The Monge-Ampère-Kantorovich approach to reconstruction in cosmology

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.

arXiv comment6 pages, 2 figures; based on an invited lecture at the conference "Euler's Equations: 250 Years On" (see http://www.obs-nice.fr/etc7/EE250/); to be published in a special issue of Physica D containing the proceedings of that conference

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• Past and present cosmic structure in the SDSS DR7 main sample
• Iterative initial condition reconstruction
• Isobaric Reconstruction of the Baryonic Acoustic Oscillation
• A fast semi-discrete optimal transport algorithm for a unique reconstruction of the early Universe
• The Monge-Ampere equation: various forms and numerical methods
• Improving baryon acoustic oscillation measurement with the combination of cosmic voids and galaxies
• On the use of stochastic differential geometry for non-equilibrium thermodynamics modeling and control
• Biased tracer reconstruction with halo mass information
• Lagrangian reconstruction of cosmic velocity fields