Graviton non-gaussianity in $\alpha$-vacuum

We compute the leading order non-Gaussianity, i.e., the bispectrum, of the tensor perturbation in the general $\alpha$-vacuum on de Sitter space in general relativity. In addition to the well-known Bunch-Davies (BD) vacuum, there exits an infinite number of de Sitter invariant vacua represented by a real parameter $\alpha$ and a phase $\phi$, with $\alpha=0$ being the BD vacuum. They are called $\alpha$-vacua. In the standard slow-roll inflation, as de Sitter invariance no longer applies, the $\alpha$-vacua lose its relevance in the rigorous sense. Nevertheless, if we assume that the parameter $\alpha$ is only weakly dependent on the wavenumber with an appropriate UV cutoff, we may consider pseudo-$\alpha$-vacua. In the case of false vacuum inflation where the background spacetime is pure de Sitter, a non-trivial (non-BD) $\alpha$-vacuum could indeed be realized. We find an intriguing result that the bispectrum may be exponentially enhanced to be detectable by observation even if the spectrum is too small to be detected.