Graviton non-Gaussianities and Parity Violation in the EFT of Inflation

We study graviton non-Gaussianities in the EFT of Inflation. At leading (second) order in derivatives, the graviton bispectrum is fixed by Einstein gravity. There are only two contributions at third order. One of them breaks parity. They come from operators that directly involve the foliation: we then expect sizable non-Gaussianities in three-point functions involving both gravitons and scalars. However, we show that at leading order in slow roll the parity-odd operator does not modify these mixed correlators. We then identify the operators that can affect the graviton bispectrum at fourth order in derivatives. There are two operators that preserve parity. We show that one gives a scalar-tensor-tensor three-point function larger than the one computed in Maldacena, 2003 if $M^2_{\rm P}A_{\rm s}/\Lambda^2\gg 1$ (where $\Lambda$ is the scale suppressing this operator and $A_{\rm s}$ the amplitude of the scalar power spectrum). There are only two parity-odd operators at this order in derivatives.
Comment: 21 pages (15+6). v3: added a section on soft limits, a more detailed discussion on the $M^2_{\rm P}A_{\rm s}/\Lambda^2\gg 1$ limit, and comments in light of "boostless cosmological bootstrap" program