Based on 58 SLACS strong-lens early-type galaxies with direct total-mass and stellar-velocity dispersion measurements, we find that inside one effective radius massive elliptical galaxies with M eff 3 · 10 10 M ⊙ are well-approximated by a power-law ellipsoid with an average logaritmic density slope of γ ′ LD ≡ −d log(ρ tot )/d log(r) = 2.085 +0.025 −0.018 (random error on mean) for isotropic orbits with β r = 0, ±0.1 (syst.) and σ γ ′ 0.20 +0.04 −0.02 intrinsic scatter (all errors indicate the 68% CL). We find no correlation of γ ′ LD with galaxy mass (M eff ), rescaled radius (i.e. R einst /R eff ) or redshift, despite intrinsic differences in density-slope between galaxies. Based on scaling relations, the average logarithmic density slope can be derived in an alternative manner, fully independent from dynamics, yielding γ ′ SR = 1.959 ± 0.077. Agreement between the two values is reached for β r = 0.45 ± 0.25, consistent with mild radial anisotropy. This agreement supports the robustness of our results, despite the increase in mass-to-light ratio with total galaxy mass: M eff ∝ L 1.363±0.056 V,eff. We conclude that massive early-type galaxies are structurally close-to homologous with close-to isothermal total density profiles ( 10% intrinsic scatter) and have at most some mild radial anisotropy. Our results provide new observational limits on galaxy formation and evolution scenarios, covering four Gyr look-back time.

We introduce a new adaptive and fully Bayesian grid‐based method to model strong gravitational lenses with extended images. The primary goal of this method is to quantify the level of luminous and dark mass substructure in massive galaxies, through their effect on highly magnified arcs and Einstein rings. The method is adaptive on the source plane, where a Delaunay tessellation is defined according to the lens mapping of a regular grid on to the source plane. The Bayesian penalty function allows us to recover the best non‐linear potential‐model parameters and/or a grid‐based potential correction and to objectively quantify the level of regularization for both the source and potential. In addition, we implement a Nested‐Sampling technique to quantify the errors on all non‐linear mass model parameters – marginalized over all source and regularization parameters – and allow an objective ranking of different potential models in terms of the marginalized evidence. In particular, we are interested in comparing very smooth lens mass models with ones that contain mass substructures. The algorithm has been tested on a range of simulated data sets, created from a model of a realistic lens system. One of the lens systems is characterized by a smooth potential with a power‐law density profile, 12 include a Navarro, Frenk and White (NFW) dark matter substructure of different masses and at different positions and one contains two NFW dark substructures with the same mass but with different positions. Reconstruction of the source and lens potential for all of these systems shows the method is able, in a realistic scenario, to identify perturbations with masses ≳107 M⊙ when located on the Einstein ring. For positions both inside and outside of the ring, masses of at least 109 M⊙ are required (i.e. roughly the Einstein ring of the perturber needs to overlap with that of the main lens). Our method provides a fully novel and objective test of mass substructure in massive galaxies.

The mass-function of dwarf satellite galaxies that are observed around Local Group galaxies substantially differs from simulations 1-5 based on cold dark matter: the simulations predict many more dwarf galaxies than are seen. The Local Group, however, may be anomalous in this regard 6, 7 . A massive dark satellite in an early-type lens galaxy at z = 0.222 was recently found 8 using a new method based on gravitational lensing 9, 10 , suggesting that the mass fraction contained in substructure could be higher than is predicted from simulations. The lack of very low mass detections, however, prohibited any constraint on their mass function. Here we report the presence of a 1.9 ± 0.1 × 10 8 M ⊙ dark satellite in the Einstein-ring system JVAS B1938+666 (ref. 11) at z = 0.881, where M ⊙ denotes solar mass. This satellite galaxy has a mass similar to the Sagittarius 12 galaxy, which is a satellite of the Milky Way. We determine the logarithmic slope of the mass function for substructure beyond the local Universe to be α = 1.1 +0.6 −0.4 , with an average mass-fraction of f = 3.3 +3.6 −1.8 %, by combining data on both of these recently discovered galaxies. Our results are consistent with the predictions from cold dark matter simulations 13-15 at the 95 per cent confidence level,and therefore agree with the view that galaxies formed hierarchically in a Universe composed of cold dark matter.The gravitational lens system JVAS B1938+666 (ref. 11) has a bright infrared background galaxy at redshift 2.059 (ref. 16), which is gravitationally lensed into an almost complete Einstein ring of diameter ∼ 0.9 arcseconds by a massive elliptical galaxy at redshift 0. 881 (ref. 17). The bright, highly-magnified Einstein ring made this system an excellent candidate in which to to search for surface brightness anomalies caused by very low mass (dark matter) substructure in the halo around the high redshift elliptical lens galaxy. The presence of a low-mass substructure (e.g. a luminous or dark satellite galaxy; also denoted as substructure hereafter) in the lens galaxy 1

We report the detection of a dark substructure – undetected in the Hubble Space Telescope HST ACS F814W image – in the gravitational lens galaxy SDSSJ0946+1006 (the ‘double Einstein ring’), through direct gravitational imaging. The detection of a small mass concentration in the surface density maps, at 4.3 kpc from the galaxy centre, has a strong statistical significance. We confirm this detection by modelling the substructure with a tidally truncated pseudo‐Jaffe density profile; in that case the substructure mass is Msub= (3.51 ± 0.15) × 109 M⊙, precisely where also the surface density map shows a strong convergence peak (Bayes factor ; equivalent to a ∼16σ detection). The result is robust under substantial changes in the model. We set a lower limit of (M/L)V,⊙≳ 120  M⊙/LV,⊙ (3σ) inside a sphere of 0.3 kpc centred on the substructure (rtidal= 1.1 kpc). The mass and luminosity limit of this substructure are consistent with Local Group results if the substructure had a virial mass of ∼1010 M⊙ before accretion and formed at z≳ 10. Our detection implies a projected dark matter mass fraction in substructure at the radius of the inner Einstein ring of f= 2.15+2.05−1.25 per cent [68 per cent confidence level (CL)] in the mass range 4 × 106– 4 × 109 M⊙, assuming α= 1.9 ± 0.1 (with dN/dm∝m−α). Assuming a flat prior on α, between 1.0 and 3.0, increases this to f= 2.56+3.26−1.50 per cent (68 per cent CL). The likelihood ratio is ∼0.5 between these fractions and that from simulations (fN‐body≈ 0.003). Hence the inferred dark matter mass fraction in substructure, admittedly based on a single‐lens system, is large but still consistent with predictions.

We present the results of a search for galaxy substructures in a sample of 11 gravitational lens galaxies from the Sloan Lens ACS Survey. We find no significant detection of mass clumps, except for a luminous satellite in the system SDSS J0956+5110. We use these non-detections, in combination with a previous detection in the system SDSS J0946+1006, to derive constraints on the substructure mass function in massive early-type host galaxies with an average redshift z lens ∼ 0.2 and an average velocity dispersion σ eff ∼ 270 km s −1 . We perform a Bayesian inference on the substructure mass function, within a median region of about 32 kpc 2 around the Einstein radius ( R ein ∼ 4.2 kpc). We infer a mean projected substructure mass fraction f = 0.0076 +0.0208 −0.0052 at the 68 percent confidence level and a substructure mass function slope α < 2.93 at the 95 percent confidence level for a uniform prior probability density on α. For a Gaussian prior based on cold dark matter (CDM) simulations, we infer f = 0.0064 +0.0080 −0.0042 and a slope of α = 1.90 +0.098 −0.098 at the 68 percent confidence level. Since only one substructure was detected in the full sample, we have little information on the mass function slope, which is therefore poorly constrained (i.e. the Bayes factor shows no positive preference for any of the two models). The inferred fraction is consistent with the expectations from CDM simulations and with inference from flux ratio anomalies at the 68 percent confidence level.

We investigate how Einstein rings and magnified arcs are affected by small-mass dark-matter haloes placed along the line-of-sight to gravitational lens systems. By comparing the gravitational signature of line-of-sight haloes with that of substructures within the lensing galaxy, we derive a mass-redshift relation that allows us to rescale the detection threshold (i.e. lowest detectable mass) for substructures to a detection threshold for line-of-sight haloes at any redshift. We then quantify the line-of-sight contribution to the total number density of low-mass objects that can be detected through strong gravitational lensing. Finally, we assess the degeneracy between substructures and line-of-sight haloes of different mass and redshift to provide a statistical interpretation of current and future detections, with the aim of distinguishing between CDM and WDM. We find that line-of-sight haloes statistically dominate with respect to substructures, by an amount that strongly depends on the source and lens redshifts, and on the chosen dark matter model. Substructures represent about 30 percent of the total number of perturbers for low lens and source redshifts (as for the SLACS lenses), but less than 10 per cent for high redshift systems. We also find that for data with high enough signal-to-noise ratio and angular resolution, the non-linear effects arising from a double-lens-plane configuration are such that one is able to observationally recover the line-of-sight halo redshift with an absolute error precision of 0.15 at the 68 per cent confidence level.

We present the measurement of the Hubble Constant, H 0 , with three strong gravitational lens systems. We describe a blind analysis of both PG 1115+080 and HE 0435−1223 as well as an extension of our previous analysis of RXJ 1131−1231. For each lens, we combine new adaptive optics (AO) imaging from the Keck Telescope, obtained as part of the SHARP AO effort, with Hubble Space Telescope (HST ) imaging, velocity dispersion measurements, and a description of the line-of-sight mass distribution to build an accurate and precise lens mass model. This mass model is then combined with the COSMOGRAIL measured time delays in these systems to determine H 0 . We do both an AO-only and an AO+HST analysis of the systems and find that AO and HST results are consistent. After unblinding, the AO-only analysis gives H 0 = 82.8 +9.4 −8.3 km s −1 Mpc −1 for PG 1115+080, H 0 = 70.1 +5.3 −4.5 km s −1 Mpc −1 for HE 0435−1223, and H 0 = 77.0 +4.0 −4.6 km s −1 Mpc −1 for RXJ 1131−1231. The joint AOonly result for the three lenses is H 0 = 75.6 +3.2 −3.3 km s −1 Mpc −1 . The joint result of the AO+HST analysis for the three lenses is H 0 = 76.8 +2.6−2.6 km s −1 Mpc −1 . All of the above results assume a flat Λ cold dark matter cosmology with a uniform prior on Ω m in [0.05, 0.5] and H 0 in [0, 150] km s −1 Mpc −1 . This work is a collaboration of the SHARP and H0LiCOW teams, and shows that AO data can be used as the high-resolution imaging component in lens-based measurements of H 0 . The full time-delay cosmography results from a total of six strongly lensed systems are presented in a companion paper.

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