Detecting chiral gravity with the pure pseudospectrum reconstruction of
  the cosmic microwave background polarized anisotropies
  We consider the possible detection of parity violation at the linear level in
gravity using polarized anisotropies of the cosmic microwave background. Since
such a parity violation would lead to non-zero TB and EB correlations, this
makes those odd-parity angular power spectra a potential probe of parity
violation in the gravitational sector. These spectra are modeled incorporating
the impact of lensing and we explore their possible detection in the context of
small-scale (balloon-borne or ground-based) experiments and a future satellite
mission dedicated to B-mode detection. We assess the statistical uncertainties
on their reconstruction using mode-counting and a (more realistic) pure
pseudospectrum estimator approach. Those uncertainties are then translated into
constraints on the level of parity asymmetry. We found that detecting chiral
gravity is impossible for ongoing small-scale experiments. However, for a
satellite-like mission, a parity asymmetry of at least 50% could be detected at
68% of confidence level, and a parity asymmetry of 100% is measurable with at
least a confidence level of 95%. We also assess the impact of a possible
miscalibration of the orientation of the polarized detectors, leading to
spurious TB and EB cross-correlations. We show that in the context of
pseudospectrum estimation of the angular power spectra, self-calibration of
this angle could significantly reduce the statistical significance of the
measured level of parity asymmetry (by e.g. a factor ~2.4 for a miscalibration
angle of 1 degree). For chiral gravity and assuming a satellite mission
dedicated to primordial B-mode, a non detection of the TB and EB correlation
would translate into an upper bound on parity violation of 39% at 95%
confidence level for a tensor-to-scalar ratio of 0.2, excluding values of the
(imaginary) Barbero-Immirzi parameter comprised between 0.2 and 4.9 at 95% CL.
