Parity Violation of Gravitons in the CMB Bispectrum

We investigate the cosmic microwave background (CMB) bispectra of the intensity (temperature) and polarization modes induced by the graviton non-Gaussianities, which arise from the parity-conserving and parity-violating Weyl cubic terms with time-dependent coupling. By considering the time-dependent coupling, we find that even in the exact de Sitter space time, the parity violation still appears in the three-point function of the primordial gravitational waves and could become large. Through the estimation of the CMB bispectra, we demonstrate that the signals generated from the parity-conserving and parity-violating terms appear in completely different configurations of multipoles. For example, the parity-conserving non-Gaussianity induces the nonzero CMB temperature bispectrum in the configuration with $\sum_{n=1}^3 \ell_n = {\rm even}$ and, while due to the parity-violating non-Gaussianity, the CMB temperature bispectrum also appears for $\sum_{n=1}^3 \ell_n = {\rm odd}$. This signal is just good evidence of the parity violation in the non-Gaussianity of primordial gravitational waves. We find that the shape of this non-Gaussianity is similar to the so-called equilateral one and the amplitudes of these spectra at large scale are roughly estimated as $|b_{\ell \ell \ell}| \sim \ell^{-4} \times 3.2 \times 10^{-2} ({\rm GeV} / \Lambda)^2 (r / 0.1)^4$, where $\Lambda$ is an energy scale that sets the magnitude of the Weyl cubic terms (higher derivative corrections) and $r$ is a tensor-to-scalar ratio. Taking the limit for the nonlinearity parameter of the equilateral type as $f_{\rm NL}^{\rm eq} < 300$, we can obtain a bound as $\Lambda \gtrsim 3 \times 10^6 {\rm GeV}$, assuming $r=0.1$.