Abstract
In this work we develop a method to incorporate the information from galaxy kinematics into the reconstruction of the two-dimensional, projected gravitational potential of galaxy clusters. We start by deprojecting the observed line-of-sight velocity dispersions of cluster galaxies with an application of Bayes' theorem, the Richardson-Lucy method, requiring the assumption of a shape for the cluster. Assuming spherical symmetry, after the deprojection we obtain an effective galaxy pressure, i.e. the density-weighted radial velocity dispersions of the cluster galaxies, which is then related to the three-dimensional gravitational potential by using the tested assumption of a polytropic relation between the effective galaxy pressure and the density. The two-dimensional gravitational potential can finally be found by straightforward projection along the line of sight. We test the method with a numerically simulated triaxial galaxy cluster and the galaxies identified therein and perform the reconstruction for three different lines of sight, initially assuming sphericity. Expanding the gravitational potential in the cluster's geometrical ellipticities yields second-order corrections to the spherical reconstruction. By comparing our results with the projected gravitational potential directly obtained from the simulation, we show that the deviation between the projected potential obtained with our reconstruction method and the potential directly extracted from the simulation is $\lesssim10~\%$ within approximately the virial radius ($1.5\,h^{-1}\mathrm{Mpc}$) from the cluster centre in the case of a spherical cluster and remains moderate (below $10\,\%-25\,\%$) within the same radius in the case of an ellipsoidal cluster.
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