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Chern-Simons gravity coupled to the scalar sector through a generic couplingfunction $f(\phi)$ can be tested at the very high energies of the inflationaryperiod. In 1706.04627, we computed the theoretical parity breaking signaturesof the $\langle \gamma \gamma \zeta \rangle$ primordial bispectrum which mixestwo gravitons and one scalar curvature perturbation. We defined a parameter$\Pi$ which measures the level of parity breaking of the correspondingbispectrum. In this work we forecast the expected $1 \sigma$ error on $\Pi$using the cosmic microwave background (CMB) angular bispectra. We find that,given the angular resolution of an experiment like $Planck$, $\Pi \sim 10^6$ isdetectable via the measurement of $BBT$ or $BBE$ angular bispectra if thetensor-to-scalar ratio $r = 0.01$. We also show that, from the theoreticalpoint of view, $\Pi$ can be greater than $10^6$. Thus, our conclusion is that$BBT$ or $BBE$ CMB angular bispectra can become an essential observable fortesting Chern-Simons gravity in the primordial universe.

Measuring chiral gravitational waves in Chern-Simons gravity with CMB bispectra