Joint reconstruction of galaxy clusters from gravitational lensing and
  thermal gas. II. Inversion of the thermal Sunyaev-Zel'dovich effect
  This paper continues a series in which we intend to show how all observables
of galaxy clusters can be combined to recover the two-dimensional, projected
gravitational potential of individual clusters. Our goal is to develop a
non-parametric algorithm for joint cluster reconstruction taking all cluster
observables into account. In this paper, we begin with the relation between the
Compton-y parameter and the Newtonian gravitational potential, assuming
hydrostatic equilibrium and a polytropic stratification of the intracluster
gas. We show how Richardson-Lucy deconvolution can be used to convert the
intensity change of the CMB due to the thermal Sunyaev-Zel'dovich effect into
an estimate for the two-dimensional gravitational potential. Synthetic data
simulated with characteristics of the ALMA telescope show that the
two-dimensional potential of a cluster with mass 5*10^14 M_sun/h at redshift
0.2 is possible with an error of < 5% between the cluster centre and a radius r
< 0.9 Mpc/h.
