Implications of the Planck bispectrum constraints for the primordial
  trispectrum
  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.
