Glass-Like Random Catalogues for Two-Point Estimates on the Light Cone

Abstract

We introduce grlic, a publicly available Python tool for generating glass-like point distributions with a radial density profile $n(r)$ as it is observed in large-scale surveys of galaxy distributions on the past light cone. Utilising these glass-like catalogues, we assess the bias and variance of the Landy-Szalay (LS) estimator of the first three two-point correlation function (2PCF) multipoles in halo and particle catalogues created with the cosmological N-body code gevolution. Our results demonstrate that the LS estimator calculated with the glass catalogues is biased by less than $10^{-4}$ with respect to the estimate derived from Poisson-sampled random catalogues, for all multipoles considered and on all but the smallest scales. Additionally, the estimates derived from glass-like catalogues exhibit significantly smaller standard deviation $\sigma$ than estimates based on commonly used Poisson-sampled random catalogues of comparable size. The standard deviation of the estimate depends on a power of the number of objects $N_R$ in the random catalogue; we find a power law $\sigma \propto N_R^{-0.9}$ for glass-like random catalogues as opposed to $\sigma \propto N_R^{-0.48}$ using Poisson-sampled random catalogues. Given a required precision, this allows for a much reduced number of objects in the glass-like random catalogues used for the LS estimate of the 2PCF multipoles, significantly reducing the computational costs of each estimate.