A Unifying Framework for Self-consistent Gravitational Lensing and Stellar Dynamics Analyses of Early-Type Galaxies

Gravitational lensing and stellar dynamics are two independent methods, based solely on gravity, to study the mass distributions of galaxies. Both methods suffer from degeneracies, however, that are difficult to break. In this paper we present a new framework that self-consistently unifies gravitational lensing and stellar dynamics, breaking some classical degeneracies that have limited their individual usage, particularly in the study of high-redshift galaxies. For any given galaxy potential, the mapping of both the unknown lensed source brightness distribution and the stellar phase-space distribution function onto the photometric and kinematic observables can be cast as a single set of coupled linear equations, which are solved by maximizing the likelihood penalty function. The Bayesian evidence penalty function subsequently allows one to find the best potential-model parameters and to quantitatively rank potential-model families or other model assumptions (e.g., PSF). We have implemented a fast algorithm that solves for the maximum-likelihood pixelized lensed source brightness distribution and the two-integral stellar phase-space distribution function [unk], assuming axisymmetric potentials. To make the method practical, we have devised a new Monte Carlo approach to Schwarzschild's orbital superposition method, based on the superposition of two-integral (E and Lx) toroidal components, to find the maximum-likelihood two-integral distribution function in a matter of seconds in any axisymmetric potential. The nonlinear parameters of the potential are subsequently found through a hybrid MCMC and Simplex optimization of the evidence. Illustrated by the power-law potential models of Evans, we show that the inclusion of stellar kinematic constraints allows the correct linear and nonlinear model parameters to be recovered, including the potential strength, oblateness, and inclination, which, in the case of gravitational-lensing constraints only, would otherwise be fully degenerate.