Fast analytical calculation of the random pair counts for realistic survey geometry

TL;DR

This paper introduces an analytical method to efficiently compute random pair counts in galaxy surveys, eliminating the need for large random catalogues and significantly reducing computational costs while maintaining high accuracy.

Contribution

The authors derive an analytical expression for anisotropic random pair counts that accounts for survey geometry and galaxy distribution, improving efficiency over traditional methods.

Findings

Analytical calculation matches well with traditional random catalogue results.

Primary calculation takes only a few minutes on a single CPU.

Achieves monopole accuracy comparable to using 1500 times more random points.

Abstract

Galaxy clustering is a standard cosmological probe that is commonly analysed through two-point statistics. In observations, the estimation of the two-point correlation function crucially relies on counting pairs in a random catalogue. The latter contains a large number of randomly distributed points, which accounts for the survey window function. Random pair counts can also be advantageously used for modelling the window function in the observed power spectrum. Since pair counting scales as $\mathcal{O}(N^2)$, where $N$ is the number of points, the computational time to measure random pair counts can be very expensive for large surveys. In this work, we present an alternative approach for estimating those counts that does not rely on the use of a random catalogue. We derived an analytical expression for the anisotropic random-random pair counts that accounts for the galaxy radial distance distribution, survey geometry, and possible galaxy weights. Considering the cases of the VIPERS and SDSS-BOSS redshift surveys, we find that the analytical calculation is in excellent agreement with the pair counts obtained from random catalogues. The main advantage of this approach is that the primary calculation only takes a few minutes on a single CPU and it does not depend on the number of random points. Furthermore, it allows for an accuracy on the monopole equivalent to what we would otherwise obtain when using a random catalogue with about 1500 times more points than in the data at hand. We also describe and test an approximate expression for data-random pair counts that is less accurate than for random-random counts, but still provides subpercent accuracy on the monopole. The presented formalism should be very useful in accounting for the window function in next-generation surveys, which will necessitate accurate two-point window function estimates over huge observed cosmological volumes.