Fast Scalar Quadratic Maximum Likelihood Estimators for the CMB B-mode Power Spectrum

Constructing a fast and efficient estimator for the B-mode power spectrum of the cosmic microwave background (CMB) is of critical importance for CMB science. For a general CMB survey, the Quadratic Maximum Likelihood (QML) estimator for CMB polarization has been proved to be the optimal estimator with minimal uncertainties, but it is computationally very expensive. In this article, we propose two new QML methods for B-mode power spectrum estimation. We use the Smith-Zaldarriaga approach to prepare the pure B-mode map and the E-mode recycling method to obtain a leakage free B-mode map. We then use the scalar QML estimator to analyze the scalar pure B map (QML-SZ) or B-mode map (QML-TC). The QML-SZ and QML-TC estimators have similar error bars as the standard QML estimators but their computational cost is nearly one order of magnitude smaller. The basic idea is that one can construct the pure B-mode CMB map by using the E-B separation method proposed by Smith & Zaldarriaga (SZ) or the one considering the template cleaning (TC) technique, then apply the QML estimator to these scalar fields. By simulating potential observations of space-based and ground-based detectors, we test the reliability of these estimators by comparing them with the corresponding results of the traditional QML estimator and the pure B-mode pseudo-C ( l ) estimator.