Three-Dimensional Reconstruction of the Density Field: An SVD Approach to Weak-Lensing Tomography

We present a new method for constructing three-dimensional mass maps from gravitational lensing shear data. We solve the lensing inversion problem using truncation of singular values (within the context of generalized least-squares estimation) without a priori assumptions about the statistical nature of the signal. This singular value framework allows a quantitative comparison between different filtering methods: we evaluate our method beside the previously explored Wiener-filter approaches. Our method yields near-optimal angular resolution of the lensing reconstruction and allows cluster sized halos to be de-blended robustly. It allows for mass reconstructions which are two to three orders of magnitude faster than the Wiener-filter approach; in particular, we estimate that an all-sky reconstruction with arcminute resolution could be performed on a timescale of hours. We find however that linear, non-parametric reconstructions have a fundamental limitation in the resolution achieved in the redshift direction.