Computing the Three-Point Correlation Function of Galaxies in $\mathcal{O}(N^2)$ Time

TL;DR

This paper introduces an efficient algorithm for computing the galaxy three-point correlation function that scales similarly to the two-point function, enabling large-scale analysis in practical timeframes for upcoming surveys.

Contribution

The paper presents a novel algorithm that computes the multipole coefficients of the 3PCF without explicit triplet enumeration, significantly improving computational efficiency.

Findings

Algorithm scales with number density like the two-point function

Computes edge-corrected 3PCF in under an hour on modest resources

Achieves 500x speedup over naive triplet counting

Abstract

We present an algorithm that computes the multipole coefficients of the galaxy three-point correlation function (3PCF) without explicitly considering triplets of galaxies. Rather, centering on each galaxy in the survey, it expands the radially-binned density field in spherical harmonics and combines these to form the multipoles without ever requiring the relative angle between a pair about the central. This approach scales with number and number density in the same way as the two-point correlation function, allowing runtimes that are comparable, and 500 times faster than a naive triplet count. It is exact in angle and easily handles edge correction. We demonstrate the algorithm on the LasDamas SDSS-DR7 mock catalogs, computing an edge corrected 3PCF out to $90\;{\rm Mpc}/h$ in under an hour on modest computing resources. We expect this algorithm will render it possible to obtain the large-scale 3PCF for upcoming surveys such as Euclid, LSST, and DESI.