Reconstruction of the early Universe as a convex optimization problem

TL;DR

This paper presents a novel convex optimization approach using Monge-Ampere-Kantorovich theory to reconstruct the Universe's early mass distribution from present data, achieving over 60% accuracy at large scales.

Contribution

It introduces an effective optimal mass transportation method for cosmological reconstruction, integrating advanced optimization tools previously unused in this field.

Findings

Over 60% of points exactly reconstructed in simulations

Reconstruction is unique above 6 Mpc/h scales

Method is computationally efficient with cubic time complexity

Abstract

We show that the deterministic past history of the Universe can be uniquely reconstructed from the knowledge of the present mass density field, the latter being inferred from the 3D distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those scales -- a few Mpc -- where multi-streaming becomes significant. Above 6 Mpc/h we propose and implement an effective Monge-Ampere-Kantorovich method of unique reconstruction. At such scales the Zel'dovich approximation is well satisfied and reconstruction becomes an instance of optimal mass transportation, a problem which goes back to Monge (1781). After discretization into N point masses one obtains an assignment problem that can be handled by effective algorithms with not more than cubic time complexity in N and reasonable CPU time requirements. Testing against N-body cosmological simulations gives over 60% of exactly reconstructed points. We apply several interrelated tools from optimization theory that were not used in cosmological reconstruction before, such as the Monge-Ampere equation, its relation to the mass transportation problem, the Kantorovich duality and the auction algorithm for optimal assignment. Self-contained discussion of relevant notions and techniques is provided.