Stronglensing Measurement of the Massdensity Profile out to 3 Effective Radii for Earlytype Galaxies
Abstract
We measure the total massdensity profiles out to three effective radii for a sample of 63 , massive earlytype galaxies (ETGs) acting as strong gravitational lenses through a joint analysis of lensing and stellar dynamics. The compilation is selected from three galaxyscale stronglens samples including the Baryon Oscillation Spectroscopic Survey (BOSS) EmissionLine Lens Survey (BELLS), BELLS for GALaxyLy EmitteR sYstems Survey, and Strong Lensing Legacy Survey (SL2S). Utilizing the wide sourceredshift coverage (0.8–3.5) provided by these three samples, we build a statistically significant ensemble of massive ETGs for which robust mass measurements can be achieved within a broad range of Einstein radii up to three effective radii. Characterizing the threedimensional total massdensity distribution by a powerlaw profile as , we find that the average logarithmic density slope for the entire sample is (CL) with an intrinsic scatter . Further parameterizing as a function of redshift and ratio of Einstein radius to effective radius , we find the average density distributions of these massive ETGs become steeper at larger radii and later cosmic times with magnitudes and .
keywords:
gravitational lensing: strong  galaxies: elliptical  galaxies:structure1 Introduction
Believed to be the end products of the hierarchical merging scenario (Kauffmann et al., 1993; Cole et al., 2000), earlytype galaxies (ETGs) play an important role in understanding the formation and evolution of galaxies. In particular, the massdensity slope in the inner region of ETGs provide useful insight into the physical processes that regulate the mass distributions. Numerical simulations suggest that darkmatter halos across a broad range of mass scales can be well described by a universal NFW profile (Navarro et al., 1997). However, baryonic physics can significantly modify the mass distribution of an ETG, especially in the inner region, through dissipative gascooling processes and the supernovae (SN)/ active galactic nucleus (AGN) feedback. The former processes lead to a higher baryon densities and steepen the inner density profile (e.g., Gustafsson et al., 2006; Abadi et al., 2010; Velliscig et al., 2014), while the latter processes tend to heat the gas and soften the density profile (e.g., Martizzi et al., 2012; Dubois et al., 2013; Velliscig et al., 2014). Therefore, the inner massdensity profile of ETGs and its dependences on galaxy properties can be used to study the relative importance and efficiency of the aforementioned physical processes across different evolving stages.
One effective method of measuring the inner mass distributions of ETGs is through stellar dynamical modeling. For instance, Cappellari et al. (2015) analyzed twodimensional stellar kinematic data for 14 local fastrotator ETGs to infer the mass density profiles of this sample out to 4 effective radii. However, for distant ETGs (), stellar kinematic observations, especially the integral field unit (IFU) observations, become technically challenging and sometimes impossible. Alternatively, strong gravitational lensing serves as another powerful method of measuring the mass distributions of distant ETGs. Assuming a twoparameter powerlaw mass distribution model, the projected total mass constraint within the Einstein radius provided by the stronglensing data and the stellar velocity dispersion measured within a fiber or slit can be used to determine the average massdensity slope of the lens galaxy in its central region by solving the spherical Jeans equations. This joint analysis of strong lensing and stellar dynamics has been applied to a sample of lens galaxies, which leads to better understanding of the mass distribution and its evolution for massive ETGs. For instance, Koopmans et al. (2006) found that the average inner massdensity profile of massive ETGs can be well approximated by an isothermal profile (i.e. ) based on 15 lenses selected from the Sloan Lens ACS Survey (SLACS, Bolton et al., 2008). Auger et al. (2010) and Sonnenfeld et al. (2013b) suggested that the average logarithmic density slope correlates with the central surface mass density in the sense that denser galaxies have steeper slopes. Bolton et al. (2012) pointed out that inner density slope of massive ETGs evolves with galaxy redshift. Shu et al. (2015) found that the inner density slope and dark matter fraction are strongly correlated with galaxy mass/velocity dispersion.
Another important question is whether the massdensity slope evolve along radius. If such an evolution indeed exists, it could affect the interpretations of some results based on strong lenses, in particular the redshift evolution of the density slope in massive galaxies. Koopmans et al. (2006) found little correlation between the massdensity slope and the normalized radius , but their results were limited to within one effective radius and lower redshifts (). Although Sonnenfeld et al. (2013b) extended this analysis to 3 effective radii and higher redshifts (), the results were not statistically significant because only two of the 23 lens galaxies used in their work have larger than 2.
In this work, we combine BELLS and SL2S lenses with the recently discovered BELLS GALLERY lenses, which contains 7 lens galaxies with , to examine the radial dependence of the massdensity slope of massive ETGs at high redshifts (). Throughout the paper, represents the twodimensional radial coordinate and represents the threedimensional radial coordinate. We adopt a fiducial cosmological model with , , and .
2 Lens systems and the data
Strong gravitational lenses can only provide a robust measurement of the total mass within the Einstein radii. To infer the mass distribution, we assume the totalmass density distribution of a lens galaxy can be well described by a powerlaw profile as suggested by a variety of studies (Koopmans & Treu, 2003; Koopmans et al., 2006, 2009; Auger et al., 2010; Bolton et al., 2012; Sonnenfeld et al., 2013b; Shu et al., 2015), and use the stellar velocity dispersion as an extra constraint for the logarithmic slope . We then build an ETGlens compilation with a wide range of Einstein radii and study the radial dependence of for entire ETGlens population within a hierarchical Bayesian framework. This is achieved by selecting stronglens systems with similar lens masses and redshifts but a wide range of source redshifts because the Einstein radii is determined by the lens’ mass, distance to the observer, and distance to the source as , where and are the comoving distances to the lens and source respectively. Given a similar lens mass and redshift, a wide range of source redshifts corresponds to a wide range of Einstein radii.
The compilation of the 63 stronglens systems used in this work is built from the BELLS, BELLS GALLERY, and part of SL2S samples. BELLS and BELLS GALLERY stronglens systems were selected from the BOSS spectroscopic database based on detections of emission lines in the BOSS galaxy spectra that are identified to come from redshifts higher than the BOSS galaxies (Bolton, 2006; Bolton et al., 2006). Highresolution HST imaging data were acquired to confirm the lensing nature. SL2S stronglens systems were identified photometrically from the CFHT Legacy Survey by the presence of lensinglike features around galaxies and confirmed by imaging (HST or CFHT) and spectroscopic data. In total, there are 67 stronglens systems in the three samples with 25 from BELLS (Brownstein et al., 2012), 17 from BELLS GALLERY (Shu et al., 2016), and 25 from SL2S (Sonnenfeld et al., 2013a, b). The lens redshifts of the three lens samples are within a similar range of 0.3–0.65, but the source redshifts cover a wide range from 0.8 to 3.5. Because this work aims to investigate the massdensity profile of massive ETGs, we discard two BELLS GALLERY systems with multiple lens components and two SL2S systems with disklike lenses. The remaining 63 stronglens systems, each with a single massive ETG as the lens, are used in this analysis.
Figure 1a shows the distributions of the Einstein radius (, black solid line), effective radius (, black dashed line), and ratio of the Einstein radius to the effective radius (, red dashed line) for the 63 lens galaxies. The ratios populate primarily between 0.5 and 1.5 with a sharp drop below 0.5 and an extended wing up to 3.5. Ten galaxies have ratios larger than 2.0. In Figure 1b, we compare the Einstein radii () and effective radii (). We find that the Einstein radii are smaller than the effective radii for most of the BELLS and SL2S samples. But for the BELLS GALLERY samples, the Einstein radii of most systems are larger than the effective radii.
3 Joint lensing and dynamical analysis
Following previous work (e.g., Treu & Koopmans, 2004; Koopmans et al., 2006, 2009, Bolton et al. 2012), we assume the threedimensional total massdensity distribution of the lens galaxies can be described by a powerlaw profile as
(1) 
where is the logarithmic slope and is a normalization factor. We model the twodimensional light distributions of lens galaxies using a Sérsic profile as
(2) 
where is the Sérsic index, is given by for , and is the effective radius. We deproject twodimensional light distributions to infer the threedimensional light distributions by solving the Abel integral equation. The spherical Jeans Equation can be written as (Binney & Tremaine, 1987)
(3) 
where and are the tangential and radial components of velocity dispersion vector. is the number density of the stars in the galaxy, which is assumed to be proportional to the threedimensional light distributions. is the velocity dispersion anisotropy parameter (Osipkov, 1979; Merritt, 1985), which is set to be 0 in our analysis. is the total mass inside a sphere with radius .
For each lens galaxy, we use the total enclosed mass within the Einstein radius provided by stronglensing data to determine its massdensity normalization factor . The threedimensional velocity dispersion profile, , is determined by solving Equation (3). Finally we project to get the twodimensional velocity dispersion profile, , for each lens galaxy. In order to compare with observations, which measure the luminosityweighted velocity dispersion within an aperture, we further convolve with the twodimensional light distribution and an aperture weighting function to get a predicted velocity dispersion, as
(4) 
We adopt the functional form in Schwab et al. (2010) for the weighting function. An aperture radius of 1 and the mean seeing value 1.5855 of the 40 BELLS+BELLS GALLERY lens galaxies are used in .
Instead of directly comparing to the reported velocity dispersions , which generally have large uncertainties because they are determined from low signaltonoise ratio (SNR) spectroscopic data, we use , which is the curve as a function of trial velocity dispersion , to infer for each lens galaxy. As explained in Shu et al. (2012) and Bolton et al. (2012), the is obtained by fitting the same spectroscopic data, , using fewer eigenspectra to avoid overfitting. Redshift error is also marginalized in this process. We minimize , which is converted from by interpolations, to determine the bestfit value for each lens galaxy.
The logarithmic total massdensity profile slope for BELLS and BELLS GALLERY lens galaxies are obtained by this analysis, and listed in the last column of Table 1. The logarithmic total massdensity slope of SL2S lens galaxies as well as their uncertainties are directly taken from Sonnenfeld et al. (2013b), who performed a similar joint analysis.
4 The average total massdensity slope and its radial dependence
In this section, we perform a hierarchical Bayesian analysis to study the mean total massdensity slope of this lens galaxy sample and its dependences on other galaxy properties, especially the ratio of the Einstein radius to the effective radius. Here we assume the 63 lens galaxies can be treated as a single population.
We parameterize the probability density function (PDF) of as a Gaussian function
(5) 
where and are the two hyperparameters that represent the mean and intrinsic scatter of the massdensity slope. The likelihood function of and , , can be written as
(6) 
in which is simply proportional to . Notice that, for SL2S lens galaxies, we have no curve. Therefore, we use the given values of and their uncertainties in Sonnenfeld et al. (2013b) to build a simple curve for our study. Then the posterior PDF of and is obtained from the likelihood function through the Baye’s theorem as
(7) 
where is the prior. For simplicity, we adopt an uniform prior over reasonable ranges of and .
The posterior PDF contours of and , constructed from a Markov chain Monte Carlo method, is shown in Figure 2. We find that the total massdensity slope of this lens galaxy population is , and its intrinsic scatter is . It suggests that the massdensity profile in the central region of this lens galaxy population is very close to an isothermal distribution, consistent with previous findings (Koopmans et al., 2009; Auger et al., 2010; Bolton et al., 2012; Sonnenfeld et al., 2013b; Shankar et al., 2017; Mukherjee et al., 2018).
We now examine the radial dependence of in this galaxy population. To do this, we normalize the Einstein radius by the effective radius for each lens galaxy, and introduce one more hyperparameter, , so that the PDF of is now
(8) 
where is the logarithmic slope at , which is the mean value for the full lensgalaxy population, quantifies the radial dependence of , and is the intrinsic scatter. Finally we get , , and . The fitting result is shown in Figure 3. We find that, the slope has a very slight increasing trend along the radii within 3 effective radii for a fixed galaxy.
5 discussion
The time evolution of slope that galaxies with lower redshift would have steeper massdensity profile, is clearly visible in previous work(Bolton et al., 2012). For our samples, we also examine the slope evolution with redshift. Assuming this relation is linear, we write the PDF as
(9) 
where is the logarithmic slope at , which is the mean redshift value for all the samples, quantifies the redshift dependence of . The result is , indicating that the galaxies at higher redshift have shallower mass density profile, which are similar with previous work (Auger et al., 2010; Bolton et al., 2012).
In our research, by normalizing the Einstein radius by the effective radius for each lens galaxy, we find increase along radius, which is a fundamental dependence of the slope on structural properties of ETGs. However, we notice that, with redshift increasing, the ratio of would increase for pure geometrical reasons. Therefore, the existence of the time evolution of may affect the inference of the radius evolution and vice versa. So, in order to make an comprehensive analysis, we assume the slope evolutes with and the redshift of the lenses at the same time as
(10) 
where
(11) 
is the value of the slope when , and , which are the mean values of and redshift , respectively. , are the evolution factors. The posterior probability contours and the credible regions for all these parameters are shown in Figure 4. We get the parameters as follows
Besides the time evolution, we find the slope still has a increasing trend along radius, implying that the total massdensity slope would become steeper with increasing radius for a fixed galaxy.
Unlike previous work (e.g., Koopmans et al. (2006, 2009); Ruff et al. (2011); Sonnenfeld et al. (2013b), we find has a slight increasing trend along radius within 3 effective radius. This finding can be mostly attributed to the larger of BELLS GALLERY samples with higher source redshift. One possible explanation for the evolution along radius is the different strength of SN/AGN feedback processes at different radius. The SN/AGN feedback process of ETGs may toward weaker with radius increasing, which could lead the density profile at larger radius be steeper than that at smaller radius. Actually, Xu et al. (2017) have found a mild increase of with increasing using numerical simulation depending on galactic wind and AGN feedback. In their work, they used the Illustris simulation project (Genel et al., 2014; Vogelsberger et al., 2014a, b; Nelson et al., 2015; Sijacki et al., 2015) to study the massdensity slope of the inner regions of ETGs and find the simulation predicted higher central dark matter fractions, which would suppress the dominating role of baryons and thus lead to shallower total massdensity profile at smaller radius. Our observational result for the density profile is in accordance with their predictions.
6 Conclusions
In this paper, we study the total massdensity profile out to three effective radii for a sample of 63 intermediateredshift earlytype galaxies (ETGs). The sample is compiled from three galaxyscale stronglens surveys, including 25 galaxies from the BELLS, 15 galaxies from the BELLS GALLERY, and 23 galaxies from the SL2S. Assuming a powerlaw totalmass density profile of for the lens galaxies, we investigate the evolution of the slope out to 3 effective radii with a hierarchical Bayesian method by combining the strong lensing data and dynamical constraints. In our study, we include more higher redshift galaxies in the sample comparing with the SLACS. We obtain the following conclusions:

The average logarithmic density slope of our samples is , with an intrinsic scatter of . The total massdensity profile is very close to the isothermal distribution, similar with previous work.

Assuming a linear relation between and , we obtain that the evolution factor is , indicating that the total massdensity slope has a sight rise along radius for a fixed galaxy. When consider the slope evolves with and the redshift at the same time, the increasing trend still exists. We conclude that the total massdensity slope of ETGs would increase along radius within 3 effective radii for a fixed galaxy, which is in accordance with the numerical simulation depending on galactic wind and AGN feedback (Xu et al., 2017).
We acknowledge the financial support from the National Natural Science Foundation of China 11573060 and 11661161010. Y.S. has been supported by the National Natural Science Foundation of China (No. 11603032 and 11333008), the 973 program (No. 2015CB857003), and the Royal Society  K.C. Wong International Fellowship (NF170995).
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Appendix A List of strong gravitational lenses
Table 1 shows a list of all 63 strong gravitational lens systems used in this paper.
Lens Name  Slope  
(km s)  (arcsec)  (arcsec)  ()  ()  
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8) 
SDSSJ00292544  0.5869  2.4504  24145  1.34  0.49  4.82  2.030.27 
SDSSJ02013228  0.3957  2.8209  25620  1.70  2.32  5.21  1.960.17 
SDSSJ02370641  0.4859  2.2491  29089  0.65  1.05  0.97  2.320.54 
SDSSJ07423341  0.4936  2.3633  21828  1.22  0.89  3.41  1.980.20 
SDSSJ07553445  0.7224  2.6347  27252  2.05  2.89  13.52  1.720.28 
SDSSJ08562010  0.5074  2.2335  33454  0.98  0.51  2.30  2.550.23 
SDSSJ09185104  0.5811  2.4030  29849  1.60  0.57  6.85  2.140.26 
SDSSJ11102808  0.6073  2.3999  19139  0.98  1.45  2.69  1.880.30 
SDSSJ11103649  0.7330  2.5024  531165  1.16  0.39  4.48  2.560.31 
SDSSJ11160915  0.5501  2.4536  27455  1.03  0.98  2.68  2.230.25 
SDSSJ11412216  0.5858  2.7624  28544  1.27  0.44  4.18  2.220.26 
SDSSJ12014743  0.5628  2.1258  23943  1.18  0.48  3.76  2.090.27 
SDSSJ12265457  0.4980  2.7322  24826  1.37  0.56  4.20  2.060.21 
SDSSJ22281205  0.5305  2.8324  25550  1.28  0.53  3.85  2.130.28 
SDSSJ23420120  0.5270  2.2649  27443  1.11  1.75  3.05  2.300.31 
SDSSJ01510049  0.5171  1.3636  21939  0.68  2.04  1.37  2.470.26 
SDSSJ07475055  0.4384  0.8983  32860  0.75  1.27  1.83  2.580.26 
SDSSJ07474448  0.4366  0.8966  28152  0.61  2.80  1.19  2.510.31 
SDSSJ08014727  0.4831  1.5181  9824  0.49  1.82  0.63  1.540.27 
SDSSJ08305116  0.5301  1.3317  26836  1.14  1.14  4.10  2.080.25 
SDSSJ09440147  0.5390  1.1785  20434  0.73  1.87  1.85  2.040.30 
SDSSJ11590007  0.5793  1.3457  16541  0.68  2.22  1.64  2.370.51 
SDSSJ12150047  0.6423  1.2970  26245  1.37  2.10  7.95  1.790.28 
SDSSJ12213806  0.5348  1.2844  18748  0.70  2.00  1.59  2.060.39 
SDSSJ12340241  0.4900  1.0159  12231  0.53  3.40  0.98  1.900.45 
SDSSJ13180104  0.6591  1.3959  17727  0.68  2.45  1.91  1.810.30 
SDSSJ13373620  0.5643  1.1821  22535  1.39  1.88  7.23  1.730.25 
SDSSJ13493612  0.4396  0.8926  17818  0.75  0.47  1.83  1.530.23 
SDSSJ13523216  0.4634  1.0341  16121  1.82  1.85  10.33  1.460.19 
SDSSJ15222910  0.5553  1.3108  16627  0.74  1.23  1.83  1.710.26 
SDSSJ15411812  0.5603  1.1133  17424  0.64  2.19  1.61  1.820.28 
SDSSJ15421629  0.3521  1.0233  21016  1.04  2.51  2.32  1.910.21 
SDSSJ15452748  0.5218  1.2886  25037  1.21  3.91  4.60  1.920.28 
SDSSJ16012138  0.5435  1.4461  20736  0.86  1.70  2.27  2.120.24 
SDSSJ16111705  0.4766  1.2109  10923  0.58  2.64  0.97  1.460.29 
SDSSJ16311854  0.4081  1.0863  27214  1.63  0.74  6.70  1.940.17 
SDSSJ16371439  0.3910  0.8744  20830  0.65  2.17  1.16  2.190.32 
SDSSJ21220409  0.6261  1.4517  32456  1.58  2.46  9.28  2.040.25 
SDSSJ21250411  0.3632  0.9777  24717  1.20  2.69  3.31  2.040.18 
SDSSJ23030037  0.4582  0.9363  27431  1.02  1.15  3.43  2.210.27 
SL2SJ02120555  0.750  2.74  26717  1.27  1.22  5.31  2.050.09 
SL2SJ02130743  0.717  3.48  28733  2.39  1.97  16.77  1.790.12 
SL2SJ02140405  0.609  1.88  23815  1.41  1.21  6.13  1.850.07 
SL2SJ02170513  0.646  1.85  27021  1.27  0.73  5.37  2.020.09 
SL2SJ02190829  0.389  2.15  30023  1.30  0.95  3.15  2.260.08 
SL2SJ02250454  0.238  1.20  22620  1.76  2.12  3.98  1.780.10 
SL2SJ02260420  0.494  1.23  26624  1.19  0.84  4.26  2.010.12 
SL2SJ02320408  0.352  2.34  27120  1.04  1.41  1.80  2.390.10 
SL2SJ08480351  0.682  1.55  20521  0.85  0.45  2.88  1.850.14 
SL2SJ08490412  0.722  1.54  31218  1.10  0.46  5.27  2.140.06 
SL2SJ08490251  0.274  2.09  27534  1.16  1.34  1.80  2.320.17 
SL2SJ08550147  0.365  3.39  22219  1.03  0.69  1.74  2.150.11 
SL2SJ09040059  0.611  2.36  17820  1.40  2.00  5.55  1.480.11 
SL2SJ09590206  0.552  3.35  19522  0.74  0.46  1.29  2.110.16 
SL2SJ13595535  0.783  2.77  22919  1.14  1.13  4.45  1.860.14 
SL2SJ14045200  0.456  1.59  33719  2.55  2.03  15.56  1.950.06 
SL2SJ14055243  0.526  3.01  29121  1.51  0.83  5.25  2.140.08 
SL2SJ14065226  0.716  1.47  25814  0.94  0.80  3.96  2.000.07 
SL2SJ14115651  0.322  1.42  22023  0.93  0.85  1.47  2.150.15 
SL2SJ14205630  0.483  3.12  22819  1.40  1.62  4.16  1.930.11 
SL2SJ22030205  0.400  2.15  21821  1.95  0.99  7.28  1.770.09 
SL2SJ22050147  0.476  2.53  32630  1.66  0.66  6.01  2.190.09 
SL2SJ22210115  0.325  2.35  22423  1.40  1.12  3.03  1.960.13 