# Decorrelating the errors of the galaxy correlation function with compact transformation matrices

## TL;DR

This paper introduces a method using compact transformation matrices, inspired by Cholesky decomposition, to decorrelate errors in galaxy correlation functions, reducing covariance complexity and improving error estimation with fewer mocks.

## Contribution

The authors develop simple, scale-compact transformation matrices that effectively decorrelate galaxy correlation function errors, simplifying covariance estimation in cosmology.

## Key findings

- Transformations suppress off-diagonal covariances by ~95% and ~87%.
- Method reduces the number of mocks needed for Fisher matrix fitting.
- Transforms recover original correlation structure with largely decorrelated errors.

## Abstract

Covariance matrix estimation is a persistent challenge for cosmology, often requiring a large number of synthetic mock catalogues. The off-diagonal components of the covariance matrix also make it difficult to show representative error bars on the 2-point correlation function (2PCF), since errors computed from the diagonal values of the covariance matrix greatly underestimate the uncertainties. We develop a routine for decorrelating the projected and anisotropic 2PCF with simple and scale-compact transformations on the 2PCF. These transformation matrices are modeled after the Cholesky decomposition and the symmetric square root of the Fisher matrix. Using mock catalogues, we show that the transformed projected and anisotropic 2PCF recover the same structure as the original 2PCF, while producing largely decorrelated error bars. Specifically, we propose simple Cholesky based transformation matrices that suppress the off-diagonal covariances on the projected 2PCF by ~95% and that on the anisotropic 2PCF by ~87%. These transformations also serve as highly regularized models of the Fisher matrix, compressing the degrees of freedom so that one can fit for the Fisher matrix with a much smaller number of mocks.