Primordial B-mode diagnostics and self-calibrating the CMB polarization

Distortions in the primordial cosmic microwave background (CMB) along the line of sight can be modeled and described using 11 fields. These distortion fields correspond to various cosmological signals such as weak gravitational lensing of the CMB by a large-scale structure, screening from patchy reionization, and rotation of the plane of polarization due to magnetic fields or parity violating physics. Various instrumental systematics such as gain fluctuations, pixel rotation, differential gain, pointing, and differential ellipticity are also described by the same distortion model. All these distortions produce a B mode that contaminates the primordial tensor B-modes signal. In this paper we show that, apart from generating B modes, each distortion uniquely couples different modes (l(1) +/- l(2)) of the CMB anisotropies, generating < EB > correlations which for the primordial CMB are zero. We describe and implement unbiased minimum variance quadratic estimators which using the off-diagonal correlations in the CMB can extract the map of distortions. We perform Monte Carlo simulations to characterize the estimators and illustrate the level of distortions that can be detected with current and future experiments. The estimators can be used to look for cosmological signals, or to check for any residual systematics in the data. As a specific example of primordial tensor B-mode diagnostics, we compare the level of minimum detectable distortions using our method with maximum allowed distortion level for the B-modes detection. We show that for any experiment, the distortions will be detected at high significance using correlations before they would show up as spurious B modes in the power spectrum. In other words, if B modes were to be detected, the presence of any nonprimordial contribution could be checked and removed using the off-diagonal correlations in the CMB.