Multiple image gravitational lens systems, and especially quads are invaluable in determining the amount and distribution of mass in galaxies. This is usually done by mass modeling using parametric or free-form methods. An alternative way of extracti
ng information about lens mass distribution is to use lensing degeneracies and invariants. Where applicable, they allow one to make conclusions about whole classes of lenses without model fitting. Here, we use approximate, but observationally useful invariants formed by the three relative polar angles of quad images around the lens center to show that many smooth elliptical+shear lenses can reproduce the same set of quad image angles within observational error. This result allows us to show in a model-free way what the general class of smooth elliptical+shear lenses looks like in the three dimensional (3D) space of image relative angles, and that this distribution does not match that of the observed quads. We conclude that, even though smooth elliptical+shear lenses can reproduce individual quads, they cannot reproduce the quad population. What is likely needed is substructure, with clump masses larger than those responsible for flux ratio anomalies in quads, or luminous or dark nearby perturber galaxies.
It has been shown in previous work that DARKexp, which is a theoretically derived, maximum entropy, one shape parameter model for isotropic collisionless systems, provides very good fits to simulated and observed dark-matter halos. Specifically, it f
its the energy distribution, N(E), and the density profiles, including the central cusp. Here, we extend DARKexp N(E) to include the distribution in angular momentum, L^2, for spherically symmetric systems. First, we argue, based on theoretical, semi-analytical, and simulation results, that while dark-matter halos are relaxed in energy, they are not nearly as relaxed in angular momentum, which precludes using maximum entropy to uniquely derive N(E,L^2). Instead, we require that when integrating N(E,L^2) over squared angular momenta one retrieves the DARKexp N(E). Starting with a general expression for N(E,L^2) we show how the distribution of particles in L^2 is related to the shape of the velocity distribution function, VDF, and velocity anisotropy profile, beta(r). We then demonstrate that astrophysically realistic halos, as judged by the VDF shape and beta(r), must have linear or convex distributions in L^2, for each separate energy bin. The distribution in energy of the most bound particles must be nearly flat, and become more tilted in favor of radial orbits for less bound particles. These results are consistent with numerical simulations and represent an important step towards deriving the full distribution function for spherically symmetric dark-matter halos.
{Abridged} We show that the surface brightness (SB) profiles of elliptical galaxies can be parametrized using a linear superposition of 2-3 components, described by functions developed in Dhar & Williams as the 2D projections of a 3D Einasto profile.
For a sample of 23 ellipticals with -24 < Mv < -15, our multi-component models span a range of up to 10^6 in SB and 10^5 in radius, have a median rms of 0.032 mag arcsec^-2, and are statistically justified at >3{sigma}. Our models indicate that i) the central component is more concentrated than the outer component; and ii) the central component of core galaxies is much more luminous, extended and concentrated than that of cuspy galaxies, with their near exponential central profiles indicating disk-like systems whose existence must be verified spectroscopically. While such central excess components are not necessarily contrary to the notion of a mass deficit in core galaxies, we show that the existence, amount, radial extent and sign of mass deficits disagree substantially in the literature, both for a given galaxy and on an average over a sample. We discuss possible implications and suggest that SMBH binaries are unlikely to be the sole mechanism for producing the large cores. We also deduce conditions under which the 3D light density can be described with a multi-component Einasto model for both cuspy and core galaxies; indicating an universality in the functional form of the 3D density of light in galaxies and dark matter in LCDM N-body haloes. Finally, we show that our result - the outer component of the SB profiles of massive galaxies has 5 < n < 8 - could imply i) a common feature of collisionless systems; and ii) that galaxies with such n for their outer component are dark matter dominated.
Recent advances in N-body simulations of dark matter halos have shown that three-parameter models, in particular the Einasto profile characterized by d ln {rho}(r)/d ln r / r with a shape parameter {alpha} < 0.3, are able to produce better fits to th
e 3D spatial density profiles than two-parameter models like the Navarro, Frenk and White (NFW), and Moore et al. profiles. In this paper, we present for the first time an analytically motivated form for the 2D surface mass density of the Einasto family of dark matter haloes, in terms of the 3D spatial density parameters for a wide range of the shape parameter 0.1 < {alpha} < 1. Our model describes a projected (2D) Einasto profile remarkably well between 0 and (3 - 5) r_{200}, with errors less than 0.3 per cent for {alpha} < 0.3 and less than 2 per cent for {alpha} as large as 1. This model (in 2D) can thus be used to fit strong and weak lensing observations of galaxies and clusters whose total spatial (3D) density distributions are believed to be Einasto-like. Further, given the dependence of our model on the 3D parameters, one can reliably estimate structural parameters of the spatial (3D) density from 2D observations. We also consider a Sersic-like parametrization for the above family of projected Einasto profiles and observe that fits with a Sersic profile are sensitive to whether one fits the projected density in linear scale or logarithmic scale and yield widely varying results. Structural parameters of Einasto-like systems, inferred from fits with a Sersic profile, should be used with caution.
