The Monge-Amp\`ere-Kantorovich approach to reconstruction in cosmology

TL;DR

This paper introduces a new method for reconstructing large-scale cosmic motions from present-day density data, leveraging the Monge-Ampère-Kantorovich approach to improve understanding of the Universe's structure and parameters.

Contribution

It develops an approximate reconstruction technique based on gradient transfer maps, connecting fluid dynamics with cosmological data analysis.

Findings

Constraints on the mean mass density Omega_m.

Estimates of the age of the Universe.

Validation against mock cosmological catalogues.

Abstract

Motion of a continuous fluid can be decomposed into an "incompressible" rearrangement, which preserves the volume of each infinitesimal fluid element, and a gradient map that transfers fluid elements in a way unaffected by any pressure or elasticity (the polar decomposition of Y. Brenier). The Euler equation describes a system whose kinematics is dominated by the incompressible rearrangement. The opposite limit, in which the incompressible component is negligible, corresponds to the Zel'dovich approximation, a model of motion of self-gravitating fluid in cosmology. We present a method of approximate reconstruction of the large-scale proper motions of matter in the Universe from the present-day mass density field. The method is based on recovering the corresponding gradient transfer map. We discuss its algorithmics, tests of the method against mock cosmological catalogues, and its application to observational data, which result in tight constraints on the mean mass density Omega_m and age of the Universe.