Large-scale cosmic microwave background temperature and polarization cross-spectra likelihoods

We present a cross-spectra-based approach for the analysis of cosmic microwave background data at large angular scales to constrain the reionization optical depth τ, the tensor to scalar ratio r and the amplitude of the primordial scalar perturbations A_s. With respect to the pixel-based approach developed so far, using cross-spectra has the unique advantage to eliminate spurious noise bias and to give a better handle over residual systematics, allowing to efficiently combine the cosmological information encoded in cross-frequency or cross-data set spectra. We present two solutions to deal with the non-Gaussianity of the hat{C}_ell estimator distributions at large angular scales: the first one relies on an analytical parametrization of the estimator distribution, while the second one is based on modification of the Hamimache and Lewis (HL) likelihood approximation at large angular scales. The modified HL method (oHL) is powerful and complete. It allows us to deal with multipole and mode correlations for a combined temperature and polarization analysis. We validate our likelihoods on numerous simulations that include the realistic noise levels of the Wilkinson Microwave Anisotropy Probe, Planck-Low Frequency Instrument and Planck-High Frequency Instrument experiments, demonstrating their validity over a broad range of cross-spectra configurations.