Mapping the 3D dark matter potential with weak shear

We investigate the practical implementation of Taylor's three‐dimensional gravitational potential reconstruction method using weak gravitational lensing, together with the requisite reconstruction of the lensing potential. This methodology calculates the 3D gravitational potential given a knowledge of shear estimates and redshifts for a set of galaxies. We analytically estimate the noise expected in the reconstructed gravitational field taking into account the uncertainties associated with a finite survey, photometric redshift uncertainty, redshift‐space distortions and multiple scattering events. In order to implement this approach for future data analysis, we simulate the lensing distortion fields due to various mass distributions. We create catalogues of galaxies sampling this distortion in three dimensions, with realistic spatial distribution and intrinsic ellipticity for both ground‐based and space‐based surveys. Using the resulting catalogues of galaxy position and shear, we demonstrate that it is possible to reconstruct the lensing and gravitational potentials with our method. For example, we demonstrate that a typical ground‐based shear survey with redshift limit z = 1 and photometric redshifts with error Δ z = 0.05 is directly able to measure the 3D gravitational potential for mass concentrations ≳10 14 M ⊙ between 0.1 ≲ z ≲ 0.5 , and can statistically measure the potential at much lower mass limits. The intrinsic ellipticity of objects is found to be a serious source of noise for the gravitational potential, which can be overcome by Wiener filtering or examining the potential statistically over many fields. We examine the use of the 3D lensing potential to measure mass and position of clusters in 3D, and to detect clusters behind clusters.