A Fast and Accurate Analytic Method of Calculating Galaxy Two-point Correlation Functions

TL;DR

This paper introduces a new analytic method for calculating galaxy two-point correlation functions that is significantly faster and more accurate than traditional methods, especially for ideal survey geometries.

Contribution

The paper presents a novel analytic approach to compute galaxy two-point correlation functions with perfect accuracy and zero variance, drastically reducing computation time for ideal survey geometries.

Findings

Calculates RR and DR with perfect accuracy and zero variance.

Achieves 3 to 6 orders of magnitude speedup over Monte Carlo methods.

Reduces computation time to under 1 minute for 10 million galaxies.

Abstract

We have developed a new analytic method to calculate the galaxy two-point correlation functions (TPCFs) accurately and efficiently, applicable to surveys with finite, regular, and mask-free geometries. We have derived simple, accurate formulas of the normalized random-random pair counts $RR$ as functions of the survey area dimensions. We have also suggested algorithms to compute the normalized data-random pair counts $DR$ analytically. With all edge corrections fully accounted for analytically, our method computes $RR$ and $DR$ with perfect accuracy and zero variance in $O(1)$ and $O(N_{\rm g})$ time, respectively. We test our method on a galaxy catalogue from the EAGLE simulation. Our method calculates $RR+DR$ at a speed 3 to 6 orders of magnitude faster than the brute-force Monte Carlo method and 2.5 orders of magnitude faster than tree-based algorithms. For a galaxy catalogue with 10 million data points in a cube, this reduces the computation time to under 1 minute on a laptop. Our analytic method is favored over the traditional Monte Carlo method whenever applicable. Some applications in the study of correlation functions and power spectra in cosmological simulations and galaxy surveys are discussed. However, we recognize that its applicability is very limited for realistic surveys with masks, irregular shapes, and/or weighted patterns.