Back to the early Universe by a Monge-Ampere-Kantorovich mass transportation method

Reconstructing the minute density fluctuations in the early Universe that evolved into a highly clumpy matter distribution, as revealed by the present distribution of luminous matter, constitutes a major challenge of modern cosmology. A number of techniques have been devised in recent years which attempt to achieve this aim by using galaxy positions alone [8 refs.]. However, without knowledge of their velocities, this problem is not well-posed and its solution suffers frequently from lack of uniqueness. Here we make the hypothesis that the map from initial to present locations of mass elements is irrotational. Using recent mathematical work [Brenier], we then relate reconstruction to ``mass transportation'', a well-posed optimisation problem in engineering introduced by Monge in 1781. We propose a new powerful algorithm for unique reconstruction which, when tested against N-body simulations, gives excellent reconstruction down to scales of a few comoving megaparsecs and demonstrates the validity of our hypothesis. The uniqueness and the quality achieved are essential for the analysis of the large datasets expected from new-generation galaxy redshift surveys. [A version with slightly better figures is available at www.obs-nice.fr/etc7/nature.pdf