Cluster Lens Reconstruction Using Only Observed Local Data { an Improved Nite--eld Inversion Technique

Gravitational light deeection can distort the images of distant sources by its tidal eeects. The population of faint blue galaxies is at suuciently high redshift so that their images are distorted near foreground clusters, with giant luminous arcs being the most spectacular evidence for this eeect. Much weaker distortions, however, can observationally be detected by a statistical analysis of the numerous faint galaxy images, as rst demonstrated by Tyson, Valdes & Wenk. This distortion eeect can be used as a quantitative tool for the reconstruction of the surface mass density of galaxy clusters with appropriate redshifts, as was demonstrated by Kaiser & Squires. They have derived an explicit equation for this surface mass density in terms of its tidal eld. The reconstruction formula by Kaiser & Squires must be modiied because of two eeects: in its original form it applies only to weak lenses, and hence must be generalized to account for stronger lensing eeects. Second, due to the nature of the inversion formula, it produces boundary artefacts (or biases) if applied to real data which are connned to a nite eld on the sky. We discuss several possibilities to obtain inversion formulae which are exact for ideal data on a nite data eld (CCD). We demonstrate that there exists an innnite number of such nite-eld inversion formulae, which diier in their sensitivity to observational eeects (such as noise, intrinsic ellipticities of the sources, etc.). We show that, using two simple conditions, one can uniquely specify a nite-eld formula which in a well-deened sense minimizes the sensitivity to observational eeects. We then use synthetic data to compare the quality of our new reconstruction method with that of previous nite-eld inversion techniques and of the nonlinear generalization of the Kaiser & Squires method. This analysis demonstrates that our new inversion method is superior to the other previously considered nite-eld inversions, and only slightly more noisy than the Kaiser & Squires inversion, but in contrast to the latter, it lacks the boundary artefacts. We shall discuss that the lack of boundary artefacts and the slightly increased noise have the same origin, and that every nite-eld reconstruction must be more noisy than that obtained from the Kaiser & Squires method. Furthermore, the rms deviations of the reconstructed density eld from the input surface mass density are fairly homogeneously distributed over the eld. We therefore conclude that the new method developed here is the …