Cosmic microwave background temperature and polarization pseudo-Cℓestimators and covariances

We develop the pseudo-Cl method for reconstructing the Cosmic Microwave Background (CMB) temperature and polarization auto- and cross-power spectra, and estimate the pseudo-Cl covariance matrix for a realistic experiment on the cut sky. We calculate the full coupling equations for all six possible CMB power spectra, relating the observed pseudo-Cl's to the underlying all-sky Cl's, and test the reconstruction on both full-sky and cut-sky simulated CMB data sets. In particular we consider the reconstruction of the Cl from upcoming ground-based polarization experiments covering areas of a few hundred sq. degrees and find that the method is fast, unbiased and performs well over a wide range of multipoles from l=2 to l=2500. We then calculate the full covariance matrix between the modes of the pseudo-temperature and polarization power spectra, assuming that the underlying CMB fields are Gaussian randomly distributed. The complexity of the covariance matrix prohibits its rapid calculation, required for parameter estimation. Hence we present an approximation for the covariance matrix in terms of convolutions of the underlying power spectra. The coupling matrices in these expressions can be estimated by fitting to numerical simulations, circumventing direct and slow calculation, and further, inaccurate analytic approximations. We show that these coupling matrices are mostly independent of cosmology, and that the full covariance matrix for all six pseudo-Cl power spectra can be quickly and accurately calculated for any given cosmological model using this method. We compare these semi-analytic covariance matrices against simulations and find good agreement, the accuracy of which depends mainly on survey area and the range of cosmological parameters considered.