A fast semi-discrete optimal transport algorithm for a unique reconstruction of the early Universe (2012.09074v1)

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.

• The paper introduces a fast semi-discrete optimal transport algorithm that reconstructs the early universe's density field from large cosmological datasets.
• It leverages smooth convex optimization and Laguerre cell partitioning to solve the MAK problem with efficient N log N complexity.
• Results from N-body simulations demonstrate precise recovery of cosmic power spectrum features, including baryonic acoustic oscillations.

A Semi-Discrete Optimal Transport Algorithm for Reconstructing the Early Universe

The paper introduces a semi-discrete optimal transport algorithm designed to reconstruct the primordial density fluctuations of the early universe. This method is grounded in the mathematical framework of optimal transport theory, specifically addressing the computational challenges associated with solving the Monge-Ampère-Kantorovich (MAK) problem for cosmological data. Unlike traditional combinatorial solutions, which scale poorly with increasing data volumes, this algorithm leverages smooth convex optimization, rendering it computationally efficient and scalable for large cosmological datasets.

Algorithmic Overview

The developed algorithm computes the optimal transport between an initial, uniform continuous density field and a final, discrete set of point masses corresponding to the current universe's large-scale structure. By constructing Laguerre cells to partition the initial density field, the algorithm transforms the optimization problem into a smooth and convex framework amenable to a Newton method solution. This approach enables the handling of datasets far exceeding capabilities of previous methods, specifically allowing reconstruction of up to $10^7$ particles within hours on consumer-grade computers.

Numerical Implementation and Testing

This semi-discrete approach benefits from an empirical complexity of $N \log N$ and is demonstrated using cosmological $N$-body simulations from the AbacusCosmos suite. The algorithm successfully recovers initial positions of particles and reconstructs key features of the cosmic power spectrum, including the baryonic acoustic oscillations (BAO), with high precision. The results indicate robustness and accuracy in recovering density fields above spatial scales of a few megaparsecs.

Theoretical Implications

The paper's methodological advancements have significant implications for cosmology. By removing the limitations of combinatorial complexity, it offers a practical tool for precision cosmology, especially in the context of BAO, which serve as a standard ruler for cosmological distance measurements. The unique, deterministic solutions provided by this formulation potentially reduce the uncertainties in estimating cosmological parameters and offer a new mechanism for probing the cosmic acceleration and testing theories of gravity.

Future Prospects

The development of this semi-discrete optimal transport algorithm introduces new possibilities for tackling large-scale problems in cosmology efficiently. Its application is not only limited to BAO reconstruction but also extends to other cosmological phenomena, such as probing primordial non-Gaussianity and offering insights into the initial density fluctuations of the universe. Future work may involve adapting this method to accommodate redshift space distortions and integrate nonlinear halo bias, further refining cosmological reconstructions.

Overall, this paper makes a notable contribution to computational cosmology by providing an advanced algorithmic framework that balances efficiency, scalability, and precision, enabling more comprehensive explorations of the universe's formative conditions.