Implications of the Planck bispectrum constraints for the primordial trispectrum

TL;DR

The paper explores how tight Planck bispectrum constraints imply that a large trispectrum would cause observable variations in the bispectrum, affecting inflation models regardless of the number of contributing fields.

Contribution

It demonstrates that large trispectrum signals necessarily lead to significant bispectrum variations, extending the implications beyond the local model to other non-Gaussian shapes.

Findings

Large gnl implies observable fnl variations.

Constraints on gnl limit the expected fnl size.

Variance of the 3-point function is bounded by the 4-point function.

Abstract

The new Planck constraints on the local bispectrum parameter fnl are about 10^5 times tighter than the current constraints on the trispectrum parameter gnl, which means that the allowed numerical values of the second and third order terms in the perturbative expansion of the curvature perturbation are comparable. We show that a consequence of this is that if gnl is large enough to be detectable, then it will induce a large variation between the observable value of fnl and its value in a larger inflated volume. Even if there were only a few extra efoldings between the beginning of inflation and horizon crossing of our Hubble horizon, an observably large gnl means that fnl is unlikely to be as small as its current constraint, regardless of its true background value. This result is very general, it holds regardless of how many fields contributed to the curvature perturbation. We also generalise this result to other shapes of non-Gaussianity, beyond the local model. We show that the variance of the 3-point function in the squeezed limit is bounded from below by the square of the squeezed limit of the 4-point function.