Euclid: Fast two-point correlation function covariance through linear construction

TL;DR

The paper introduces a fast linear construction method for estimating the covariance matrix of the two-point galaxy correlation function, significantly reducing computational costs while maintaining accuracy.

Contribution

A novel linear construction approach that estimates covariance matrices efficiently using small random catalogs, validated with simulations and applicable to large datasets.

Findings

Unbiased covariance estimates with small random catalogs.

Speed-up factor of 14 compared to standard methods.

Effective for large-scale galaxy correlation analyses.

Abstract

We present a method for fast evaluation of the covariance matrix for a two-point galaxy correlation function (2PCF) measured with the Landy-Szalay estimator. The standard way of evaluating the covariance matrix consists in running the estimator on a large number of mock catalogs, and evaluating their sample covariance. With large random catalog sizes (data-to-random objects ratio M>>1) the computational cost of the standard method is dominated by that of counting the data-random and random-random pairs, while the uncertainty of the estimate is dominated by that of data-data pairs. We present a method called Linear Construction (LC), where the covariance is estimated for small random catalogs of size M = 1 and M = 2, and the covariance for arbitrary M is constructed as a linear combination of these. We validate the method with PINOCCHIO simulations in range r = 20-200 Mpc/h, and show that the covariance estimate is unbiased. With M = 50 and with 2 Mpc/h bins, the theoretical speed-up of the method is a factor of 14. We discuss the impact on the precision matrix and parameter estimation, and derive a formula for the covariance of covariance.