Renormalization group computation of likelihood functions for cosmological data sets

TL;DR

This paper introduces a renormalization group method to efficiently compute likelihood functions for large-scale cosmological data, significantly reducing computational complexity without relying on Monte Carlo sampling.

Contribution

The paper presents a novel RG-based approach for likelihood computation in cosmology, enabling near-exact results with linear time complexity and no Monte Carlo methods.

Findings

Allows near-exact likelihood computation in O(N) time for Gaussian fields

No sampling or Monte Carlo methods required

Applicable to non-linear and non-Gaussian regimes in a perturbative framework

Abstract

I show how a renormalization group (RG) method can be used to incrementally integrate the information in cosmological large-scale structure data sets (including CMB, galaxy redshift surveys, etc.). I show numerical tests for Gaussian fields, where the method allows arbitrarily close to exact computation of the likelihood function in order $\sim N$ time, even for problems with no symmetry, compared to $N^3$ for brute force linear algebra (where $N$ is the number of data points -- to be fair, methods already exist to solve the Gaussian problem in at worst $N \log N$ time, and this method will not necessarily be faster in practice). The method requires no sampling or other Monte Carlo (random) element. Non-linearity/non-Gaussianity can be accounted for to the extent that terms generated by integrating out small scale modes can be projected onto a sufficient basis, e.g., at least in the sufficiently perturbative regime. The formulas to evaluate are straightforward and require no understanding of quantum field theory, but this paper may also serve as a pedagogical introduction to Wilsonian RG for astronomers.