A fast semi-discrete optimal transport algorithm for a unique reconstruction of the early Universe

TL;DR

This paper introduces an efficient, scalable algorithm based on optimal transport theory for reconstructing the early Universe's density fluctuations, capable of handling millions of particles quickly and accurately.

Contribution

The authors develop a novel semi-discrete optimal transport algorithm that scales to millions of particles, enabling fast and precise reconstruction of the primordial density field.

Findings

Reconstructed initial particle positions for 10^7 particles within hours.

Successfully recovered subtle features like baryonic acoustic oscillations.

Algorithm outperforms existing methods in scalability and accuracy.

Abstract

We leverage powerful mathematical tools stemming from optimal transport theory and transform them into an efficient algorithm to reconstruct the fluctuations of the primordial density field, built on solving the Monge-Amp\`ere-Kantorovich equation. Our algorithm computes the optimal transport between an initial uniform continuous density field, partitioned into Laguerre cells, and a final input set of discrete point masses, linking the early to the late Universe. While existing early universe reconstruction algorithms based on fully discrete combinatorial methods are limited to a few hundred thousand points, our algorithm scales up well beyond this limit, since it takes the form of a well-posed smooth convex optimization problem, solved using a Newton method. We run our algorithm on cosmological $N$-body simulations, from the AbacusCosmos suite, and reconstruct the initial positions of $\mathcal{O}(10^7)$ particles within a few hours with an off-the-shelf personal computer. We show that our method allows a unique, fast and precise recovery of subtle features of the initial power spectrum, such as the baryonic acoustic oscillations.