Hybrid estimation of cosmic microwave background polarization power spectra

This paper generalizes the hybrid power spectrum estimation method developed in a previous paper to the estimation of polarization power spectra of the cosmic microwave background radiation. The hybrid power spectrum estimator is unbiased and we show that it is close to optimal at all multipoles, provided the pixel noise satisfies certain constraints. Furthermore, the hybrid estimator is computationally fast and can easily be incorporated in a Monte Carlo chain for Planck-sized data sets, which may be essential to account for realistic noise covariance and systematic errors. Simple formulae are given for the covariance matrices, including uncorrelated inhomogeneous instrumental noise, and these are tested extensively against numerical simulations. We compare the behaviour of simple pseudo-Cℓ (PCL) estimates with maximum-likelihood estimates at low multipoles. For realistic sky cuts, maximum-likelihood estimates reduce very significantly the mixing of E and B modes. To achieve limits on the scalar–tensor ratio of r≪ 0.1 from sky maps with realistic sky cuts, maximum-likelihood methods, or PCL estimators based on unambiguous E and B modes, will be essential.