Inflationary trispectrum from graviton exchange

TL;DR

This paper computes the inflationary trispectrum arising from graviton exchange in single-field slow-roll models, using the in-in formalism in the uniform curvature gauge to evaluate the graviton–scalar–scalar vertex and the resulting four-point function of curvature perturbations. The graviton-exchange contribution, proportional to the tensor-to-scalar ratio $r$, is found to be of the same order as the previously computed scalar-contact contribution, and remains finite after combinations of time integrals. In the counter-collinear limit, the trispectrum becomes locally parametrized with $\tau_{\text{NL}}^{local} \sim r$, and a detailed numerical analysis shows that the total trispectrum can reach $|\tau_{\text{NL}}| \approx 1.218 r$, indicating $\tau_{\text{NL}}$ is generically of order $r$. The work establishes that the total trispectrum in single-field slow-roll inflation is bounded by $\tau_{\text{NL}} \sim r$, challenges the simple $\tau_{\text{NL}} \sim f_{\text{NL}}^2$ expectation, and provides results applicable to multi-field scenarios with flat field space.

Abstract

We compute the connected four-point correlation function of the primordial curvature perturbation generated during inflation with standard kinetic terms, where the correlation is established via exchange of a graviton between two pairs of scalar fluctuations. Any such correlation yields a contribution to the scalar trispectrum of the order of the tensor to scalar ratio r. This contribution is numerically one order of magnitude larger than the one previously calculated on the basis of scalar perturbations interacting at a point and satisfies a simple relation in the limit where the momentum of the graviton which is exchanged becomes much smaller than the external momenta. We conclude that the total non-linearity parameter generated by single-field models of slow-roll inflation is at maximum tauNL ~ r.

Figures (2)

• Figure 1: Possible planar momentum quadrilaterals. Note, however, that there is no need for the quadrilateral to lie in a plane, and in general it is a three-dimensional object. In (a), there is no particular relationship among the sides of the quadrilateral. This is the general case. In (b), the side associated with $\bm{\mathrm{{k}}}_4$ is taken to zero length, causing the quadrilateral to degenerate to a triangle. In (c), adjacent sides possess equal magnitudes and their directions are becoming opposite, a limit we refer to as the folded kite. In (d), opposite sides possess equal magnitudes, yielding a parallelogram. The shapes described in (c) and (d), are therefore precisely dual to each other, and describe the same physics.
• Figure 2: In (a), exchange of a graviton (represented by a wavy line) leads to correlations among four scalar fluctuations (represented by straight lines). In the limit where the graviton which is exchanged becomes extremely soft, so that $|\bm{\mathrm{{k}}}_1 + \bm{\mathrm{{k}}}_2| \rightarrow 0$, one can think of such correlations as being mediated by fluctuations on top of a modified background which carries a classical gravitational wave. The same interpretation holds in (b), where the exchange is mediated by a scalar particle.