Implications of the Planck bispectrum constraints for the primordial trispectrum

The new Planck constraints on the local bispectrum parameter are about 105 times tighter than the current constraints on the trispectrum parameter , 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 is large enough to be detectable, then it will induce a large variation between the observable value of and its value in a larger inflated volume. Even if there were only a few extra e-foldings between the beginning of inflation and horizon crossing of our Hubble horizon, an observably large means that 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.