We use stellar masses, surface photometry, strong lensing masses, and stellar velocity dispersions (σ e/2 ) to investigate empirical correlations for the definitive sample of 73 early-type galaxies (ETGs) that are strong gravitational lenses from the SLACS survey. The traditional correlations (Fundamental Plane [FP] and its projections) are consistent with those found for non-lens galaxies, supporting the thesis that SLACS lens galaxies are representative of massive ETGs (dimensional mass M dim = 10 11 − 10 12 M ⊙ ). The addition of high-precision strong lensing estimates of the total mass allows us to gain further insights into their internal structure: i) the average slope of the total mass density profile (ρ tot ∝ r −γ ′ ) is γ ′ = 2.078 ± 0.027 with an intrinsic scatter of 0.16 ± 0.02; ii) γ ′ correlates with effective radius (r e ) and central mass density, in the sense that denser galaxies have steeper profiles; iii) the dark matter fraction within r e /2 is a monotonically increasing function of galaxy mass and size (due to a mass-dependent central cold dark matter distribution or to baryonic dark matter -stellar remnants or low mass stars -if the IMF is non-universal and its normalization increases with mass); iv) the dimensional mass M dim ≡ 5r e σ 2 e/2 /G is proportional to the total (lensing) mass M re/2 , and both increase more rapidly than stellar mass M * (M * ∝ M 0.8 re/2 ); v) the Mass Plane (MP), obtained by replacing surface brightness with surface mass density in the FP, is found to be tighter and closer to the virial relation than the FP and the M * P, indicating that the scatter of those relations is dominated by stellar populations effects; vi) we construct the Fundamental Hyper-Plane by adding stellar masses to the MP and find the M * coefficient to be consistent with zero and no residual intrinsic scatter. Our results demonstrate that the dynamical structure of ETGs is not scale invariant and that it is fully specified by M re/2 , r e , and σ e/2 . Although the basic trends can be explained qualitatively in terms of varying star formation efficiency as a function of halo mass and as the result of dry and wet mergers, reproducing quantitatively the observed correlations and their tightness may be a significant challenge for galaxy formation models.

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In this paper, we describe a procedure for modelling strong lensing galaxy clusters with parametric methods, and to rank models quantitatively using the Bayesian evidence. We use a publicly-available Markov Chain Monte-Carlo (MCMC) sampler ("BayeSys"), allowing us to avoid local minima in the likelihood functions.To illustrate the power of the MCMC technique, we simulate three clusters of galaxies, each composed of a cluster-scale halo and a set of perturbing galaxyscale subhalos. We ray-trace three light beams through each model to produce a catalogue of multiple images, and then use the MCMC sampler to recover the model parameters in the three different lensing configurations.We find that, for typical HST-quality imaging data, the total mass in the Einstein radius is recovered with ∼ 1% to 5% error according to the considered lensing configuration. However, we find that the mass of the galaxies is strongly degenerate with the cluster mass when no multiple images appear in the cluster centre. The mass of the galaxies is generally recovered with a 20% error, due largely to the poorly constrained cut-off radius.Finally, we describe how to rank models quantitatively using the Bayesian evidence. We confirm the ability of strong lensing to constrain the mass profile in the central region of galaxy clusters in this way. Ultimately, such a method applied to strong lensing clusters with a very large number of multiple images may provide unique geometrical constraints on cosmology.The implementation of the MCMC sampler used in this paper has been done within the framework of the lenstool software package, which is publicly available. ‡ PACS numbers: 90 98.62.Sb 07.05.Kf 98.65.Cw 95.35.+d ‡