Abstract
ABSTRACT This paper explores methods for constructing low multipole temperature and polarization 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 polarization 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 generalized to produce low multipole joint temperature-polarization (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, τ, from the HFI. Our best estimate of τ comes from combining the low multipole SRoll2momento (TTTEEE) likelihood with the CamSpec high multipole likelihood and is $\tau = 0.0627^{+0.0050}_{-0.0058}$. This is consistent with the SRoll2 team’s determination of τ, though slightly higher by ∼0.5σ, mainly because of our joint treatment of temperature and polarization.
Highlights
Over the last decade, observations of the cosmic microwave background (CMB) (Hinshaw et al 2013; Planck Collaboration 2020d; Henning et al 2018; Aiola et al 2020), together with measurements of the baryon acoustic oscillation scale from large galaxy surveys (Gil-Marín et al 2020; Bautista et al 2020) and many other cosmological observations have transformed cosmology into a high precision science
All of the likelihoods are based on a quadratic cross spectrum (QCS) estimator, which we use to measure the foreground cleaned cross-spectra at low multipoles (2 ≤ l ≤ 29) from the SRoll1 and SRoll2 temperature and polarisation maps4
Dust cleaning in temperature using 353 GHz or higher frequencies removes almost all of the foreground emission at low multipoles at 143 GHz leaving noise-free CMB signal over most of the sky, as discussed in detail in EG19
Summary
Observations of the cosmic microwave background (CMB) (Hinshaw et al 2013; Planck Collaboration 2020d; Henning et al 2018; Aiola et al 2020), together with measurements of the baryon acoustic oscillation scale from large galaxy surveys (Gil-Marín et al 2020; Bautista et al 2020) and many other cosmological observations have transformed cosmology into a high precision science. Systematics in the data may bias the results if they cannot be modelled with fidelity and included in the likelihood These issues are of particular importance for the measurement of the optical depth to reionization τ from Planck temperature and polarisation CMB maps. We apply three likelihood approximations to measure the optical depth to reionization from Planck HFI maps. Assuming the base six parameter ΛCDM model, and adding the Plik high multipole likelihood, the best fit values of τ are higher than those of Eqs. All of the likelihoods are based on a quadratic cross spectrum (QCS) estimator, which we use to measure the foreground cleaned cross-spectra at low multipoles (2 ≤ l ≤ 29) from the SRoll and SRoll temperature and polarisation maps.
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