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