ECLIPSE: a fast Quadratic Maximum Likelihood estimator for CMB intensity and polarization power spectra

We present ECLIPSE (Efficient Cmb poLarization and Intensity Power Spectra Estimator), an optimized implementation of the Quadratic Maximum Likelihood (QML) method for the estimation of the power spectra of the Cosmic Microwave Background (CMB). This approach allows one to reduce significantly the computational costs associated to this technique, allowing to estimate the power spectra up to higher multipoles than previous implementations. In particular, for a resolution of $N_\mathrm{side}=64$, $\ell_{\mathrm{max}}=192$ and a typical Galactic mask, the number of operations can be reduced by approximately a factor of 1000 in a full analysis including intensity and polarization with respect to an efficient direct implementation of the method. In addition, if one is interested in studying only polarization, it is possible to obtain the power spectra of the E and B modes with a further reduction of computational resources without degrading the results. We also show that for experiments observing a small fraction of the sky, the Fisher matrix becomes singular and, in this case, the standard QML can not be applied. To solve this problem, we have developed a binned version of the method that is unbiased and of minimum variance. We also test the robustness of the QML estimator when the assumed fiducial model differs from that of the sky and show the performance of an iterative approach. Finally, we present a comparison of the results obtained by QML and a pseudo-$C_{\ell}$ estimator (NaMaster) for a next-generation satellite, showing that, as expected, QML produces significantly smaller errors at low multipoles. The ECLIPSE fast QML code developed in this work will be made publicly available.