A compressed sensing approach to 3D weak lensing

Laboratoire AIM, UMR CEA-CNRS-Paris 7, Irfu, SAp/SEDI, Service d’Astrophysique, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France
e-mail: adrienne.leonard@cea.fr; francois-xavier.dupe@lif.univ-mrs.fr

Received: 6 July 2011
Accepted: 29 November 2011

Abstract

Context. Weak gravitational lensing is an ideal probe of the dark universe. In recent years, several linear methods have been developed to reconstruct the density distribution in the Universe in three dimensions, making use of photometric redshift information to determine the radial distribution of lensed sources.

Aims. We aim to address three key problems seen in these methods; namely, the bias in the redshifts of detected objects, the line-of-sight smearing seen in reconstructions, and the damping of the amplitude of the reconstruction relative to the underlying density. We also aim to detect structures at higher redshifts than have previously been achieved, and to improve the line-of-sight resolution of our reconstructions.

Methods. We considered the problem under the framework of compressed sensing (CS). Under the assumption that the data are sparse or compressible in an appropriate dictionary, we constructed a robust estimator and employ state-of-the-art convex optimisation methods to reconstruct the density contrast. For simplicity in implementation, and as a proof of concept of our method, we reduced the problem to one dimension, considering the reconstruction along each line of sight independently. We also assumed an idealised survey in which the redshifts of sources are known.

Results. Despite the loss of information inherent in our one-dimensional implementation, we demonstrate that our method is able to accurately reproduce cluster haloes up to a redshift of z_cl = 1.0, deeper than state-of-the-art linear methods. We directly compare our method with these linear methods, and demonstrate minimal radial smearing and redshift bias in our reconstructions, as well as a reduced damping of the reconstruction amplitude as compared to the linear methods. In addition, the CS framework allows us to consider an underdetermined inverse problem, thereby allowing us to reconstruct the density contrast at finer resolution than the input data.

Conclusions. The CS approach allows us to recover the density distribution more accurately than current state-of-the-art linear methods. Specifically, it addresses three key problem areas inherent in linear methods. Moreover, we are able to achieve super-resolution and increased high-redshift sensitivity in our reconstructions.