Reconstructing the projected gravitational potential of galaxy clusters from galaxy kinematics

We have developed a method for reconstructing the two-dimensional, projected gravitational potential of galaxy clusters from observed line-of-sight velocity dispersions of cluster galaxies. It is the second in an intended series of papers aiming at a unique reconstruction method for cluster potentials that combine lensing, X-ray, Sunyaev-Zel’dovich and kinematic data. The observed galaxy velocity dispersions are deprojected using the Richardson-Lucy algorithm. The obtained radial velocity dispersions are then related to the gravitational potential by using the tested assumption of a polytropic relation between the effective galaxy pressure and the density. Once the gravitational potential is obtained in three dimensions, projection along the line of sight yields the two-dimensional potential. For simplicity we adopt spherical symmetry and a known profile for the anisotropy parameter of the galaxy velocity dispersions. We tested the method with a numerically simulated galaxy cluster and the galaxies identified therein and performed the reconstruction for three different lines of sight. We extracted a projected velocity-dispersion profile from the simulated cluster and passed it through our algorithm, showing that the deviation between the true and the reconstructed gravitational potential is ≲10% within ≈ 1.5 h-1 Mpc from the cluster centre