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.