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

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.