Chern-Simons gravitational term coupled to a spectator field

The Chern-Simons gravitational term is typically coupled to the inflaton field during inflation. In this work, we explore an alternative scenario where the Chern-Simons term is minimally coupled to a massive spectator field, $\sigma$, within a multi-field inflationary framework where the spectator field is linearly coupled to the inflaton. We derive the corresponding parity-violating cubic interaction Lagrangian involving the curvature, spectator and tensor perturbations and compute how it affects primordial scalar-tensor bispectra. We find that these correlators yield distinctive parity-odd shapes. Perturbativity and consistency bounds are derived, constraining the couplings and the amplitude of the associated non-Gaussianities.