Unfolding the matter distribution using 3-D weak gravitational lensing

Combining redshift and galaxy shape information offers new exciting ways of exploiting the gravitational lensing effect for studying the large scales of the cosmos. One application is the three-dimensional reconstruction of the matter density distribution which is explored in this paper. We give a generalisation of an already known minimum-variance estimator of the 3-D matter density distribution that facilitates the combination of thin redshift slices of sources with samples of broad redshift distributions for an optimal reconstruction. We show how, in principle, intrinsic alignments of source ellipticities or shear/intrinsic alignment correlations can be accommodated, albeit these effects are not the focus of this paper. We describe an efficient and fast way to implement the estimator on a contemporary desktop computer. Analytic estimates for the noise and biases in the reconstruction are given. The bias -- a spread and shift of structures in radial direction -- can be expressed in terms of a radial PSF, comprising the limitations of the reconstruction due to source shot-noise and the unavoidably broad lensing kernel. We conclude that a 3-D mass-density reconstruction on galaxy cluster scales is feasible but, for foreseeable surveys, a map with a S/N>~3 threshold is limited to structures with M200>~10^14 Msol/h, or 7x10^14 Msol/h, at low to moderate redshifts (z=0.1 or 0.6). However, we find that a heavily smoothed full-sky map of the very large-scale density field may also be possible as the S/N of reconstructed modes increases towards larger scales. Future improvements of the method can be obtained by including higher-order lensing information (flexion) which could also be implemented into our algorithm. [ABRIDGED]