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 ${\sum }_{n=1}^3{\mathrm{\ell }}_n=\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}$ and, while due to the parity-violating non-Gaussianity, the CMB temperature bispectrum also appears for ${\sum }_{n=1}^3{\mathrm{\ell }}_n=\mathrm{o}\mathrm{d}\mathrm{d}$. 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 $f_{\mathrm{N}\mathrm{L}}^{\mathrm{e}\mathrm{q}}<300$, we can obtain a bound as $\mathrm{\Lambda }\gtrsim 3\times {10}^6\mathrm{G}\mathrm{e}\mathrm{V}$, assuming $r=0.1$.