Inference of the optical depth to reionization from low multipole temperature and polarisation Planck data

This paper explores methods for constructing low multipole temperature and polarisation likelihoods from maps of the cosmic microwave background anisotropies that have complex noise properties and partial sky coverage. We use Planck 2018 High Frequency Instrument (HFI) and updated SRoll2 temperature and polarisation maps to test our methods. We present three likelihood approximations based on quadratic cross spectrum estimators: (i) a variant of the simulation-based likelihood (SimBaL) techniques used in the Planck legacy papers to produce a low multipole EE likelihood; (ii) a semi-analytical likelihood approximation (momento) based on the principle of maximum entropy; (iii) a density-estimation `likelihood-free scheme (DELFI). Approaches (ii) and (iii) can be generalised to produce low multipole joint temperature-polarisation (TTTEEE) likelihoods. We present extensive tests of these methods on simulations with realistic correlated noise. We then analyse the Planck data and confirm the robustness of our method and likelihoods on multiple inter- and intra-frequency detector set combinations of SRoll2 maps. The three likelihood techniques give consistent results and support a low value of the optical depth to reoinization, tau, from the HFI. Our best estimate of tau comes from combining the low multipole SRoll2 momento (TTTEEE) likelihood with the CamSpec high multipole likelihood and is tau = 0.0627+0.0050-0.0058. This is consistent with the SRoll2 teams determination of tau, though slightly higher by 0.5 sigma, mainly because of our joint treatment of temperature and polarisation.