# On the regularity of the covariance matrix of a discretized scalar field on the sphere

## TL;DR

This paper analyzes the regularity and rank constraints of covariance matrices of discretized scalar fields on the sphere, with applications to CMB data analysis, comparing different pixelization schemes and their numerical stability.

## Contribution

It provides analytical expressions for covariance matrix rank constraints considering pixelization, symmetries, and masks, and evaluates their impact on CMB data analysis.

## Key findings

- HEALPix pixelization yields covariance matrices closer to maximum rank.
- Numerical precision and noise influence matrix regularity and rank.
- Analytical rank constraints assist in regularizing covariance matrices for CMB analysis.

## Abstract

We present a comprehensive study of the regularity of the covariance matrix of a discretized field on the sphere. In a particular situation, the rank of the matrix depends on the number of pixels, the number of spherical harmonics, the symmetries of the pixelization scheme and the presence of a mask. Taking into account the above mentioned components, we provide analytical expressions that constrain the rank of the matrix. They are obtained by expanding the determinant of the covariance matrix as a sum of determinants of matrices made up of spherical harmonics. We investigate these constraints for five different pixelizations that have been used in the context of Cosmic Microwave Background (CMB) data analysis: Cube, Icosahedron, Igloo, GLESP and HEALPix, finding that, at least in the considered cases, the HEALPix pixelization tends to provide a covariance matrix with a rank closer to the maximum expected theoretical value than the other pixelizations. The effect of the propagation of numerical errors in the regularity of the covariance matrix is also studied for different computational precisions, as well as the effect of adding a certain level of noise in order to regularize the matrix. In addition, we investigate the application of the previous results to a particular example that requires the inversion of the covariance matrix: the estimation of the CMB temperature power spectrum through the Quadratic Maximum Likelihood algorithm. Finally, some general considerations in order to achieve a regular covariance matrix are also presented.