# Disconnected pseudo-$C_\ell$ covariances for projected large-scale structure data

## TL;DR

This paper introduces efficient approximate methods to compute the disconnected Gaussian covariance matrix for large-scale structure data, enabling accurate cosmological inference while significantly reducing computational costs.

## Contribution

It presents scalable algorithms for estimating the Gaussian covariance matrix of power spectra on curved and flat skies, improving computational efficiency for large datasets.

## Key findings

- Methods achieve $O(
obreak \, \ell_{\rm max}^3)$ scaling
- Accurate recovery of cosmological parameters using approximate covariance
- Limitations on reliability at the largest scales and for B modes

## Abstract

The disconnected part of the power spectrum covariance matrix (also known as the "Gaussian" covariance) is the dominant contribution on large scales for galaxy clustering and weak lensing datasets. The presence of a complicated sky mask causes non-trivial correlations between different Fourier/harmonic modes, which must be accurately characterized in order to obtain reliable cosmological constraints. This is particularly relevant for galaxy survey data. Unfortunately, an exact calculation of these correlations involves $O(\ell_{\rm max}^6)$ operations that become computationally impractical very quickly. We present an implementation of approximate methods to estimate the Gaussian covariance matrix of power spectra involving spin-0 and spin-2 flat- and curved-sky fields, expanding on existing algorithms. These methods achieve an $O(\ell_{\rm max}^3)$ scaling, which makes the computation of the covariance matrix as fast as the computation of the power spectrum itself. We quantify the accuracy of these methods on large-scale structure and weak lensing data, making use of a large number of Gaussian but otherwise realistic simulations. We show that, using the approximate covariance matrix, we are able to recover the true posterior distribution of cosmological parameters to high accuracy. We also quantify the shortcomings of these methods, which become unreliable on the very largest scales, as well as for covariance matrix elements involving cosmic shear $B$ modes. The algorithms presented here are implemented in the public code NaMaster (https://github.com/LSSTDESC/NaMaster).