THE SL2S GALAXY-SCALE LENS SAMPLE. IV. THE DEPENDENCE OF THE TOTAL MASS DENSITY PROFILE OF EARLY-TYPE GALAXIES ON REDSHIFT, STELLAR MASS, AND SIZE

The formation and evolution of early-type galaxies (ETGs) is still an open question. Though frequently labeled as "red and dead" and traditionally thought to form in a "monolithic collapse" followed by "passive" pure luminosity evolution, over the past decades a far more complicated history has emerged (e.g., Renzini 2006 and references therein). ETGs are thought to harbor supermassive black holes at their centers (Ferrarese & Merritt 2000; Gebhardt et al. 2000) which regulate the conversion of gas into stars (De Lucia et al. 2006). Traces of recent star formation are ubiquitously found when sensitive diagnostics are applied (Treu et al. 2002; Kaviraj 2010). Episodes of tidal disturbances and interactions with other systems occur with remarkable frequency even at recent times (e.g., Malin & Carter 1983; van Dokkum 2005; Tal et al. 2009; Atkinson et al. 2013). Their structural properties evolve in the sense that their sizes appear to grow with time at fixed stellar mass (van Dokkum et al. 2008; Damjanov et al. 2011; Newman et al. 2012a; Huertas-Company et al. 2013; Carollo et al. 2013). The mode of star formation seems to be different from that found in spiral galaxies, resulting in a different stellar initial mass function (IMF; Treu et al. 2010; van Dokkum & Conroy 2010; Auger et al. 2010b; Brewer et al. 2012; Cappellari et al. 2013). Finally, from a demographic point of view, their number density has been found to have evolved significantly since z ∼ 2 (e.g., Ilbert et al. 2013).

Reproducing these observations is an enormous challenge for theoretical models. Major and minor mergers are thought to be the main processes driving their structural and morphological evolution, but it is not clear if they can account for the observed evolution while reproducing all of the observables (Nipoti et al. 2009a; Hopkins et al. 2010; Oser et al. 2012; Remus et al. 2013).

Gravitational lensing, by itself and in combination with other probes, can be used to great effect to measure the mass profiles of ETGs, both in the nearby universe and at cosmological distances (Treu & Koopmans 2002a, 2002b, 2004; Rusin et al. 2003; Rusin & Kochanek 2005; Koopmans et al. 2006; Jiang & Kochanek 2007; Gavazzi et al. 2007; Auger et al. 2010a; Lagattuta et al. 2010). Until recently, however, this approach was severely limited by the small size of the known samples of strong gravitational lenses. This has motivated a number of dedicated searches which have, over the past decade, increased the sample of known strong gravitational lens systems by more than an order of magnitude (e.g., Browne et al. 2003; Bolton et al. 2008; Faure et al. 2008; Treu et al. 2011).

In spite of all this progress, the number of known lenses at z ∼ 0.5 and above is still a severe limitation. Increasing this sample and using it as a tool to understand the formation and evolution of massive galaxies is the main goal of our Strong Lensing Legacy Survey (SL2S) galaxy-scale lens search (Gavazzi et al. 2012) and other independent searches based on a variety of methods (Brownstein et al. 2012; Marshall et al. 2009; Negrello et al. 2010; Pawase et al. 2012; Inada et al. 2012; González-Nuevo et al. 2012; Wardlow et al. 2013; Vieira et al. 2013).

In our pilot SL2S paper (Ruff et al. 2011), we measured the evolution of the density slope of massive ETGs by combining lensing and dynamics measurements of a sample of just 11 SL2S lenses with similar measurements taken from the literature (Treu & Koopmans 2004; Koopmans et al. 2009; Auger et al. 2010b), finding tentative evidence that the density profile of massive ETGs, on average, steepens with cosmic time. This trend was later confirmed qualitatively in an independent study by Bolton et al. (2012) and agrees with the theoretical work by Dubois et al. (2013). However, the picture is still not clear: the observed trend is tentative at best, while different theoretical studies find contrasting evolutionary trends (Johansson et al. 2012; Remus et al. 2013). More data and better models are needed to make progress.

In order to clarify the observational picture, we have collected a much larger sample of objects, more than tripling the sample of secure lenses with all of the necessary measurements, with respect to our pilot study. Photometric and strong lensing measurements for this expanded sample are presented in a companion paper (Sonnenfeld et al. 2013, hereafter Paper III). In this paper, we present spectroscopic data for the same objects. Deflector and source redshifts are used to convert the geometry of the lens system into measurements of a physical mass within a physical aperture. Stellar velocity dispersions are used as an independent constraint on the gravitational potential of the lens, allowing for more diagnostic power on the structure of our targets.

The combination of the photometric, lensing, and spectroscopic data is used in this paper to study the cosmic evolution of the slope of the average mass density profile of massive ETGs. This is achieved by fitting power-law density profiles ($\rho (r) \propto r^{-\gamma ^{\prime }}$; γ' ≈ 2 in the local universe) to the measured Einstein radii and velocity dispersions of our lenses. Such a measurement of γ' is a good proxy for the mean density slope within the effective radius. The goal of this paper is to measure trends of γ' with redshift, in continuity with our previous work (Ruff et al. 2011), as well as with other structural properties of massive ETGs, such as stellar mass and size. Such measurements will help us understand the structural evolution of ETGs from z = 0.8 to present times.

This paper is organized as follows. We briefly summarize the relevant features of the SL2S galaxy-scale lens sample in Section 2, and show in detail the spectroscopic data set and the measurements of redshifts and velocity dispersions of our lenses in Section 3. In Section 4, we discuss the properties of SL2S lenses in relation to lenses from independent surveys. In Section 5, we briefly explain how lensing and kinematics measurements are combined to infer the density slope γ' and discuss the physical meaning of such measurements. In Section 6, we combine individual γ' measurements to infer trends of this parameter across the population of ETGs. After a discussion of our results in Section 7, we conclude in Section 8.

Throughout this paper, magnitudes are given in the AB system. We assume a concordance cosmology with matter and dark energy density Ω_m = 0.3, Ω_Λ = 0.7, and Hubble constant H₀ = 70 km s⁻¹ Mpc⁻¹.

The gravitational lenses studied in this paper were discovered as part of the SL2S (Cabanac et al. 2007) with a procedure described in detail in Gavazzi et al. (2012). Lens candidates are identified in imaging data from the Canada–France–Hawaii Telescope (CFHT) Legacy Survey and then followed up with Hubble Space Telescope high-resolution imaging and/or spectroscopy. In Paper III, we ranked the candidates, assigning them a grade indicating their likelihood of being strong lenses with the following scheme: grade A for definite lenses, grade B for probable lenses, grade C for possible lenses, and grade X for non-lenses. A summary with the number of systems in each category is given in Table 1. In this paper, we analyze all lenses with spectroscopic data that have not been ruled out as grade X systems.

The SL2S spectroscopic campaign was started in 2006. The goal of our spectroscopic observations is to measure the lens and source redshifts and lens velocity dispersion for all our systems. Different telescopes (Keck, Very Large Telescope (VLT), and Gemini), instruments (LRIS, DEIMOS, X-Shooter, and GNIRS), and setups have been used to achieve this goal, reflecting technical advances throughout the years and the optimization of our strategy. In what follows, we describe the procedure used to measure the three key spectroscopic observables. A summary of all measurements is given in Table 2.

3.1. Deflector Redshifts and Velocity Dispersions

The typical brightness of our lenses is around i ∼ 20. With an 8 m class telescope, their redshift can be measured from their optical absorption lines with ∼10 minutes of exposure time, while a measurement of their velocity dispersion typically takes from 30 to 120 minutes. Optical spectroscopy data come from three different instruments.

For most of the systems, we have data obtained with the LRIS spectrograph at Keck (Oke et al. 1995). The wavelength coverage of LRIS is typically in the range of 3500–8000 Å for data taken before 2009 and extends up to 10, 000 Å for later data, after the installation of the new detector with much reduced fringing patterns (Rockosi et al. 2010). The spectral resolution is about 140 km s⁻¹ FWHM on the red side of the spectrograph. Data reduction for LRIS spectra was performed with a pipeline written by M. W. Auger.

For a set of 13 systems, we have VLT observations with the instrument X-Shooter.⁹ X-Shooter has both a higher resolution (∼50 km s⁻¹) and a longer wavelength coverage (from 3500 Å up to 25,000 Å) than LRIS. X-Shooter spectra were reduced with the default ESO pipeline.¹⁰ The observations were done by nodding along a long slit of width 09 for the UVB and VIS arms and 10 for the near-infrared arm.

Finally, six systems presented here have data obtained with DEIMOS at Keck (Faber et al. 2003). The grating used in all DEIMOS observations is the 600ZD, with a wavelength range between 4500 Å and 9500 Å and a spectral resolution of about 160 km s⁻¹. DEIMOS data were reduced with the DEEP2 pipeline (Cooper et al. 2012; Newman et al. 2012b).

Both redshifts and velocity dispersions are measured by fitting stellar templates, broadened with a velocity kernel, to the observed spectra. This is done in practice with a Monte Carlo Markov chain adaptation of the velocity dispersion fitting code by van der Marel (1994), written by M. W. Auger, and described by Suyu et al. (2010). We used seven different templates of G and F stars, which should provide an adequate description of the stars in red passive galaxies, taken from the Indo US stellar library. The code also fits for an additive polynomial continuum, to accommodate for template mismatch effects or imperfections in the instrumental response correction. In most cases, a polynomial of order five is used.

The rest-frame wavelength range typically used in our fits is 3850–5250 Å, which brackets important absorption lines such as Ca K,H at 3934, 3967 Å, the G-band absorption complex around 4300 Å, and Mgb at 5175 Å. Depending on the redshift of the target and the instrument used, this is not always allowed because part of the wavelength region can fall outside the spectral coverage allowed by the detector, or because of Telluric absorption. In those cases, the fitted rest-frame region is extended.

Systematic uncertainties in the velocity dispersion measurements are estimated by varying the fitted wavelength region and order of the polynomial continuum. These are typically on the order of 20 km s⁻¹ and are then summed in quadrature to the statistical uncertainty.

All of the optical spectra of our systems are shown in Figure 1.

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3.2. Source Spectroscopy

Measuring the redshift of a lensed background source is important not only for determining the geometry of the gravitational lens system, but also to confirm that the arc is actually in the background relative to the lens. The arcs of the lensed sources are relatively faint in broadband photometry (g ∼ 24), implying that their continuum radiation cannot be detected in most cases. However, the sources are selected to be blue (Gavazzi et al. 2012) and are often associated with emission lines from star formation and/or nuclear activity. The typical redshifts of our arcs are in the range of 1 < z < 3. This means that optical spectroscopy can effectively detect emission from the [O ii] doublet at 3727–3729 Å for the lowest redshift sources, or Lyα for objects at z > 2.5 or so. This is the case for roughly half of the systems observed. The remaining half does not show detectable emission lines in the observed optical part of the spectrum, either because the most important lines fall in the near-infrared or because emission is too weak. Emission lines from the arcs can be easily distinguished from features in the lens because they are spatially offset from the lens light.

X-Shooter observations proved to be remarkably efficient in measuring source redshifts. This is in virtue of its wavelength range that extends through the near-infrared up to 25, 000 Å and its medium resolution that limits the degrading effect of emission lines from the atmosphere. Of 13 systems observed with X-Shooter, 12 of them yielded a redshift of the background source, all of which show at least two identified lines.

In addition, for four systems we have near-infrared spectroscopic observations with the instrument GNIRS on Gemini North (PI: Marshall, GN-2012B-Q-78, PI: Sonnenfeld, GN-2013A-Q-91), used in cross-dispersed mode, covering the wavelength range 10, 000–25, 000 Å at once. Of the four systems observed, two of them show two emission lines from the background source.

In most cases, when only optical spectroscopy is available, only one emission line is detected over the whole spectrum. The [O ii] doublet can be easily identified even with relatively low-resolution spectrographs. The identification of the Lyα line is less trivial. Lyα is typically the brightest emission line in the rest-frame wavelength range 1000–3000 Å when present, but other emission lines like C iv λ1546, O iii] λ1666 or C iii] λ1908 can sometimes be seen. When we detect an emission line close to the blue end of the spectrum, it could, in principle, be any of those lines. However, a detection of one of the above lines and a non-detection of the other ones is quite unlikely, unless C iii] λ1908 falls right at the blue edge of the observed spectrum. In that case though, we should expect to observe the [O ii] doublet at redder wavelengths. This case is never encountered; therefore, in all cases when we detect an unresolved emission line bluer than 6000 Å, and no other lines, we can safely assume it is Lyα. The system SL2SJ022357-065142 is a particular case; we detected an emission line spatially associated with the background source at 9065 Å, with a 5σ significance. Given the low signal-to-noise ratio (S/N), the line is both compatible with being the [O ii] doublet or an individual line. Possible other lines are [O iii] λ5007 and H-β, which cannot be ruled out. Therefore, we do not claim redshift measurements for that source; deeper data are needed to establish whether the line is the [O ii] doublet or not.

The two-dimensional (2D) spectra around all of the detected emission lines for all of the systems are shown in Figure 2. Note that for some systems, the line emission is multiply imaged on both sides of the foreground object. This provides a decisive clue on the lens nature of those systems, which is important when ranking our targets by their likelihood of being lenses (Paper III).

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Finally, six background sources are bright enough to be visible with continuum radiation and several absorption/emission features can be identified in their spectra. The absorption line spectra of these sources are plotted in Figure 3.

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Despite our efforts in acquiring spectroscopic data for our lenses, 7 of the 36 grade A lenses with spectroscopic follow-up have no measured source redshifts. In Paper II, Ruff et al. (2011) made use of photometric data together with lensing cross-section arguments to estimate source redshifts with a technique called photogeometric redshift. Here, the fraction of lenses with no source redshift is small compared to the sample size; therefore, it is not essential to include them in the analysis through the use of this method.

In Paper III, we presented effective radii, magnitudes, stellar masses, and Einstein radii of our lenses. Here, we complement this information with lens and source redshifts, and lens velocity dispersions. It is possible at this point to look at the distribution of our lenses in the parameter space defined by these quantities. Since our scientific goal is to measure the evolution in the mean density slope with time, it is very important to assess whether other observables appear to evolve in our sample. In Figure 4, we plot the effective radii, stellar masses, and velocity dispersions as a function of redshift for all of our objects, and also for lenses from other surveys. Throughout this paper, when dealing with stellar masses, we refer to values measured from stellar population synthesis fitting based on a Salpeter IMF. For a fair comparison, all velocity dispersions, which are measured within rectangular apertures of arbitrary sizes, are transformed into velocity dispersions within a circular aperture, σ_e2, with radius R_eff/2 following the prescription of Jørgensen et al. (1995). The values of σ_e2 for individual SL2S lenses are reported in Table 3.

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Table 3. Lensing and Dynamics

Name	zd	Reff	REin	σe2	$\log {M_*^{\mathrm{Salp}}/M_\odot }$	γ'	Notes
		(kpc)	(kpc)	(km s−1)
SL2SJ021247−055552	0.750	8.92	9.33	267 ± 17	11.45 ± 0.17	2.05 ± 0.09
SL2SJ021325−074355	0.717	17.67	17.22	287 ± 33	11.97 ± 0.19	1.79 ± 0.12
SL2SJ021411−040502	0.609	6.29	9.48	238 ± 15	11.60 ± 0.14	1.85 ± 0.07
SL2SJ021737−051329	0.646	4.27	8.80	270 ± 21	11.53 ± 0.16	2.02 ± 0.09
SL2SJ021902−082934	0.389	3.01	6.88	300 ± 23	11.50 ± 0.10	2.26 ± 0.08
SL2SJ022511−045433	0.238	8.59	6.65	226 ± 20	11.81 ± 0.09	1.78 ± 0.10
SL2SJ022610−042011	0.494	6.44	7.23	266 ± 24	11.73 ± 0.11	2.01 ± 0.12
SL2SJ023251−040823	0.352	4.78	5.15	271 ± 20	11.36 ± 0.09	2.39 ± 0.10
SL2SJ084847−035103	0.682	3.21	6.02	205 ± 21	11.24 ± 0.16	1.85 ± 0.14
SL2SJ084909−041226	0.722	3.55	7.94	312 ± 18	11.63 ± 0.13	2.14 ± 0.06
SL2SJ084959−025142	0.274	6.11	4.84	275 ± 34	11.52 ± 0.09	2.32 ± 0.17
SL2SJ085019−034710	0.337	1.35	4.48	307 ± 25	11.14 ± 0.09	2.45 ± 0.07	Disky
SL2SJ085540−014730	0.365	3.48	5.21	222 ± 19	11.11 ± 0.10	2.15 ± 0.11
SL2SJ090407−005952	0.611	16.81	9.47	178 ± 20	11.55 ± 0.12	1.48 ± 0.11
SL2SJ095921+020638	0.552	3.47	4.73	195 ± 22	11.28 ± 0.11	2.11 ± 0.16
SL2SJ135949+553550	0.783	13.08	8.52	229 ± 19	11.41 ± 0.15	1.86 ± 0.14
SL2SJ140454+520024	0.456	11.78	14.80	337 ± 19	12.10 ± 0.10	1.95 ± 0.06
SL2SJ140546+524311	0.526	4.58	9.48	291 ± 21	11.67 ± 0.11	2.14 ± 0.08
SL2SJ140650+522619	0.716	4.35	6.79	258 ± 14	11.60 ± 0.15	2.00 ± 0.07
SL2SJ141137+565119	0.322	3.04	4.34	220 ± 23	11.28 ± 0.09	2.15 ± 0.15
SL2SJ142059+563007	0.483	7.86	8.39	228 ± 19	11.76 ± 0.10	1.93 ± 0.11
SL2SJ220329+020518	0.400	3.86	10.49	218 ± 21	11.26 ± 0.10	1.77 ± 0.09
SL2SJ220506+014703	0.476	3.93	9.87	326 ± 30	11.51 ± 0.10	2.19 ± 0.09
SL2SJ221326−000946	0.338	2.41	5.17	177 ± 15	10.99 ± 0.10	1.89 ± 0.09	Disky
SL2SJ222148+011542	0.325	5.27	6.59	224 ± 23	11.55 ± 0.09	1.96 ± 0.13

Note. Summary of lensing and dynamics measurements.

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SL2S lenses do not appear to differ from objects from independent lensing surveys in the average values of R_eff, M_*, and σ_e2. As far as trends with redshift within the SL2S sample are concerned, there is a mild increase of the stellar mass with z that will need to be taken into account when discussing the evolution of the mass profile of these objects.

As an additional test, we examine the correlation between mass and effective radius for SL2S, Sloan Lens Advanced Camera for Surveys (SLACS), and Lenses Structure and Dynamics (LSD) lenses and check it against non-lens galaxies. The goal is to make sure that these surveys do not preferentially select lenses with a larger or smaller size than typical ETGs of their mass. The mass–radius relation is seen to evolve with time (e.g., Damjanov et al. 2011; Newman et al. 2012a; Cimatti et al. 2012). We correct for this evolution by considering effective radii evolved to z = 0 assuming the trend measured by Newman et al. (2012a): log R_eff(z = 0) = log R_eff + 0.26z. Effective radii defined in this way are plotted against measured stellar masses in Figure 5, together with the mass–radius relation measured by Newman et al. (2012a) for low-redshift Sloan Digital Sky Survey (SDSS) galaxies. Points in the plot of Figure 5 should not be considered as evolutionary tracks of individual objects, as galaxies grow in mass as well as in size. For a given object, its redshift-evolved size, R_eff(z = 0), is equivalent to its measured effective radius rescaled by the average size of galaxies at its redshift and at a reference mass. This allows us to promptly display in a single plot how our lenses compare, in terms of size, to other galaxies of the same mass, regardless of redshift. We see from Figure 5 that lenses from all surveys lie nicely around the relation found for non-lenses, indicating that our sample of lenses does not appear special when compared to the more general population of galaxies of their redshift.

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We now proceed to combine lensing measurements with stellar kinematics information to infer the total mass density profile of each lens galaxy. We follow the now standard procedure in lensing and dynamics studies (Treu & Koopmans 2002a), as used by Ruff et al. (2011). We model the total (dark matter + stars) mass profile as a spherical power law $\rho (r) \propto r^{-\gamma ^{\prime }}$ in the kinematic analysis. The free parameters of the model are the slope γ' and the mass normalization. For a given model, we calculate the line of sight velocity dispersion within the rectangular aperture of our observation, broadened by the seeing, through the spherical Jeans equation. We assume isotropic orbits and a de Vaucouleurs profile for the distribution of tracers (de Vaucouleurs 1948), with effective radius fixed to the observed one. We then compare the model to the observed velocity dispersion and Einstein radius to derive posterior probability densities for the free parameters. In spite of the clear approximations, the method has been shown to be very robust when compared to results of more sophisticated models (e.g., Barnabè et al. 2011).

The data required for this inference are the Einstein radius of the lens, the redshift of both the deflector galaxy and the lensed source, and the velocity dispersion of the lens. Of the 39 grade A lenses of the SL2S sample, 25 have all of the required data. For the few systems with two or more independent measurements of the velocity dispersion, we use the weighted average. The inferred values of γ' are reported in Table 3.

5.1. The Meaning of γ'

Before analyzing the measurements in a statistical sense, we need to understand what physical properties the quantity γ' is most sensitive to. Observations (Sonnenfeld et al. 2012) and simple arguments (galaxies have a finite mass) suggest that the true density profile deviates from a pure power law, particularly at large radii. Thus, our power-law fits to the lensing and kinematics data must be interpreted as an approximation of the average density slope over a radial range explored by our data. For a typical lens, since both the Einstein radius and the velocity dispersion probe the region within the effective radius, we expect that the inferred γ' will be close to the mean density slope within R_eff, as suggested by Dutton & Treu (2013).

However, we would like to be more quantitative and explore the two following questions: what kind of average over the true density profile ρ(r) best reproduces the lensing+dynamics γ'? How sensitive to the ratio R_Ein/R_eff is the measured γ' for a fixed galaxy mass profile? The former issue is relevant when comparing theoretical models to lensing and dynamics measurements. The latter is important when trying to measure trends of γ' with redshift; the ratio R_Ein/R_eff typically increases for purely geometrical reasons, and a dependence of γ' on R_Ein/R_eff could in principle bias the inference on the evolution of the slope. In order to answer these questions, we simulate γ' measurements on a broad range of model mass profiles and compare these with the true density slopes. We consider a pure de Vaucouleurs profile, a sum of a de Vaucouleurs profile with a Navarro, Frenk, and White (Navarro et al. 1997) profile with two values of the dark matter mass fraction f_DM within the three-dimensional (3D) effective radius, and the most probable total density profile from the bulge + halo decomposition of the gravitational lens SDSSJ0946+1006 by Sonnenfeld et al. (2012). None of these model profiles is a pure power law. We emphasize that the range of models is chosen to be broader than what is likely to be found in real galaxies based on the detailed analysis of SLACS systems by Barnabè et al. (2011).

We again use the spherical Jeans equation to calculate the central velocity dispersion for each of these model galaxies and then fit power-law density profiles with fixed total projected mass within different Einstein radii. These simulated measurements of γ' are plotted in Figure 6 as a function of R_Ein/R_eff for each model profile. In the same plot, we show the local logarithmic density slope −dlog ρ/dlog r as a function of r, and also the mass-weighted density slope within radius r

$$\begin{equation} \langle \gamma ^{\prime }(r)\rangle_M = \frac{1}{M(r)}\int _0^{r} \gamma ^{\prime }(r^{\prime })4\pi r^2 \rho (r^{\prime })dr^{\prime }, \end{equation} \tag{ 1 }$$

which has been suggested by Dutton & Treu (2013) to be a good proxy for the lensing + dynamics γ'.

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Figure 6 shows that measurements of γ' (triangles) are remarkably independent of the ratio of the Einstein radius to the effective radius, for all models. This is an important result; it means that the physical interpretation of γ' measurements will be stable against different lenses having different values of R_Ein/R_eff. Excluding the pure de Vaucouleurs model, which is ruled out on many grounds (mass-follows light models fail to reproduce lensing and dynamical data, for example Koopmans & Treu 2003), the difference between the mass-weighted slope and the lensing and dynamics slope is generally smaller than the typical measurement errors on γ' of ∼0.1, particularly in the region 0.5R_eff < r < R_eff. However, the radius at which γ' and the mass-weighted slope are closest is slightly different for different mass profiles, and so it is difficult to interpret γ' precisely in terms of a mass-weighted slope within a fixed radius. For very accurate comparisons with lensing and dynamical data, we recommend simulating a lensing and dynamics measurement of the models.

The main goal of this work is to establish whether, and to what extent, γ' varies with redshift across the population of ETGs. It is useful to first study the trends of γ' on basic parameters (Section 6.1) in order to gain insights about the ingredients that will have to be considered in Section 6.2 to carry out a rigorous statistical analysis.

6.1. Qualitative Exploration of the Dependency of γ' on Other Parameters

Figure 7 shows the individual lens γ' values as a function of z for SL2S galaxies, as well as lenses from the SLACS (Auger et al. 2010a) and LSD (Treu & Koopmans 2004) surveys. A trend of γ' with z is clearly visible, with lower redshift objects having a systematically steeper slope than higher redshift ones, as previously found by Ruff et al. (2011) and Bolton et al. (2012). Before making more quantitative statements on the time evolution of γ', we would like to check whether the density slope correlates with quantities other than redshift. Galaxies grow in mass and size during their evolution, and a variation of γ' with time might be the result of a more fundamental dependence of the slope on structural properties of ETGs. Dependences of γ' on the effective radius and the stellar velocity dispersion were explored by Auger et al. (2010a), finding an anticorrelation with the former and no significant correlation with the latter. Here we consider the stellar mass, plotted against γ' in Figure 8. A weak trend is visible, with more massive galaxies having a shallower slope. However, the stellar mass is a rather steep function of redshift in our sample (see Figure 4) and the trend seen in Figure 8 might just be the result of this selection function. In fact, if we fit for a linear dependence of γ' on both z and M_*, we find that our data are consistent with γ' being independent of M_* at fixed z.

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A quantity that is expected to correlate with γ' is the stellar mass density, $\Sigma _* = M_*/(2\pi R_{\mathrm{eff}}^2)$; galaxies with a more concentrated stellar distribution should have a steeper overall density profile. This was pointed out by Auger et al. (2010a) and Dutton & Treu (2013) and is seen in our data, as shown in Figure 9. Therefore, it is important to account for a dependence of γ' on Σ_*, or on the two independent variables on which this quantity depends, R_eff and M_*, when fitting for the time dependence of the density slope. This is done in the next section.

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6.2. Quantitative Inference

In this section, we aim to quantify how the mean density slope 〈γ'〉 depends on galaxy properties and on lookback time. The population of ETGs is known to be well-described by two parameters, as revealed by the existence of the fundamental plane relation (Djorgovski & Davis 1987; Dressler et al. 1987). Two parameters are then probably sufficient to capture the variation of γ' across the population of ETGs. For our analysis, we focus on stellar mass and effective radius (this also includes dependencies on stellar mass density, which is believed to be an important parameter driving γ', as discussed above). Our objective is then to measure the trends in γ' across the 3D space defined by (z, M_*, R_eff). This is done with a simple but rigorous Bayesian inference method. We assume that the values of the slope γ' of our lenses are drawn from a Gaussian distribution with mean given by

$$\begin{equation} \langle \gamma ^{\prime } \rangle = \gamma ^{\prime }_0 + \alpha (z-0.3) + \beta (\log {M_*} - 11.5) + \xi \log {(R_{\mathrm{eff}}/5)}\ \ \ \end{equation} \tag{ 2 }$$

and dispersion $\sigma _{\gamma ^{\prime }}$. The stellar mass is in solar units and the effective radius in kiloparsecs. We also assume that individual stellar masses, M_{*, i}, are drawn from a parent distribution that we approximate as a Gaussian:

$$\begin{equation} {\rm Pr}(M_{*,i}) = \frac{1}{\sigma _{M_{*}}\sqrt{2\pi }} \exp {\left[- \frac{\left(\log {M_{*,\mathit {i}}} - \mu _{M_*}^{\mathrm{(Samp)}}(z_{\mathit {i}})\right)}{2\sigma _{M_{*}}^{2\mathrm{(Samp)}}} \right]}.\ \ \ \ \ \end{equation} \tag{ 3 }$$

To account for selection effects, we allow for a different mean stellar mass and dispersion for lenses of different surveys. We also let the mean stellar mass be a function of redshift. This choice reflects the clear trend of stellar mass with redshift seen in Figure 4 for both the SLACS and the SL2S samples, which in turn is determined by SLACS and SL2S both being magnitude-limited samples. The parameter describing the mean stellar mass is then

$$\begin{equation} \mu _{M_*}^{\mathrm{(SLACS)}} = \zeta ^{\mathrm{(SLACS)}} (z_i - 0.2) + \log {M_{*,0}}^{\mathrm{(SLACS)}} \end{equation} \tag{ 4 }$$

for SLACS galaxies and

$$\begin{equation} \mu _{M_*}^{\mathrm{(SL2S)}} = \zeta ^{\mathrm{(SL2S)}}(z_i - 0.5) + \log {M_{*,0}}^{\mathrm{(SL2S)}} \end{equation} \tag{ 5 }$$

for SL2S and LSD galaxies. We assume flat priors on all the model parameters and fit for them with a Markov chain Monte Carlo following Kelly (2007). The stellar masses considered in this model are those measured in Paper III assuming a Salpeter IMF. The full posterior probability distribution function is shown in Figure 10 and the median, 16th, and 84th percentile of the probability distribution for the individual parameters, obtained by marginalizing over the remaining parameters, is given in Table 4. The fit is done first with SL2S galaxies only and then repeated by adding SLACS and LSD lenses. For six lenses of the SLACS sample, Auger et al. (2010a) warn that their velocity dispersions might be significantly incorrect, and we conservatively exclude them from our fit. These are SDSSJ0029−0055, SDSSJ0737+3216, SDSSJ0819+4534, SDSSJ0935−0003, SDSSJ1213+6708, and SDSSJ1614+4522.

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Table 4. Linear Model with Scatter

Parameter	SL2S	SL2S +	Notes
	Only	SLACS + LSD
log M*, 0(SL2S)	$11.50_{-0.05}^{+0.05}$	$11.49_{-0.05}^{+0.05}$	Mean stellar mass at z = 0.5, SL2S sample
ζ(SL2S)	$0.35_{-0.33}^{+0.34}$	$0.38_{-0.26}^{+0.26}$	Linear dependence of mean stellar mass on redshift, SL2S sample
$\sigma _{M_*}^{\mathrm{(SL2S)}}$	$0.25_{-0.04}^{+0.05}$	$0.23_{-0.04}^{+0.04}$	Scatter in mean stellar mass, SL2S sample
log M*, 0(SLACS)	...	$11.59_{-0.03}^{+0.03}$	Mean stellar mass at z = 0.2, SLACS sample
ζ(SLACS)	...	$2.35_{-0.39}^{+0.39}$	Linear dependence of mean stellar mass on redshift, SLACS sample
$\sigma _{M_*}^{\mathrm{(SLACS)}}$	...	$0.17_{-0.02}^{+0.02}$	Scatter in mean stellar mass, SLACS sample
α	$-0.13_{-0.24}^{+0.24}$	$-0.31_{-0.10}^{+0.09}$	Linear dependence of γ' on redshift
β	$0.31_{-0.23}^{+0.23}$	$0.40_{-0.15}^{+0.16}$	Linear dependence of γ' on log M*
ξ	$-0.67_{-0.20}^{+0.20}$	$-0.76_{-0.15}^{+0.15}$	Linear dependence of γ' on log Reff
γ0	$2.05_{-0.06}^{+0.06}$	$2.08_{-0.02}^{+0.02}$	Mean slope at z = 0.3, log M* = 11.5, Reff = 5 kpc
$\sigma _{\gamma ^{\prime }}$	$0.14_{-0.03}^{+0.04}$	$0.12_{-0.02}^{+0.02}$	Scatter in the γ' distribution

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By using only the 25 SL2S lenses for which γ' measurements are possible, we are able to detect a trend of 〈γ'〉 with R_eff at the 3σ level and a dependence on M_* at the 1σ level; at fixed z and M_*, galaxies with a smaller effective radius have a steeper density profile. Similarly, at fixed R_eff, galaxies with a larger stellar mass have a marginally larger γ'. If we add 53 lenses from SLACS and 4 lenses from the LSD survey, the trends with M_* and R_eff are confirmed at a higher significance, and we detect a dependence of 〈γ'〉 on redshift at the 3σ level. Lower redshift objects appear to have a steeper slope than higher redshift counterparts at fixed mass and size. Incidentally, the median value of ξ, the parameter describing the linear dependence of 〈γ'〉 on log R_eff, is nearly −2 times β, the parameter describing the dependence on log M_*. This suggests that 〈γ'〉 grows roughly as $\beta \log {(M_*/R_{\mathrm{eff}}^2)}$, which is equivalent to the stellar mass density. It appears then that the dependence of γ' on the structure of ETGs can be well summarized with a dependence on stellar mass density, leaving little dependence on M_* or R_eff individually. This confirms and extends the trend with surface mass density seen by Auger et al. (2010a) and Dutton & Treu (2013).

We then repeated the fit allowing only for a dependence of 〈γ'〉 on redshift and stellar mass density:

$$\begin{equation} \langle \gamma ^{\prime } \rangle = \gamma _0 + \alpha (z - 0.3) + \eta (\log {\Sigma _*} - 9.0). \end{equation} \tag{ 6 }$$

This model has one less free parameter with respect to Equation (2). Our inference on the parameter describing the dependence on Σ_* is η = 0.38 ± 0.07, and the scatter in γ' is $\sigma _{\gamma ^{\prime }} = 0.12\pm 0.02$, the same value measured for the more general model of Equation (2). This is again suggesting that the dependence of γ' on the stellar mass density might be of a more fundamental nature than dependences on mass and size separately.

The main result of the previous section is that the ensemble average total mass density slope of galaxies of a fixed stellar mass increases with cosmic time (i.e., decreases with redshift). This trend with redshift is detected at the 3σ confidence level and is in good agreement with previous results from Ruff et al. (2011) and Bolton et al. (2012).

Before discussing the physical interpretation of this result, however, it is important to emphasize that what we are measuring is how the population mean density slope changes in the (z, M_*, R_eff) space within the population of ETGs, and not how γ' changes along the lifetime of an individual galaxy, dγ'/dz. In order to infer the latter quantity, we need to evaluate the variation of γ' along the evolutionary track of the galaxy as this moves in the (z, M_*, R_eff) space. This requires knowing how both mass and size of the galaxy change with time, since the slope depends on these parameters. More formally,

$$\begin{eqnarray} \frac{{d}\gamma ^{\prime }(z,\log {M_*},\log {R_{\mathrm{eff}}})}{{d} z} &=& \frac{\partial \gamma ^{\prime }}{\partial z} + \frac{\partial \gamma ^{\prime }}{\partial \log M_*}\frac{{d}\log M_*}{{d}z} \nonumber\\ &&+ \frac{\partial \gamma ^{\prime }}{\partial \log {R_{\mathrm{eff}}}}\frac{{d}\log {R_{\mathrm{eff}}}}{{d}z}. \end{eqnarray} \tag{ 7 }$$

In a parallel with fluid mechanics, our description of the population of galaxies of Section 5 is Eulerian, while Equation (7) is a Lagrangian specification of the change in time of the mean slope of an individual galaxy, providing a more straightforward way to physically understand the evolution of ETGs.

With all of these terms entering Equation (7), it is no longer clear if the density slope is indeed getting steeper with time for individual objects. In particular, we have observed that γ' depends significantly on stellar mass density (and thus effective radius). It is then crucial to consider all the terms of the equation before reaching a conclusion. Fortunately, this can be done by combining our measurements with results from the literature.

In the context of our model specified in Equation (2), the partial derivatives introduced above can be identified and evaluated as follows:

$$\begin{equation} \frac{\partial \gamma ^{\prime }}{\partial z} = \alpha =-0.31\pm 0.10, \end{equation} \tag{ 8 }$$

$$\begin{equation} \frac{\partial \gamma ^{\prime }}{\partial \log {M_*}} = \beta = 0.40\pm 0.16, \end{equation} \tag{ 9 }$$

$$\begin{equation} \frac{\partial \gamma ^{\prime }}{\partial \log {R_{\mathrm{eff}}}} = \xi = -0.76\pm 0.15. \end{equation} \tag{ 10 }$$

Note that we are not considering the effects of scatter; we are assuming that the change in γ' is the same as that of a galaxy that evolves while staying at the mean γ' as it moves through the (z, M_*, R_eff) space. By doing so, the evolution in the slope that we derive from Equation (7) will be representative of the mean change in γ' over the population, while individual objects can have different evolutionary tracks, within the limits allowed by our constraints on $\sigma _{\gamma ^{\prime }}$.

The remaining quantities to be estimated are the rate of mass and size growth. In the hierarchical merging picture, ETGs are expected to grow in stellar mass with time; therefore, dM_*/dz < 0. Observationally, we know massive ETGs grow at most by a factor of two in stellar mass since z = 1 (see, e.g., Lin et al. 2013 and references therein). Thus, we can conservatively take the mean between zero and two, even though we will show below that our conclusions are virtually insensitive to this choice:

$$\begin{equation} \frac{{d}\log {M_*}}{{d}z} =-0.15\pm 0.15. \end{equation} \tag{ 11 }$$

The effective radius grows as a result of the growth in mass, but is itself an evolving quantity at fixed M_* (Damjanov et al. 2011; Newman et al. 2012a; Cimatti et al. 2012; Poggianti et al. 2013): R_eff = R_eff(z, M_*(z)). We assume that ETGs grow while staying on the observed M_* − R_eff relation at all times. Then we can write

$$\begin{equation} \frac{{d}\log {R_{\mathrm{eff}}}}{{d}z} = \frac{\partial \log {R_{\mathrm{eff}}}}{\partial z} + \frac{\partial \log {R_{\mathrm{eff}}}}{\partial \log {M_*}}\frac{{d}\log {M_*}}{{d}z} \end{equation} \tag{ 12 }$$

and use the values measured by Newman et al. (2012a), ∂log R_eff/∂z = −0.26 ± 0.02 and ∂log R_eff/∂log M_* = 0.59 ± 0.07.

Plugging these values into Equation (7), we find that

$$\begin{eqnarray} \frac{{d}\gamma ^{\prime }}{{d}z} &=& (-0.31\pm 0.10) + (0.40\pm 0.15)(-0.15\pm 0.15) \nonumber\\ &&+ (-0.76\pm 0.15)[(-0.26\pm 0.02) \\ &&+ (-0.15\pm 0.15)(0.59\pm 0.07)] =-0.10\pm 0.12.\nonumber \end{eqnarray} \tag{ 13 }$$

Note that dγ'/dz has little dependence on the mass growth rate dlog M_*/dz, which is the most poorly known quantity in this model. To be more quantitative, we plot in Figure 11 the total derivative dγ'/dz as a function of dlog M_*/dz and show that for any plausible value spanning over an order of magnitude, the answer is unchanged. Different assumptions on the evolution of the size–mass relation do not significantly change our result. For instance, Damjanov et al. (2011) find a more rapid evolution of R_eff than Newman et al. (2012a), leading to dγ'/dz = 0.06 ± 0.15, consistent with no change of the total mass density profile with time.

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Thus, the key result is that when considering all of the terms of Equation (7), we find that, on average, individual ETGs grow at approximately constant density slope. The observed redshift dependence of γ' at fixed mass and size can then be understood as the result of the evolution of the size–mass relation and by the dependency of γ' on the stellar mass density. Qualitatively, in this picture, an individual galaxy grows in stellar mass and size so as to decrease its central stellar mass density. During this process, the slope of its total mass density profile does not vary significantly. However, the other galaxies that now find themselves to have the original stellar mass and effective radius of this galaxy originally had a steeper mass density profile, thus giving rise to the observed trend in ∂γ'/∂z.

This is illustrated in Figure 12 where we show a possible scenario consistent with the observations. The evolutionary tracks of two representative galaxies between z = 1 and z = 0 are shown as solid black arrows in the multi-dimensional parameter space of stellar mass, effective radius, effective density, and slope of the mass density profile γ'. The two galaxies are chosen so that at z = 1, one has the same mass and effective radius that the other has at z = 0. Mass and size are evolved following Equations (11) and (12). We then assign γ' at z = 0 based on the observed correlation with size and stellar mass (effectively with effective stellar mass density, since β ≈ −2ξ) and assume it does not evolve for an individual galaxy. The apparent evolution of γ' at fixed M_* and R_eff is consistent with the measured value ∂γ'/∂z = −0.31 ± 0.10, and is dictated by a difference in the initial stellar density of their progenitors being larger for the more massive object.

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In the context of simple one-parameter stellar profiles (e.g., de Vaucouleurs), this difference in γ' at fixed mass and size for galaxies at different redshift must be ascribed to corresponding differences in the underlying dark matter distribution. The implications of our results for the dark matter profiles of ETGs will be explored in an upcoming paper (A. Sonnenfeld et al., in preparation).

An important assumption at the basis of our analysis is that scaling relations of γ' with mass and size measured at low redshift can be used to predict the evolution of the slope for higher redshift objects. This assumption holds well if the evolutionary tracks of higher redshift galaxies stay on parts of the parameter space probed by the lower redshift systems. At first approximation, this seems to be the case for the galaxies in our sample. Figure 5 shows the positions of our lenses in the M_* − R_eff space, where the effective radius of each object is renormalized by the average R_eff of galaxies at its redshift. Under our assumptions, objects evolve along lines parallel to the mass–size relation (dashed line) toward higher masses. There is significant overlap between the high-z SL2S–LSD sample and the lower redshift SLACS sample, implying that SLACS galaxies are informative on the evolution in γ' of SL2S–LSD objects. Alternatively, one could rely on extrapolations of the scaling relations for γ'.

A more quantitative explanation of our findings would require a detailed comparison with the theoretical model and is beyond the scope of this work. However, we can check, at least qualitatively, how our result compares with published predictions. Nipoti et al. (2009b) studied the impact of dissipationless (dry) mergers on γ' finding that for an individual galaxy, the slope tends to get shallower with time. Johansson et al. (2012) looked at the evolution in the slope on nine ETGs in cosmological simulations, finding no clear trend in the redshift range explored by our data. Their simulations include both dry and dissipational (wet) mergers. Remus et al. (2013) examined simulated ETGs in a cosmological framework and in binary mergers. They found slopes that become shallower in time, asymptotically approaching the value γ' ≈ 2.1 as observed in our data. They also detected a correlation between the amount of in situ star formation and slope, with γ' being larger in systems that experienced more star formation events. Finally, Dubois et al. (2013) produced zoomed cosmological simulations of ETGs with or without active galactic nucleus (AGN) feedback. They found that the total density slope becomes steeper with time. They also observed that galaxies with strong AGN feedback have a shallower profile than systems with no AGN feedback and interpreted this result with the AGN shutting off in situ star formation. Qualitatively, our data is not in stark contrast with any of these models.

A more quantitative comparison is required to find out whether the models work in detail. This is left for future work. The combination of constraints from the evolution of the size stellar mass relation obtained via traditional studies of large samples of ETGs, and our own detailed measurements of the evolution of their internal structure, should provide a stringent test for evolutionary models of ETGs, and thus help us improve our understanding of the baryonic and dark matter physics relevant at kpc scales.

We have presented spectroscopic observations from the Keck, VLT, and Gemini Telescopes of a sample of 53 lenses and lens candidates from the SL2S survey. We measured stellar velocity dispersions for 47 of them, and redshifts of both lens and background source for 35 of them. Thirty-six systems are confirmed grade A lenses and 25 of these were able to be used for a lensing and stellar dynamics analysis. We have shown how spectroscopic observations can be used in combination with ground-based imaging with good seeing (∼07) to confirm gravitational lens candidates by the presence of multiply imaged emission lines from the lensed background source. We have also shown how SL2S lenses are comparable with lenses from other surveys in terms of their size, mass, and velocity dispersion, and lie on the same M_*–R_eff relation as non-lens galaxies.

By fitting a power-law density profile ($\rho (r) \propto r^{-\gamma ^{\prime }}$) to the lensing and stellar kinematics data of SL2S, SLACS, and LSD lenses, we measured the dependence of γ' on redshift, stellar mass, and galaxy size, over the ranges z ≈ 0.1–1.0, log M_*/M_☉ ≈ 11 − 12, R_eff = 1–20 kpc.

Our main results can be summarized as follows.

• 1.  
  In the context of power-law models for the density profile $\rho _{\rm tot}\propto r^{-\gamma ^{\prime }}$, the (logarithmic) density slope γ' of the SL2S lenses is approximately—but not exactly—that of a single isothermal sphere (γ' = 2), consistent with previous studies of lenses in different samples. This can be understood as the result of the combination of a stellar mass density profile that falls off more steeply than the dark matter halo. The relative scaling of the two conspires to produce the power-law index close to isothermal ("bulge-halo" conspiracy).
• 2.  
  At a given redshift, the mass density slope γ' depends on the surface stellar mass density $\Sigma _*=M_*/2R_{\mathrm{eff}}^2$, in the sense that galaxies with denser stars also have steeper total mass density profiles (∂γ'/∂log Σ_* = 0.38 ± 0.07).
• 3.  
  At fixed M_* and R_eff, 〈γ'〉 depends on redshift, in the sense that galaxies at a lower redshift have, on average, a steeper average slope (∂γ'/∂z = −0.31 ± 0.10).
• 4.  
  Once the dependencies of γ' on redshift and surface stellar mass density are taken into account, less than 6% intrinsic scatter is left ($\sigma _\gamma ^{\prime }=0.12\pm 0.02$).
• 5.  
  The average redshift evolution of γ' for an individual galaxy is consistent with zero: dγ'/dz = −0.10  ±  0.12. This result is obtained by combining our measured dependencies of 〈γ'〉 on redshift stellar mass and effective radii with the observed evolution of the size stellar mass relation taken from the literature.

The key result of this work is that the dependency of 〈γ'〉 on redshift and stellar mass density does not imply that massive ETGs change their mass density profile over the second half of the lifetime. In fact, at least qualitatively, the observed dependencies can be understood as the result of two effects. Individual galaxies grow in stellar mass and decrease in density over the redshift range 1–0, while apparently largely preserving their total mass density profiles. This could be explained by the addition of stellar mass in the outer part of the galaxies in quantities that are sufficient to explain the decrease in stellar mass density but insufficient to alter the total mass density profile, since the regions are already dominated by dark matter. As shown by Nipoti et al. (2012), the growth in size during this period is slow enough that it could perhaps be explained by the infall of dark matter and stars via a drizzle of minor mergers, with material of decreasing density tracking the decreasing cosmic density. This process needs to happen while substantially preserving the total mass density profile.

Alternatively, the evolution at constant slope can be interpreted as the combined effect of the decrease in stellar mass density and a variation in the dark matter profile (either a steepening or a decrease of the central dark matter distribution). The latter effect would be responsible for the term ∂γ'/∂z.

Checking whether or not these scenarios can work quantitatively requires detailed comparisons with theoretical calculations, which are beyond the scope of this paper.

The second important result of this work is that the total mass density profile of ETGs depends on their stellar mass density, with very little scatter. Qualitatively this makes sense, as we expect that the more concentrated stellar distributions should have been able to contract the overall profile the most. Presumably, this difference may trace back to differences in past star formation efficiency or merger history. Therefore, the tightness of the observed correlation should provide interesting constraints on these crucial ingredients of our understanding of ETGs.

We thank our friends of the SLACS and SL2S collaborations for many useful and insightful discussions over the course of the past years. We thank V. N. Bennert and M. Bradac for their help in our observational campaign. T.T. thanks S. W. Allen and B. Poggianti for useful discussions. R.G. acknowledges support from the Centre National des Etudes Spatiales (CNES). P.J.M. acknowledges support from the Royal Society in the form of a research fellowship. T.T. acknowledges support from the NSF through CAREER award NSF-0642621 and from the Packard Foundation through a Packard Research Fellowship. This research is based on XSHOOTER observations made with ESO Telescopes at the Paranal Observatory under program IDs 086.B-0407(A) and 089.B-0057(A). This research is based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, and with WIRCam, a joint project of CFHT, Taiwan, Korea, Canada, and France, at the Canada–France–Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre. The authors would like to thank S. Arnouts, L. Van waerbeke, and G. Morrison for giving access to the WIRCam data collected in W1 and W4 as part of additional CFHT programs. We are particularly thankful to Terapix for the data reduction of this dataset. This research is supported by NASA through Hubble Space Telescope programs GO-10876, GO-11289, and GO-11588, and in part by the National Science Foundation under grant No. PHY99-07949, and is based on observations made with the NASA/ESA Hubble Space Telescope and obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555, and at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.