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 less than or similar to 10% within approximate to 1.5 h(-1) Mpc from the cluster centre.