CMB E/B decomposition of incomplete sky: a pixel space approach

CMB polarization signal may be decomposed into gradient-like (E) and curl-like (B) mode. We have investigated E/B decomposition in pixel space. We find E/B mixing due to incomplete sky is localized in pixel-space, and negligible in the regions far away from the masked area. By estimating the expected local leakage power, we have diagnosed ambiguous pixels. Our criteria for ambiguous pixels (i.e. r_c) is associated with the tensor-to-scalar ratio of B mode power spectrum, which the leakage power is comparable to. By setting r_c to a lower value, we may reduce leakage level, but reduce sky fraction at the same time. Therefore, we have solved \partial \Delta C_l/\partial r_c=0, and obtained the optimal r_c, which minimizes the estimation uncertainty, given a foreground mask and noise level. We have applied our method to a simulated map blocked by a foreground (diffuse + point source) mask. Our simulation shows leakage power is smaller than primordial (i.e. unlensed) B mode power spectrum of tensor-to-scalar ratio r\sim 10^{-3} at wide range of multipoles (50\lesssim l \lesssim 2000), while allowing us to retain sky fraction ~ 0.48.