Optimal Boltzmann hierarchies with non-vanishing spatial curvature

TL;DR

This paper examines the impact of spatial curvature on the optimal Boltzmann hierarchy used in cosmological perturbation theory, showing it remains accurate for temperature and polarization spectra but introduces significant errors for tensor polarization.

Contribution

It analyzes how the assumption of separability in the optimal hierarchy affects accuracy in curved universes, providing corrected error estimates.

Findings

Optimal hierarchy yields very accurate temperature and polarization spectra despite curvature.

Errors in tensor polarization spectra can reach about 50% of the curvature parameter.

Separable eigenfunction assumption introduces errors proportional to spatial curvature.

Abstract

Within cosmological perturbation theory, the cosmic microwave background anisotropies are usually computed from a Boltzmann hierarchy coupled to the perturbed Einstein equations. In this setup, one set of multipoles describes the temperature anisotropies, while two other sets, of electric and magnetic types, describe the polarization anisotropies. In order to reduce the number of multipoles types needed for polarization, and thus to speed up the numerical resolution, an optimal hierarchy has been proposed in the literature for Einstein-Boltzmann codes. However, it has been recently shown that the separability between directional and orbital eigenfunctions employed in the optimal hierarchy is not correct in the presence of spatial curvature. We investigate how the assumption of separability affects the optimal hierarchy, and show that it introduces relative errors of order $\Omega_K$ with respect to the full hierarchy. Despite of that, we show that the optimal hierarchy still gives extremely good results for temperature and polarization angular spectra, with relative errors that are much smaller than cosmic variance even for curvatures as large as $|\Omega_K|=0.1$. Still, we find that the polarization angular spectra from tensor perturbations are significantly altered when using the optimal hierarchy, leading to errors that are typically of order $50 |\Omega_K| \%$ on that component.