The inflationary trispectrum

TL;DR

The paper computes the primordial trispectrum generated during slow-roll inflation using the $\delta N$ formalism and a full fourth-order action for perturbations, evaluated with Schwinger-Keldysh techniques. It derives the form-factor $\mathcal{M}_4$ governing the momentum dependence of the connected four-point function, and establishes a robust upper bound on the horizon-crossing contribution to the trispectrum: $|\Delta\tau_{NL}| \lesssim \dfrac{r}{50}$ with $r<1$, rendering it unobservable by current and near-future CMB experiments. In the single-field case, Maldacena's consistency relation implies further suppression in the squeezed limit, since $\zeta$ is conserved on superhorizon scales and the leading trispectrum contributions are slow-roll suppressed. The results indicate that any detectable primordial trispectrum would have to originate from superhorizon gravitational effects, guiding observational strategies toward late-time evolution rather than horizon-crossing quantum interactions.

Abstract

We calculate the trispectrum of the primordial curvature perturbation generated by an epoch of slow-roll inflation in the early universe, and demonstrate that the non-gaussian signature imprinted at horizon crossing is unobservably small, of order tau_NL < r/50, where r < 1 is the tensor-to-scalar ratio. Therefore any primordial non-gaussianity observed in future microwave background experiments is likely to have been synthesized by gravitational effects on superhorizon scales. We discuss the application of Maldacena's consistency condition to the trispectrum.

Figures (1)

• Figure 1: Connected $(a)$ and disconnected $(b)$ contributions to the four-point function of the $\{ \delta\phi \}$. The disconnected contribution factorizes into a product of two copies of the power spectrum, and contains no information concerning primordial non-gaussianities beyond those encoded in loop corrections, which are expected to be unobservably small Sloth:2006az.