Steps towards nonlinear cluster inversion through gravitational distortions II. Generalization of the Kaiser and Squires method.

The weak distortions of high-redshift galaxies caused by gravitational light deflection near clusters of galaxies can be used to reconstruct the projected (two-dimensional) surface mass density of intermediate redshift clusters. This technique, pioneered by Tyson et al., and Kaiser & Squires, is reconsidered in the present paper, where we generalize the inversion equation found by Kaiser & Squires (KS) in several respect. Adopting a different smoothing procedure for the discreetly sampled data (individual galaxy images), we effectively reduce the shot noise in the KS procedure. In particular we show that the best density reconstructions are obtained if the smoothing scale is adapted to the `strength of the signal', which yields a better resolution near the center of the cluster where the distortions are strongest. Furthermore, we point out the importance of boundary effects and demonstrate their disastrous impact on rectangular data fields (CCDs) with large side ratio. Most important, however, is the generalization of the KS method to critical clusters, i.e., to such clusters which are capable of producing multiple images and giant luminous arcs. The corresponding modifications of the inversion procedure are severe; in particular, the resulting inversion equation is much more difficult to solve. As we pointed out in a previous paper (Schneider & Seitz), there exists a local degeneracy if the cluster is critical. We have developed an iteration procedure to solve the inversion equation, which we demonstrate to yield a very accurate reconstruction of the cluster mass distribution. In particular, the mass within the inner few arcminutes of the cluster can be determined with an error of only a few percent, thus demonstrating the potential applicability and accuracy of this method for cluster mass determinations.