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 large-scale structure, screening from patchy
reionization, 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, differential
ellipticity are also described by the same distortion model. All these
distortions produce B-mode that contaminate the primordial tensor B-modes
signal. In this paper we show that apart from generating B-modes, each
distortion uniquely couples different modes $(\bfl_1\ne \bfl_2)$ 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.
