Large-scale CMB temperature and polarization cross-spectra likelihoods

We present a cross-spectra based approach for the analysis of CMB data at large angular scales to constrain the reionization optical depth $\tau$, 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-dataset 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\&Lewis likelihood approximation at large angular scales. The modified HL method (oHL) is powerful and complete. It allows 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 \wmap, \planck-LFI and \planck-HFI experiments, demonstrating their validity over a broad range of cross-spectra configurations.