Quadratic estimator for CMB cross-correlation

The quadratic maximum likelihood estimator can be used to reconstruct the cosmic microwave background (CMB) power spectra with minimal error bars. Still, it requires an accurate estimate of the data set's noise covariance matrix in order to be corrected for spurious bias. We describe an extension of this method to cross-correlation, thus removing noise bias and mitigating the impact of systematic effects, providing that they are uncorrelated. This estimator is tested on two simulation surveys at large and intermediate angular scales, respectively corresponding to satellite and ground-based CMB experiments. The analysis focuses on polarization maps, over a wide range of noise levels from 0. 1 to 50 arcmin. We show how this estimator minimizes the increase of variance due to polarization leakage between E and B modes. We compare this method with the pure pseudospectrum formalism, which is computationally faster but less optimal, especially on large angular scales.