiplies the previously discussedlensing cross-section term in the selection correction: S ( η i | ξ i , λ ) = AX lens √ πσ sel exp (cid:26) − ( R Ein ( ω i ) − R sel ) σ sel (cid:27) (20)where R Ein is a function of the lens parameters ω i and A is a normalization constant. Here λ = { R sel , σ sel } are hyper-parameters describing the selection function,which can be different for SL2S and SLACS surveys.Note that there is no source brightness term in Equa-tion 20, which we anticipated being important in deter-mining the detection probability of a strong lens. Thisis because the source brightness is not directly modeledin the hierarchical Bayesian inference framework intro-duced in the previous section. The term Equation 20should then be considered as the effective selection func-tion, obtained by marginalizing over all possible valuesof the source brightness. To illustrate what a selectionfunction of the form given by Equation 20 correspondsto in terms of stellar and dark matter mass, we show inFigure 7 how the Einstein radius of a typical lens changesas a function of M ∗ and M DM5 , the other parameters be-ing fixed. A Gaussian selection function in the Einsteinradius implies that only lenses that occupy a band in thelog M ∗ − log M DM5 plane can be observed. In the sameplot we show the lensing cross-section depends on M ∗ and M DM5 . As expected, larger masses correspond tolarger lensing cross-sections. RESULTS, NFW HALOS
Before proceeding to analyze the most general case, wefocus in this section on models with a fixed dark matterslope γ DM = 1, corresponding to an NFW profile. Thiswill indicate whether we can get an adequate descrip-tion of the evolution of the structure of massive galax-ies with a simple dark matter model. We need to ex-plore the posterior probability distribution Pr( τ | d ) viaMarkov Chain Monte Carlo (MCMC). This requires eval-uating, for each lens and at each step of the chain, thelikelihood term Pr( d i | ω i ) given by Equation 5 and inte- grating over all possible values of the lens parameters ω i ,as given by Equation 4. The integration over log α IMF can be performed analytically, because both the likeli-hood Equation 5 and the parent distribution Equation 14are Gaussian in log α IMF . Integrals over z and log R e are trivial, because lens redshift and effective radius areassumed to be known exactly. We are left with two-dimensional integrals over log M ∗ and log M DM5 . Thisis a computationally expensive operation, because M ∗ and M DM5 enter the likelihood and the selection func-tion term S ( η i | ξ i , λ ) through the Einstein radius andthe velocity dispersion, which are in general non-analyticfunctions of these parameters. In order to speed up thecomputation, we sample the likelihood term beforehandfor each galaxy and then perform the integrals in Equa-tion 4 via Monte Carlo integration at each step of thechain, evaluating the integrand by importance sampling(see e.g. Suyu et al. 2010; Busha et al. 2011, and refer-ences therein). For both computational and physical rea-sons (our lenses have a finite amount of stars and darkmatter), we truncate the distribution Equation 14 be-tween 10 . . M ∗ and between 10 . . M DM5 . In order to be self-consistent, at each stepof the chain all probability terms must be normalized tounity. The term Pr( η i | ξ i , θ ) requires particular care, asit contains a term, S , that is non-analytic in the modelparameters. The following equality should hold Z d η i Pr( η i | ξ i , θ ) = 1 (21)for each set of values of ξ i and θ . This is an implicitequation for the normalization constant in Equation 14,which we solve via Monte Carlo integration.We assume a uniform prior on all model hyper-parameters. We sample the posterior probability distri-bution with an MCMC with 100000 points, using PyMC(Patil et al. 2010). The median, 16th and 84th percentileof the posterior probability distribution function (PDF)of each parameter, marginalized over the other parame-ters, are listed in Table 5. The inference on the hyper-parameters describing the dependent variables ( M DM5 and α IMF ), ξ , is plotted in Figures 8 and 9.The parameters explored by this model are numerous.Among the results of this analysis we highlight the fol-lowing. Under the assumption that dark matter halos ofall ETGs have an NFW profile: • The average projected dark matter mass within 5kpc in massive ETGs is log M = 10 . +0 . − . . • We find marginal evidence for an anticorrelationbetween dark matter mass enclosed within 5 kpc( M DM5 ) and stellar mass density Σ ∗ (parameter ξ DM < M DM5 and redshift (parameter ζ DM < β DM is consis-tent with zero). • The IMF normalization is consistent with an IMFslightly heavier than Salpeter: log α IMF , = 0 . ± . • The IMF normalization correlates positively with2 Sonnenfeld et al.
TABLE 5Bayesian hierarchical inference: the hyper-parameters. NFW model.
With S No S Parameter description µ (SL2S) ∗ , . +0 . − . . +0 . − . Mean stellar mass at z = 0 .
5, SL2S sample ζ (SL2S) ∗ . +0 . − . . +0 . − . Linear dependence of mean stellar mass on redshift, SL2S sample σ (SL2S) ∗ . +0 . − . . +0 . − . Scatter in mean stellar mass, SL2S sample µ (SLACS) ∗ , . +0 . − . . +0 . − . Mean stellar mass at z = 0 .
2, SLACS sample ζ (SLACS) ∗ . +0 . − . . +0 . − . Linear dependence of mean stellar mass on redshift, SLACS sample σ (SLACS) ∗ . +0 . − . . +0 . − . Scatter in mean stellar mass, SLACS sample µ (SL2S) R, . +0 . − . . +0 . − . Mean effective radius at z = 0 .
5, log M ∗ = 11 .
5, SL2S sample ζ (SL2S) R . +0 . − . . +0 . − . Linear dependence of mean effective radius on redshift, SL2S sample β (SL2S) R . +0 . − . . +0 . − . Linear dependence of mean effective radius on stellar mass, SL2S sample σ (SL2S) R . +0 . − . . +0 . − . Scatter in mean effective radius, SL2S sample µ (SLACS) R, . +0 . − . . +0 . − . Mean effective radius at z = 0 .
2, log M ∗ = 11 .
5, SLACS sample ζ (SLACS) R . +0 . − . . +0 . − . Linear dependence of mean effective radius on redshift, SLACS sample β (SLACS) R . +0 . − . . +0 . − . Linear dependence of mean effective radius on stellar mass, SLACS sample σ (SLACS) R . +0 . − . . +0 . − . Scatter in mean effective radius, SLACS sample ζ DM . +0 . − . . +0 . − . Linear dependence of log M DM5 on redshift. β DM . +0 . − . − . +0 . − . Linear dependence of log M DM5 on log M ∗ . ξ DM − . +0 . − . − . +0 . − . Linear dependence of log M DM5 on log Σ ∗ log M DM , . +0 . − . . +0 . − . Mean M DM5 at z = 0 .
3, log M ∗ = 11 . R eff = 5 kpc σ DM . +0 . − . . +0 . − . Scatter in the M DM5 distribution ζ IMF − . +0 . − . − . +0 . − . Linear dependence of IMF normalization on redshift. β IMF . +0 . − . . +0 . − . Linear dependence of IMF normalization on log M ∗ . ξ IMF . +0 . − . . +0 . − . Linear dependence of IMF normalization on log Σ ∗ log α IMF , . +0 . − . . +0 . − . Mean IMF normalization at z = 0 .
3, log M ∗ = 11 . R eff = 5 kpc σ IMF . +0 . − . . +0 . − . Scatter in the IMF normalization distribution R (SL2S)sel . +0 . − . · · · Mean observable Einstein radius, SL2S sample σ (SL2S)sel . +0 . − . · · · Dispersion in observable Einstein radius, SL2S sample R (SLACS)sel . +0 . − . · · · Mean observable Einstein radius, SLACS sample σ (SLACS)sel . +0 . − . · · · Dispersion in observable Einstein radius, SLACS sample
Note . — Median, 16th and 84th percentile of the posterior probability distribution function of each model hyper-parameter, marginalized over the other parameters. Results are reported for the full case and ignoring the selectionfunction. stellar mass ( β IMF = 0 . ± . ω i | λ ), we also show the posterior PDF obtainedexcluding it from our analysis. Such model without theselection function correction strictly describes the popu-lation of strong lenses used in our analysis, as opposed tothe general population of massive ETGs. The posteriorPDF of the model without the selection function term isconsistent with the more sophisticated model taking intoaccount selection effects. Nevertheless, the inferred prop-erties of the dark matter halos are slightly different in thetwo cases. By not accounting for the selection functionwe detect a strong dependence of the dark matter masswith redshift, as the parameter ζ DM is larger than zerowith more than 3 − σ significance for this model (emptycontours in Figure 8). A positive value of ζ DM meansthat lenses at lower redshift have preferentially smallerdark matter masses than lenses at higher redshift. At thesame time, the average dark matter mass at the referencepoint z = 0 .
3, log M ∗ = 11 .
5, Σ ∗ = Σ is smaller withrespect to the full analysis: log M = 10 . +0 . − . . Given the nature of our strong lens sample, with lenses fromthe SLACS survey dominating the low-redshift part ofthe sample and SL2S lenses populating the high-redshiftend, this result implies that SLACS lenses have on av-erage smaller dark matter masses than similar lenses athigher redshift. Since the trend in redshift of M DM5 isgreatly reduced when selection effects are taken into ac-count, this suggests that the lower dark matter massesmeasured for SLACS lenses is not necessarily related toan intrinsic difference between ETGs at low and interme-diate redshift, but might just be the result of the SLACSsurvey selecting preferentially lenses in smaller dark mat-ter halos. We further investigated this aspect by repeat-ing the analysis for SLACS and SL2S lenses separately,with and without the selection function term. We con-firmed that the SLACS sample is more sensitive to selec-tion effects. In particular it is the Einstein radius selec-tion term of Equation 20 that drives the offset betweenthe model with Pr( ω i | λ ) and the one without. Accordingto our model, the detection efficiency in the SLACS sur-vey, which we fit directly for, is a Gaussian in R Ein withmean R sel = 0 . +0 . − . and dispersion σ sel = 0 . +0 . − . .This is a peaked function in R Ein that favors the detec-ark matter content and stellar IMF of massive early-type galaxies 13
Fig. 8.—
Model hyper-parameters describing the dark matter mass within a shell of radius r eff . Empty contours: inference with noselection function term.
Filled contours: including the selection function term. The different levels represent the 68%, 95% and 99 . tion of lenses with smaller Einstein radii and thereforesmaller dark matter masses.We chose to parametrize the dark matter content withthe dark matter mass projected within 5 kpc. Manystudies, both observational and theoretical, focus insteadon the mass enclosed within the effective radius, M DM e .For a better comparison with the literature it is thenuseful to check what our results imply for this quantity.As we show in Appendix A, M DM e increases with M ∗ with a power smaller than unity and has a strong anti-correlation with stellar mass density, meaning that ETGsare not homologous systems. Evolution of individual objects
The above analysis reveals how key quantities of mas-sive ETGs scale with redshift, stellar mass and stellarmass density. In order to gain a better understanding ofthe evolution of ETGs, it is useful to consider the evolu-tion of individual objects along their evolutionary tracks.In a fluid kinematics analogy, we would like to transi-tion from an Eulerian description of the fields M DM5 and α IMF at fixed M ∗ , z and R e , which is given by the anal-ysis presented above, to a Lagrangian description of theevolution of these quantities along the history of indi- vidual galaxies. While the latter quantity is not directlyobservable, it can be inferred with the formalism intro-duced in Paper IV, which connects the observed scalingrelations with external constraints on the evolution ofsize and stellar mass. We denote with d/dz the deriva-tive with respect to redshift along the evolutionary trackof an individual galaxy. Then we can write d log M DM5 dz = ∂ log M DM5 ∂z + ∂ log M DM5 ∂ log M ∗ d log M ∗ dz + ∂ log M DM5 ∂ log Σ ∗ d log Σ ∗ dz (22)and d log α IMF dz = ∂ log M DM5 ∂z + ∂ log M DM5 ∂ log M ∗ d log M ∗ dz + ∂ log α IMF ∂ log Σ ∗ d log Σ ∗ dz . (23)This is the evolution in M DM5 and α IMF of a galaxyfor which these quantities scale with z , M ∗ and Σ ∗ inthe same way as the population averages µ DM and µ IMF .The presence of scatter will in general modify the picture,but we expect the above expressions to be correct to first4 Sonnenfeld et al.
Fig. 9.—
IMF parameters. The red dots indicate the parameter values corresponding to a universal IMF.
Empty contours: inferencewith no selection function term.
Filled contours: including the selection function term. order. Equations 22 and 23 hold as long as the popula-tion of ETGs is not significantly polluted by the forma-tion of new objects that enter the sample in the redshiftrange that we consider. Current estimates show that thenumber density of massive galaxies evolves very modestlybelow redshift 1 (Cassata et al. 2013; Ilbert et al. 2013;Muzzin et al. 2013).The partial derivatives in Equation 22 can be identifiedwith the parameters ζ DM , β DM and ξ DM measured in ouranalysis, while those in Equation 23 are matched to ζ IMF , β IMF and ξ IMF . The two total derivatives, d log M ∗ /dz and d log Σ ∗ /dz are the rate of change in stellar massand stellar mass density of an individual galaxy. Thelatter depends on the former, and on the evolution ofthe effective radius as well: d log Σ ∗ dz = d log M ∗ dz − d log R e dz . (24)As in Paper IV, we can evaluated d log R e /dz by combin-ing constraints from the redshift and mass dependence of R e , assuming again that individual galaxies evolve in thesame way as the average: d log R e dz = ∂ log R e ∂z + ∂ log R e ∂ log M ∗ d log M ∗ dz . (25) For the scaling of effective radius with mass, wetake the value measured by Newman et al. (2012): ∂ log R e /∂ log M ∗ = 0 . ± .
07. The redshift de-pendence has been measured by a number of au-thors (e.g. Damjanov et al. 2011; Newman et al. 2012;Cimatti et al. 2012; Huertas-Company et al. 2013), withsignificant scatter between the measurements. Here wetake ∂ log R e /∂z to be the mean between these measure-ments, and use the standard deviation as an estimateof its uncertainty: ∂ log R e /∂z = − . ± .
08. Withthis prescription we evaluate the derivatives Equation 22and Equation 23, which we plot in Figure 10 and Fig-ure 11 as a function of the, unknown, mass growth rate d log M ∗ /dz . The uncertainties on the derived evo-lution of enclosed dark matter and IMF normalizationare relatively large, in part due to the uncertainty onthe mass-size relation and its evolution. We expect thedark matter enclosed within 5 kpc to exhibit little changeover time, since most of the matter accreted in the laterphases of the evolution of an ETGs will likely grow theoutskirts of the galaxy. We also expect the IMF nor-malization to show little change over time, because asignificant change of the IMF would require the accre-tion or formation of stars with an extremely differentIMF from the preexisting population, a scenario at oddsark matter content and stellar IMF of massive early-type galaxies 15 Fig. 10.—
Rate of change in projected dark matter enclosedwithin a cylinder of radius 5 kpc along the evolutionary track ofan individual galaxy, calculated from Equation 22, as a functionof the growth rate in stellar mass. An NFW profile for the darkmatter halo is assumed and selection effects are taken into account.The different colors represent the 68%, 95% and 99% probabilityregions.
Fig. 11.—
Rate of change in the IMF normalization along theevolutionary track of an individual galaxy, calculated from Equa-tion 23, as a function of the growth rate in stellar mass. An NFWprofile for the dark matter halo is assumed and selection effects aretaken into account. with our current knowledge of stellar populations in theUniverse. Our measurements are consistent with theseexpectations, though with the current data we are un-able to make precise statements in this regard.In Figure 12 we plot the evolution in dark matter massinferred from the population model without the selectionfunction term – that is to say, assuming that the stronglenses from both the SL2S and SLACS survey are anunbiased sample of the general population of early-typegalaxies. Under this assumption, the data require darkmatter masses to decrease with time at a significant rate,with more than 3- σ confidence, in sharp contrast withthe result plotted in Figure 10, which does take the se-lection function into account. It is difficult to imagine aphysical scenario in which the stellar mass increases by amodest amount while at the same time a comparable, orlarger, amount of dark matter is ejected from the inner 5 Fig. 12.—
Rate of change in projected dark matter enclosedwithin a cylinder of radius 5 kpc along the evolutionary track ofan individual galaxy, calculated from Equation 22, as a function ofthe growth rate in stellar mass, inferred ignoring the selection func-tion term. An NFW profile for the dark matter halo is assumed.The different colors represent the 68%, 95% and 99% probabilityregions. kpc of a galaxy. We believe that the implausible scenarioof Figure 12 is an indication that the selection functiondoes indeed need to be included in the modeling. (How-ever, as we will show below, the lack of selection functionmodeling in our previous work does not actually affectthe conclusions of papers I-IV.) RESULTS, FREE INNER SLOPE
The results of the analysis presented in Section 6 de-pend on the assumption of a fixed NFW shape for thedark matter profile of all ETGs. Here we relax that as-sumption and consider gNFW profiles instead, with den-sity profile given by Equation 1. We impose that indi-vidual values of the inner dark matter slope lie in therange 0 . < γ DM < .
8, as we expect the dark matterdensity profile to be shallower than the total density pro-file, which is measured to be close to isothermal ( γ ′ ≈ R Ein and the value of M DM5 is obtained by extrapolating the Einstein mass to 5 kpcassuming a value of γ DM . Different lenses have differentvalues of R Ein , therefore the direction of the degeneracycontour between M DM5 and γ DM will be different for eachlens, depending on the amount of extrapolation requiredto match the mass at 5 kpc from the mass at R Ein . If thescatter in M DM5 across the population of massive ETGsis small, then it is possible to rule out extreme values ofthe dark matter slope by simply multiplying the prob-6 Sonnenfeld et al.ability distribution for individual lenses, which is whatour hierarchical Bayesian model effectively does.The posterior PDF for the parameters describing thepopulation distribution of dark matter halos and IMFnormalizations is plotted in Figure 13 and Figure 14,while the median and 68% confidence interval is listed inTable 6 for all the inferred parameters.The average inner dark matter slope inferred in ouranalysis is consistent with γ DM = 1 corresponding toan NFW profile, though with a significant uncertainty: γ = 0 . +0 . − . . The scatter in the slope is not well con-strained and can be as large as σ γ ∼ .
6. The inferenceon the parameters describing the dark matter mass andIMF normalization is very similar to the NFW case: mildanticorrelation between M DM5 and stellar mass densityand a positive correlation with redshift, no strong cor-relation of M DM5 with stellar mass, strong correlationbetween α IMF and stellar mass. The main difference is asmaller scatter in M DM5 in the gNFW case.To better illustrate the degeneracies in the model weplot in Figure 15 the projection of the posterior PDFon the parameters describing the average dark mattermass, slope and IMF normalization for galaxies at z =0 .
3, log M ∗ and R e = 5kpc. We can see a significantdegeneracy between the IMF normalization and both thedark matter mass and density slope. As discussed byAuger et al. (2010a) these degeneracies are expected ina study of this nature and illustrate how independentconstraints on the stellar IMF can help determine theproperties of the dark matter halos of ETGs.In continuity with the work of Section 6.1, we can cal-culate the rate of change of M DM5 and α IMF along theevolutionary tracks of individual galaxies in the gNFWcase. These are plotted in Figure 16 and Figure 17. Thesame operation is trivial for the dark matter slope, sincewe are assuming that the average slope is constant acrossthe whole population of massive galaxies. The mea-surements on the dark matter mass and IMF normal-ization are consistent with no evolution, similarly to thesimpler NFW case. DISCUSSION
In Paper IV we studied the evolution of the total den-sity profile of massive ETGs. We found that the popu-lation average slope of the density profile, γ ′ , increaseswith decreasing redshift, at fixed M ∗ and R e , and in-creases with Σ ∗ . We also showed how γ ′ stays more orless constant along the evolution of individual galaxiesbetween z = 1 and z = 0. The goal of the present paperis to understand what changes in the internal structureare responsible for the observed correlations of γ ′ with z , M ∗ and R e . The main steps forward in this workcompared to Paper IV are 1) the use of a more phys-ically realistic density profile, composed of a spheroidand halo instead of a single power-law component, and2) a treatment of the lensing selection function, explicitlyaccounted for when deriving our results. The latter is animportant point, as it allows us to make accurate state-ments on the general population of massive galaxies, andnot only on the population of lenses.The analysis carried out in this paper is split into twoparts: first we fix the inner slope of the dark matterhalo to γ DM = 1, then we relax this assumption. Theinference on the population distribution of dark matter masses and stellar IMF normalization is consistent in thetwo cases, as the average dark matter slope inferred inSection 7 is very close to that of an NFW profile. Wefound that the dark matter mass enclosed within 5 kpcanticorrelates with the stellar mass density and positivelycorrelates with redshift. These correlations mirror thetrends of the slope of the total density profile γ ′ with Σ ∗ and z measured in Paper IV. At fixed redshift, galaxieswith a more compact stellar distribution (larger Σ ∗ ) tendto have smaller dark matter masses. Stellar mass densityis in turn related to the formation and evolution history.We know for example that minor dry mergers tend todecrease the concentration of stars by building up anextended envelope of accreted stars (Naab et al. 2009).Galaxies with a more extended stellar component thenmight be systems that have gone through more mergerevents than the average. It would then be interesting totest whether in simulations such systems are found tohave larger central dark matter masses, at fixed radius,as suggested by our data.One important point is that the inference on the evo-lution of the dark matter mass within 5 kpc dependssignificantly on the selection function. In particular, ouranalysis reveals how SLACS lenses have preferentiallysmaller dark matter masses with respect to the popula-tion average. Our work is the first to explicitly fit forthe selection function in deriving the properties of early-type galaxies from strong lensing measurements. Theway the selection function correction is implemented is bydescribing the distribution function of lenses as a prod-uct between the general distribution of massive galax-ies and the probability of detecting them in lensing sur-veys. The latter term is in turn the product betweenthe lensing cross-section and an Einstein radius selectionterm, which describes the different probability of detect-ing strong lenses of different Einstein radii. Accordingto the works of Arneson et al. (2012) and Gavazzi et al.(2014), dedicated to the selection function of SLACS-likesurveys and SL2S respectively, the Einstein radius seemsto be the main quantity determining the detection prob-ability. Of the two terms in the selection function, theEinstein radius selection is the dominant one while thelensing cross section correction has little effect on the re-sults of our analysis. Strong lenses are drawn from thehigh mass end of the population of galaxies. At fixedstellar mass, the difference between the distribution ofstrong lenses and the general distribution of galaxies issmall compared to the scatter in the population. Eventhough lenses with radically different density profiles canhave significantly different cross-sections, as shown byMandelbaum et al. (2009), the lensing cross-section biasis in practice small because of the small intrinsic scatterin density profile across the population of ETGs (consis-tent with the small scatter of the mass plane, Auger et al.2010a; Nipoti et al. 2008).In light of this result it is important to verify the im-pact of the selection function on the measurement of theredshift evolution of the slope of the density profile car-ried out in Paper IV, which was based on the same sam-ple of lenses used here. As we show in Appendix B, theresults of Paper IV are robust to selection function ef-fects. As a further test, we checked whether the galaxiesdescribed in our population model lie on the Fundamen-tal Plane relation. As shown in Appendix C, that is theark matter content and stellar IMF of massive early-type galaxies 17 Fig. 13.—
Model hyper-parameters describing the dark matter mass within a shell of radius r eff and inner slope, for a gNFW dark matterhalo. Empty contours: inference with no selection function term.
Filled contours: including the selection function term. The differentlevels represent the 68%, 95% and 99 .
7% enclosed probability regions. case.The results presented in this work are all based onthe assumption of a fixed de Vaucouleurs profile with aspatially constant mass-to-light ratio for the stellar dis-tribution and an isotropic velocity dispersion tensor. Ifany of these assumptions break down, for example withan evolving stellar profile or orbital anisotropy, then theinference might suffer from biases. Studies of ETGswith more complex dynamical models that fit for or-bital anisotropy have found no evidence for significantanisotropies (e.g. Cappellari et al. 2013a). It seems un-likely that allowing for anisotropy would bring signifi-cant changes to our results. We tested for the effectof fixing the light profile to a de Vaucouleurs modelby repeating the analysis of SL2S lenses with both aHernquist (Hernquist 1990) and a Jaffe (Jaffe 1983) pro-file for the stars, and found no difference in the re-sults. The effect of assuming a spatially constant mass-to-light ratio can be more subtle. In particular, if thestars accreted in merger events, which are thought tobe the main drivers of the size growth of ETGs, havea lighter IMF or even a smaller mass-to-light ratio withrespect to the pre-existing stellar population, then thelight distribution of the stellar component will have a shallower profile than its mass distribution. Indeedsome observations suggest that the mass-to-light ratiodecreases with increasing radius in early-type galaxies(e.g. Szomoru et al. 2013; Mart´ın-Navarro et al. 2014).In particular, Szomoru et al. (2013) estimate the half-mass radius to be ∼
25% smaller than the half-light ra-dius. At fixed light profile, a galaxy with a negative gra-dient in the mass-to-light ratio has a steeper density pro-file than a model with constant
M/L , and thus requiresless stellar mass and more dark matter to produce theslope of the total density profile measured with lensingand dynamics. If not taken into account, such a gradientin the mass-to-light ratio would then lead to an overes-timate the IMF normalization and an underestimate ofthe dark matter mass. More detailed data is necessaryto rule out this possibility. Nevertheless, if we repeat theanalysis assuming a stellar half-mass radius 25% smallerthan the half-light radius for each lens, as suggested bythe observations of Szomoru et al. (2013), we find resultsconsistent with the original analysis.In this work we explored correlations between the darkmatter mass and stellar IMF with redshift, stellar massand size. We know a more significant correlation mustexist between dark matter mass and the environment of8 Sonnenfeld et al.
Fig. 14.—
IMF normalization hyper-parameters, for a gNFW dark matter halo. The red dot indicates the parameter values correspondingto a universal IMF.
Empty contours: inference with no selection function term.
Filled contours: including the selection function term.
Fig. 15.—
Model hyper-parameters describing the average darkmatter mass within 5 kpc, average dark matter slope and averageIMF normalization, for galaxies at z = 0 .
3, log M ∗ = 11 . R e =5kpc. Empty contours: inference with no selection function term.
Filled contours: including the selection function term. the lens, since ETGs at the center of clusters and largegroups have larger projected dark matter masses thanour lenses.
Fig. 16.—
Rate of change in projected dark matter mass withina cylinder of radius 5 kpc along the evolutionary track of an in-dividual galaxy, calculated from Equation 22, as a function of thegrowth rate in stellar mass. The dark matter halo is described witha gNFW profile. Selection effects are taken into account
We leave the exploration of correlations with the en-vironment to future work, when better data and a moreextended sample of lenses will be available, covering aark matter content and stellar IMF of massive early-type galaxies 19
TABLE 6Bayesian hierarchical inference: the hyper-parameters. gNFW model.
With S No S Parameter descriptionlog µ (SL2S) ∗ , . +0 . − . . +0 . − . Mean stellar mass at z = 0 .
5, SL2S sample ζ (SL2S) ∗ . +0 . − . . +0 . − . Linear dependence of mean stellar mass on redshift, SL2S sample σ (SL2S) ∗ . +0 . − . . +0 . − . Scatter in mean stellar mass, SL2S samplelog µ (SLACS) ∗ , . +0 . − . . +0 . − . Mean stellar mass at z = 0 .
2, SLACS sample ζ (SLACS) ∗ . +0 . − . . +0 . − . Linear dependence of mean stellar mass on redshift, SLACS sample σ (SLACS) ∗ . +0 . − . . +0 . − . Scatter in mean stellar mass, SLACS samplelog µ (SL2S) R, . +0 . − . . +0 . − . Mean effective radius at z = 0 .
5, log M ∗ = 11 .
5, SL2S sample ζ (SL2S) R . +0 . − . . +0 . − . Linear dependence of mean effective radius on redshift, SL2S sample β (SL2S) R . +0 . − . . +0 . − . Linear dependence of mean effective radius on stellar mass, SL2S sample σ (SL2S) R . +0 . − . . +0 . − . Scatter in mean effective radius, SL2S samplelog µ (SLACS) R, . +0 . − . . +0 . − . Mean effective radius at z = 0 .
2, log M ∗ = 11 .
5, SLACS sample ζ (SLACS) R . +0 . − . . +0 . − . Linear dependence of mean effective radius on redshift, SLACS sample β (SLACS) R . +0 . − . . +0 . − . Linear dependence of mean effective radius on stellar mass, SLACS sample σ (SLACS) R . +0 . − . . +0 . − . Scatter in mean effective radius, SLACS sample γ . +0 . − . . +0 . − . Mean γ DM at z = 0 .
3, log M ∗ = 11 . R eff = 5 kpc σ γ . +0 . − . . +0 . − . Scatter in the γ DM distribution ζ DM . +0 . − . . +0 . − . Linear dependence of M DM5 on redshift. β DM . +0 . − . − . +0 . − . Linear dependence of M DM5 on log M ∗ . ξ DM − . +0 . − . − . +0 . − . Linear dependence of M DM5 on log Σ ∗ log M DM , . +0 . − . . +0 . − . Mean M DM5 at z = 0 .
3, log M ∗ = 11 . R eff = 5 kpc σ DM . +0 . − . . +0 . − . Scatter in the M DM5 distribution ζ IMF − . +0 . − . − . +0 . − . Linear dependence of IMF normalization on redshift. β IMF . +0 . − . . +0 . − . Linear dependence of IMF normalization on log M ∗ . ξ IMF . +0 . − . . +0 . − . Linear dependence of IMF normalization on log Σ ∗ log α IMF , . +0 . − . . +0 . − . Mean IMF normalization at z = 0 .
3, log M ∗ = 11 . R eff = 5 kpc σ IMF . +0 . − . . +0 . − . Scatter in the IMF normalization distribution R (SL2S)sel . +0 . − . · · · Mean observable Einstein radius, SL2S sample σ (SL2S)sel . +0 . − . · · · Dispersion in observable Einstein radius, SL2S sample R (SLACS)sel . +0 . − . · · · Mean observable Einstein radius, SLACS sample σ (SLACS)sel . +0 . − . · · · Dispersion in observable Einstein radius, SLACS sample
Note . — Median, 16th and 84th percentile of the posterior probability distribution function of each model hyper-parameter, marginalized over the other parameters.
Fig. 17.—
Rate of change in the IMF normalization along theevolutionary track of an individual galaxy, calculated from Equa-tion 23, as a function of the growth rate in stellar mass. The darkmatter halo is described with a gNFW profile. Selection effects aretaken into account broader range of environments.
Comparison with previous works
The inner dark matter slope of ETGs has been mea-sured in a limited number of cases. Sonnenfeld et al.(2012) measured γ DM = 1 . ± . z = 0 .
222 ETG from theSLACS sample. This value is slightly larger than theaverage inferred here, but is not implausible given thelarge scatter in γ DM of the population allowed by ourdata.Grillo (2012) found γ DM = 1 . ± . . +0 . − . as describedby Dutton & Treu (2014). In our work we let the IMFnormalization be a free parameter and find a marginallyshallower average dark matter slope and an IMF slightlyheavier than Salpeter. Given that most mass enclosedwithin the Einstein radius is stellar, a small changein the IMF can result in a significant change in thedark matter. Indeed, if we repeat our analysis impos-0 Sonnenfeld et al.ing a Salpeter IMF, we find much steeper dark mat-ter slopes, consistent with the result of Grillo (2012).Oguri et al. (2014) fitted for an average mass profile ofETG lenses in a similar way to the analysis of Grillo(2012) but using a larger sample of lenses and includ-ing constraints from gravitational microlensing data for afew of them. They measured the dark matter slope to be γ DM = 1 . +0 . − . , the dark matter fraction to be around30% and find an IMF normalization slightly smaller thana Salpeter IMF. While dark matter fraction and IMF nor-malization are in good agreement with our findings, theslope of the dark matter halo measured by Oguri et al.(2014) is significantly larger. Even though the lensesused in the analysis of Oguri et al. (2014) are for themost part the same ones used here, there are two im-portant differences between the two works. The firstdifference is that Oguri et al. (2014) used microlensingdata for a few system and no stellar kinematics infor-mation. The second difference is that we allowed forscatter in the population of galaxies, while Oguri et al.(2014) assumed a fixed inner slope and scaling with stel-lar mass of the dark matter halo, and fixed stellar IMFfor all systems. It is possible that by allowing for scatterthe inference on the dark matter slope would be consis-tent with our results. Dutton & Treu (2014) find thatETGs of the mass range log M ∗ ∼ . γ DM = 0 . +0 . − . for SDSSJ0936+0913 and γ DM = 0 . +0 . − . for SDSSJ0912+0029. Cappellari et al.(2013b) put constraints on the dark matter fractions ofa large number of local ETGs from the ATLAS 3D sam-ple finding an average fraction of 13% within a sphere ofradius R e , corresponding to f DM e ∼
25% for an NFWprofile, consistent with our results.Concerning the IMF of ETGs and its variations withgalaxy mass, a large number of works have been pub-lished in recent years. Robust constraints on the IMFof individual systems are only available for a very lim-ited number of objects. Sonnenfeld et al. (2012) showedthat a Chabrier IMF is ruled out at 95% confidence levelin SDSSJ0946+1006, a much more massive (log M ∗ ∼ .
6) ETG. Spiniello et al. (2012) found preference fora Salpeter IMF over a Chabrier IMF for a very mas-sive lens galaxy in a group-scale halo. Barnab`e et al.(2013) find an IMF close to Salpeter for two SLACSlenses. These results are consistent with our work. Mi-crolensing provides an independent way to determinethe absolute value of the stellar mass-to-light ratio andtherefore the IMF mismatch parameter and the darkmatter fraction. Recent works by Oguri et al. (2014)and Schechter et al. (2014) find an IMF consistent withSalpeter and Jim´enez-Vicente et al. (2014) find a pro-jected dark matter fraction consistent with our results.A Salpeter IMF appears to be preferred over Chabriereven at z ∼ . M ∗ < .
8. Ifwe extrapolate our results down to the typical masses ofspiral bulges, we find IMFs consistent with their results. SUMMARY AND CONCLUSIONS
We re-examined the SL2S sample of ETG lenses, ex-tending the sample of grade A lenses and lenses usablefor a joint lensing and stellar dynamics analysis with theuse of key spectroscopic data recently acquired. We thenused SL2S and SLACS lenses to explore two componentmass models describing the stellar spheroid and darkmatter halo of massive ETGs. We fit for the distribu-tion function of dark matter masses, dark matter innerslopes and stellar IMF normalization across the popula-tion of massive ETGs with a Bayesian hierarchical infer-ence method that allows for scatter in the population andtakes into account the selection function, i.e. the map-ping between the general population of massive galaxiesand our sample of lenses. This is the most statistically ro-bust attempt at describing the population of ETGs withgravitational lensing data. We found the following. • The projected dark matter mass within 5 kpc, M DM5 , correlates with redshift and anti-correlateswith stellar mass density. The average dark mat-ter mass for galaxies at z = 0 .
3, stellar mass oflog M ∗ = 11 . R e = 5 kpc is h log M DM5 i = 10 . ± . (stat) ± . (syst) . • SLACS lenses appear to have slightly smaller darkmatter masses than the population average forgalaxies of similar mass, size and redshift. • The time evolution of the dark matter mass for in-dividual objects, inferred by tracing the dark mat-ter mass for galaxies of average mass and size ateach redshift, is consistent with a mass within theinner 5 kpc that is constant with time. Correctingark matter content and stellar IMF of massive early-type galaxies 21for the selection function is critical for recoveringthis result. • The average inner slope of the dark matter halos ofour lenses is consistent with that of an NFW pro-file. We were unable to test for correlations of theslope with redshift, stellar mass or size because theuncertainties are too large with the current data.Spatially extended stellar kinematics data wouldhelp better constrain the dark matter slope. • The IMF normalization is close to that of a SalpeterIMF and is heavier for galaxies with larger stellarmass, in agreement with previous studies.Our finding of central dark matter content anti-correlating with stellar mass density can be interpretedas the result of more compact galaxies living in darkmatter halos of smaller mass. Stellar mass density isbelieved to be closely related to the assembly historyof a galaxy: mergers that are predominantly dry con-tribute to create an extended envelope of stars, thereforegalaxies with larger size might have undergone signifi-cantly more mergers with respect to more compact ob-jects of similar mass. Our result then seems to agree withthe notion that mergers are more frequent in larger ha-los (Fakhouri & Ma 2009), as well as with recent claimsof correlation between environmental density and size ofmassive ETGs (Cooper et al. 2012; Lani et al. 2013).Current and future surveys such as the Dark En-ergy Survey, the Large Synoptic Survey Telescope, andEuclid will provide tens of thousands of new lenses(Oguri & Marshall 2010). Hierarchical Bayesian infer-ence will allow to optimally combine the informationfrom such a large number of systems and enable us toprobe further the interplay between dark matter andbaryons.We thank our friends of the SLACS and SL2S col-laborations for many useful and insightful discussionsover the course of the past years. We thank V.N. Ben-nert and A. Pancoast for their help in our observationalcampaign. AS acknowledges support by a UCSB DeanGraduate Fellowship. RG acknowledges support fromthe Centre National des Etudes Spatiales (CNES). The work of PJM was supported in part by the U.S. De-partment of Energy under contract number DE-AC02-76SF00515. TT acknowledges support from the NSFthrough CAREER award NSF-0642621, and from thePackard Foundation through a Packard Research Fel-lowship. CN acknowledges financial support from PRINMIUR 2010-2011, project “The Chemical and DynamicalEvolution of the Milky Way and Local Group Galax-ies”, prot. 2010LY5N2T. This research is based onXSHOOTER observations made with ESO Telescopesat the Paranal Observatory under program ID 092.B-0663. This research is based on observations obtainedwith MegaPrime/MegaCam, a joint project of CFHTand CEA/DAPNIA, and with WIRCam, a joint projectof CFHT, Taiwan, Korea, Canada and France, at theCanada-France-Hawaii Telescope (CFHT) which is oper-ated by the National Research Council (NRC) of Canada,the Institut National des Sciences de l’Univers of theCentre National de la Recherche Scientifique (CNRS) ofFrance, and the University of Hawaii. This work is basedin part on data products produced at TERAPIX and theCanadian Astronomy Data Centre. The authors wouldlike to thank S. Arnouts, L. Van waerbeke and G. Mor-rison for giving access to the WIRCam data collectedin W1 and W4 as part of additional CFHT programs.We are particularly thankful to Terapix for the data re-duction of this dataset. This research is supported byNASA through Hubble Space Telescope programs GO-10876, GO-11289, GO-11588 and in part by the NationalScience Foundation under Grant No. PHY99-07949, andis based on observations made with the NASA/ESA Hub-ble Space Telescope and obtained at the Space TelescopeScience Institute, which is operated by the Associationof Universities for Research in Astronomy, Inc., underNASA contract NAS 5-26555, and at the W.M. Keck Ob-servatory, which is operated as a scientific partnershipamong the California Institute of Technology, the Uni-versity of California and the National Aeronautics andSpace Administration. The Observatory was made pos-sible by the generous financial support of the W.M. KeckFoundation. The authors wish to recognize and acknowl-edge the very significant cultural role and reverence thatthe summit of Mauna Kea has always had within the in-digenous Hawaiian community. We are most fortunate tohave the opportunity to conduct observations from thismountain.
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DARK MATTER ENCLOSED WITHIN R e We want to derive what our findings on the variation of M DM5 across the population of ETGs correspond to interms of the projected dark matter enclosed within the effective radius, M DM e . Let us derive how M DM e scales withredshift, stellar mass and stellar mass density. For simplicity we restrict ourselves to the NFW case. For galaxies with R e = 5kpc, M DM5 = M DM e by definition. Therefore for these galaxies the variation with z of the dark matter massprojected within the effective radius, at fixed stellar mass and stellar mass density, is described exactly by ζ DM : ∂ log M DM e ∂z = ζ DM = 1 . +0 . − . . (A1)Let us consider the variation of M DM e with stellar mass, at fixed redshift and stellar mass density. In order for thestellar mass density to be fixed, at a variation in stellar mass δ log M ∗ must correspond a variation in effective radius δ log R e = 0 . δ log M ∗ . At fixed dark matter content, a variation in effective radius introduces a change in M DM e . Inparticular for a galaxy with R e = 5kpc and an NFW dark matter halo with r s = 10 R e , δ log M DM e ≈ . δ log R e . (A2)Then, at fixed stellar mass density and redshift, the variation in M DM e with stellar mass is given by the sum of aterm describing the increase in halo mass, captured by the hyper-parameter β DM , and a term due to the increase ineffective radius: ∂ log M DM e ∂ log M ∗ = β DM + 0 .
80 = 0 . ± . . (A3)Finally a similar argument shows that, at fixed redshift and stellar mass, a variation in stellar mass density corre-sponds to a change in M DM e given by ∂ log M DM e ∂ log M ∗ = ξ DM − .
80 = − . +0 . − . . (A4)For homologous systems, ∂ log M DM e /∂ log M ∗ = 1 and ∂ log M DM e /∂ log Σ ∗ = 0. The fact that the values we measureare inconsistent with these implies that ETGs are not homologous systems. RELATION TO POWER-LAW MODELS
In Paper IV we measured the slope of the density profile and its variation across the population of strong lenses,assuming a power-law form for the density profile. Here we are fitting a model consisting of a stellar spheroid anda dark matter halo to the same exact set of lenses. Are the results from the two analyses consistent? Additionally,in this work we take into account the lensing selection function. What would be the effect of the selection functionon the analysis of Paper IV? We can answer both these questions by generating mock samples of lenses from thepopulation distribution inferred here, and then analyzing them with the same method of assuming power-law densityprofiles that we used in Paper IV. We generated mock ensembles of 80 lenses, uniformly distributed in redshift between z = 0 . z = 0 .
8, with a Gaussian distribution in stellar mass centered at µ ∗ = 11 . σ ∗ = 0 . σ (SL2S) R , and dark matter masses were drawn from a Gaussian with mean givenby Equation 15 and dispersion σ DM . For simplicity we assumed NFW profiles for the dark matter halos, since theinference with free inner slope is consistent with that assuming NFW profiles. The values of the hyper-parametersdescribing effective radius and dark matter distributions were drawn from the posterior PDF obtained from the fit4 Sonnenfeld et al. TABLE 7Power-law model parameters.
With Pr( ω i | λ ) No Pr( ω i | λ ) Paper IV α − . +0 . − . − . +0 . − . − . +0 . − . β . +0 . − . . +0 . − . . +0 . − . ξ − . +0 . − . − . +0 . − . − . +0 . − . γ . +0 . − . . +0 . − . . +0 . − . σ γ . +0 . − . . +0 . − . . +0 . − . Note . — Fit of a Gaussian distribution in den-sity slope with mean given by Equation B1 anddispersion σ ′ γ to mock populations of lenses drawnfrom the two component model of Section 6. described in Section 6. For each ensemble we drew one set of hyper-parameters, and then drew the individual values ofeffective radii and dark matter masses. We then simulated measurements of the density slope γ ′ and added noise. Thiswas done in Paper IV by fitting a power-law density profile to the measured central velocity dispersion and Einsteinradius. In our case we can calculate the model velocity dispersion while the Einstein radius is simply set equal tothe effective radius. We have shown in Paper IV that the ratio between the Einstein radius and the effective radiushas little impact on the measurement of γ ′ . Each mock sample is then fit with the same model for the populationdistribution of γ ′ used in Paper IV, which consists of a Gaussian distribution with mean given by h γ ′ i = γ ′ + α ′ ( z − .
3) + β ′ (log M ∗ − .
5) + ξ ′ log R e / σ ′ γ . For each mock realization, we fit for the parameters of this distribution with MCMC, to givethe posterior PDF for the Paper IV model parameters given the mock data. This allows us to perform the posteriorpredictive checks we need. For our test statistic, we predict the marginalized PDFs for the Paper IV model parameters,by considering the average of these quantities over the ensemble. Results from this exercise are reported in Table 7.The parameters recovered in this way are well consistent with the values measured in Paper IV, with the exception ofthe mean density slope, γ . The slope measured for mocks generated from our two component model is systematicallyshallower than the value measured directly on the lenses of our sample. This discrepancy reflects the inability ofreproducing relatively large values of the density slope ( γ ′ > .
2) with sums of de Vaucouleurs and NFW profiles,as discussed in Section 3. However, the key trends with z , M ∗ and R e are recovered, meaning that the conclusionsof Paper IV, namely that γ ′ correlates with Σ ∗ and anticorrelates with z , are perfectly consistent with the presentwork. Furthermore, there is little difference between the values of the power-law parameters obtained by fitting mockscreated by taking the selection function into account or not. This is an important result, as it implies that the resultsof Paper IV are robust with respect to selection effects. A POSTERIOR PREDICTIVE TEST
Our hierarchical Bayesian model provides us with the the posterior probability distribution in the hyper-parametersdescribing the population of massive galaxies. One way to verify whether the inferred model is a realistic one is todraw mock observations from the posterior probability distribution and compare them with real galaxies. In particularit is interesting to check if mock galaxies drawn from our model lie on the Fundamental Plane. For simplicity, weconsider the stellar mass Fundamental Plane (Hyde & Bernardi 2009):log (cid:18) R e kpc (cid:19) = a log (cid:16) σ km s − (cid:17) − . b log (cid:18) M ∗ πR (cid:19) + c, (C1)where σ is the central velocity dispersion measured within an aperture R e /
8. Hyde & Bernardi (2009) measured a = 1 . b = 0 . c = 4 . ∼ . z = 0 . Fig. 18.—
Stellar mass Fundamental Plane from mock observations generated from the posterior probability distribution function ofsections 6 and 7. The coefficient of the Fundamental Plane relation are notnot