Improving the accuracy of mass reconstructions from weak lensing: from the shear map to the mass distribution

In this paper we provide a statistical analysis of the parameter-free method often used in weak lensing mass reconstructions. It is found that a proper assessment of the errors involved in such a non-local analysis requires the study of the relevant two-point correlation functions. After calculating the two-point correlation function for the reduced shear, we determine the expected error on the inferred mass distribution and on other related quantities, such as the total mass, and derive the error power spectrum. This allows us to optimize the reconstruction method, with respect to the kernel used in the inversion procedure. The optimization process can also be carried out on the basis of a variational principle. In particular, we find that curl-free kernels are bound to lead to more accurate mass reconstructions. Our analytical results clarify the arguments and the numerical simulations by Seitz & Schneider (1996).