Optimal Boltzmann hierarchies with non-vanishing spatial curvature
  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.
