Properties and use of CMB power spectrum likelihoods

TL;DR

This paper evaluates the accuracy and optimality of likelihood approximation methods for small-scale CMB observations, focusing on covariance estimation, estimator efficiency, and combining different likelihood approaches for improved parameter inference.

Contribution

It analyzes the optimality of hybrid pseudo-C_l estimators, assesses covariance estimation from simulations, and proposes combining approximate and exact likelihoods across scales.

Findings

Hybrid estimators have non-negligible information loss but limited impact on parameters.

Covariance estimation requires a substantial number of simulations, improved by analytic approximations.

Combining high-ell and low-ell likelihoods enhances accuracy across all scales.

Abstract

Fast robust methods for calculating likelihoods from CMB observations on small scales generally rely on approximations based on a set of power spectrum estimators and their covariances. We investigate the optimality of these approximation, how accurate the covariance needs to be, and how to estimate the covariance from simulations. For a simple case with azimuthal symmetry we compare optimality of hybrid pseudo-C_l CMB power spectrum estimators with the exact result, indicating that the loss of information is not negligible, but neither is it enough to have a large effect on standard parameter constraints. We then discuss the number of samples required to estimate the covariance from simulations, with and without a good analytic approximation, and assess the use of shrinkage estimators. Finally we discuss how to combine an approximate high-ell likelihood with a more exact low-ell harmonic-space likelihood as a practical method for accurate likelihood calculation on all scales.