Steps towards nonlinear cluster inversion through gravitational distortions. I. Basic considerations and circular clusters.

The distortion of images of faint, high-redshift galaxies by light deflection at foreground clusters of galaxies can be used to determine the (projected) mass distribution of the clusters. In the case of strong distortions, which lead to arcs in clusters, the position of the arc and/or its radius of curvature yields an estimate for the total mass inside a circle traced out by the arc. Weak distortions, which can be observed to much larger angular separations, can be used to determine the mass profile. In the case of weak distortions, an approximation which identifies the observed distortion with the shear produced by the lens can be made; this (linear) approximation breaks down, however, if one wants to probe the center of the cluster, i.e., approach the region within which giant arcs can be formed. The methods developed hitherto, the most advanced of which is due to Kaiser & Squires, rely on the linear approximation and thus cannot yield reliable results for regions close to the center of the cluster. The purpose of this paper is to provide a theoretical basis to extend the Kaiser & Squires method into the nonlinear regime, thereby making it more powerful, since the nonlinear distortions provide strong constraints on the mass profile, which in combination with the weak distortions should yield more reliable cluster inversions. We discuss the statistical properties of the observable image ellipticities and provide several methods to determine the local distortion by the lens from observed galaxy images, some of which rely only on the orientations of the images, not on their ellipticities. Analytic approximations for the error in the locally determined distortion are provided. An invariance transformation of the density profile of the cluster is derived which leaves the observable distortion invariant; this transformation differs from that valid in the linear approximation where it simply corresponds to the addition of a homogeneous matter sheet. We then investigate the inversion of spherical clusters; in particular, we show the precision with which the center of a spherical cluster can be determined, both with and without usage of the absolute value of the image ellipticities. We show that inclusion of the absolute value of the ellipticities increases the accuracy of the center position considerably. Finally, we study the fully nonlinear inversion of spherical clusters in some detail, to illustrate the difficulties one has to anticipate in nonlinear reconstructions of realistic two-dimensional matter distributions. We compare the direct nonlinear inversion with the method of fitting parametrized mass profiles to the distortion data, and point out relative strengths and shortcomings. In particular, we emphasize the influence of the aforementioned global invariance transformation and an additional local degeneracy.