Shapes of gravity: tensor non-Gaussianity and massive spin-2 fields

If the graviton is the only high spin particle present during inflation, then the form of the observable tensor three-point function is fixed by de Sitter symmetry at leading order in slow-roll, regardless of the theory, to be a linear combination of two possible shapes. This is because there are only a fixed number of possible on-shell cubic structures through which the graviton can self-interact. If additional massive spin-2 degrees of freedom are present, more cubic interaction structures are possible, including those containing interactions between the new fields and the graviton, and self-interactions of the new fields. We study, in a model-independent way, how these interactions can lead to new shapes for the tensor bispectrum. In general, these shapes cannot be computed analytically, but for the case where the only new field is a partially massless spin-2 field we give simple expressions. It is possible for the contribution from additional spin-2 fields to be larger than the intrinsic Einstein gravity bispectrum and provides a mechanism for enhancing the size of the graviton bispectrum relative to the graviton power spectrum.