Separating E and B types of polarization on an incomplete sky

Detection of magnetic-type ($B$-type) polarization in the Cosmic Microwave Background (CMB) radiation plays a crucial role in probing the relic gravitational wave (RGW) background. In this paper, we propose a new method to deconstruct a polarization map on an incomplete sky in real space into purely electric and magnetic polarization type maps, ${\mathcal{E}}(\hat{\gamma})$ and ${\mathcal{B}}(\hat{\gamma})$, respectively. The main properties of our approach are as follows: Firstly, the fields ${\mathcal{E}}(\hat{\gamma})$ and ${\mathcal{B}}(\hat{\gamma})$ are constructed in real space with a minimal loss of information. This loss of information arises due to the removal of a narrow edge of the constructed map in order to remove various numerical errors, including those arising from finite pixel size. Secondly, this method is fast and can be efficiently applied to high resolution maps due to the use of the fast spherical harmonics transformation. Thirdly, the constructed fields, ${\mathcal{E}}(\hat{\gamma})$ and ${\mathcal{B}}(\hat{\gamma})$, are scalar fields. For this reason various techniques developed to deal with temperature anisotropy maps can be directly applied to analyze these fields. As a concrete example, we construct and analyze an unbiased estimator for the power spectrum of the $B$-mode of polarization $C_{\ell}^{BB}$. Basing our results on the performance of this estimator, we discuss the RGW detection ability of two future ground-based CMB experiments, QUIET and POLARBEAR.