ENCORE: An $\mathcal{O}(N_{\rm g}^2)$ Estimator for Galaxy $N$-Point Correlation Functions

TL;DR

This paper introduces a fast, scalable algorithm for computing N-point correlation functions of 3D density fields, applicable to galaxy surveys and continuous fields, enabling more efficient cosmological analysis.

Contribution

The paper presents the ENCORE algorithm, which reduces the computational complexity of N-point correlation functions to (Ng^2) for arbitrary N, with practical implementation and GPU acceleration.

Findings

Efficient computation of 3PCF to 6PCF within 100 CPU-hours.

Algorithm scales linearly with the number of particles for higher N.

Implementation includes corrections for survey geometry and GPU acceleration.

Abstract

We present a new algorithm for efficiently computing the $N$-point correlation functions (NPCFs) of a 3D density field for arbitrary $N$. This can be applied both to a discrete spectroscopic galaxy survey and a continuous field. By expanding the statistics in a separable basis of isotropic functions built from spherical harmonics, the NPCFs can be estimated by counting pairs of particles in space, leading to an algorithm with complexity $\mathcal{O}(N_{\rm g}^2)$ for $N_{\rm g}$ particles, or $\mathcal{O}\left(N_\mathrm{FFT}\log N_\mathrm{FFT}\right)$ when using a Fast Fourier Transform with $N_\mathrm{FFT}$ grid-points. In practice, the rate-limiting step for $N>3$ will often be the summation of the histogrammed spherical harmonic coefficients, particularly if the number of radial and angular bins is large. In this case, the algorithm scales linearly with $N_{\rm g}$. The approach is implemented in the ENCORE code, which can compute the 3PCF, 4PCF, 5PCF, and 6PCF of a BOSS-like galaxy survey in $\sim$ $100$ CPU-hours, including the corrections necessary for non-uniform survey geometries. We discuss the implementation in depth, along with its GPU acceleration, and provide practical demonstration on realistic galaxy catalogs. Our approach can be straightforwardly applied to current and future datasets to unlock the potential of constraining cosmology from the higher-point functions.