Non-Gaussianity of Inflationary Gravitational Waves from the Field Equation

We demonstrate equivalence of the in-in formalism and Greens function method for calculating the bispectrum of primordial gravitational waves generated by vacuum fluctuations of the metric. The tree-level bispectrum from the field equation, $B_h$, agrees with the results obtained previously using the in-in formalism exactly. Characterising non-Gaussianity of the fluctuations using the ratio $B_h/P^2_h$ in the equilateral configuration, where $P_h$ is the power spectrum of scale-invariant gravitational waves, we show that it is much weaker than in models with spectator gauge fields. We also calculate the tree-level bispectrum of two right-handed and one left-handed gravitational wave using Greens function, reproducing the results from in-in formalism, and show that it can be as large as the bispectrum of three right-handed gravitational waves.