Reconstruction of the two-dimensional gravitational potential of galaxy
  clusters from X-ray and Sunyaev-Zel'dovich measurements
  The mass of galaxy clusters is not a direct observable, nonetheless it is
commonly used to probe cosmological models. Based on the combination of all
main cluster observables, that is, the X-ray emission, the thermal
Sunyaev-Zel'dovich (SZ) signal, the velocity dispersion of the cluster
galaxies, and gravitational lensing, the gravitational potential of galaxy
clusters can be jointly reconstructed. We derive the two main ingredients
required for this joint reconstruction: the potentials individually
reconstructed from the observables and their covariance matrices, which act as
a weight in the joint reconstruction. We show here the method to derive these
quantities. The result of the joint reconstruction applied to a real cluster
will be discussed in a forthcoming paper. We apply the Richardson-Lucy
deprojection algorithm to data on a two-dimensional (2D) grid. We first test
the 2D deprojection algorithm on a $\beta$-profile. Assuming hydrostatic
equilibrium, we further reconstruct the gravitational potential of a simulated
galaxy cluster based on synthetic SZ and X-ray data. We then reconstruct the
projected gravitational potential of the massive and dynamically active cluster
Abell 2142, based on the X-ray observations collected with XMM-Newton and the
SZ observations from the Planck satellite. Finally, we compute the covariance
matrix of the projected reconstructed potential of the cluster Abell 2142 based
on the X-ray measurements collected with XMM-Newton. The gravitational
potentials of the simulated cluster recovered from synthetic X-ray and SZ data
are consistent, even though the potential reconstructed from X-rays shows
larger deviations from the true potential. Regarding Abell 2142, the projected
gravitational cluster potentials recovered from SZ and X-ray data reproduce
well the projected potential inferred from gravitational-lensing observations.
(abridged)
