Trispectrum versus bispectrum in single-field inflation

In the standard slow-roll inflationary cosmology, quantum fluctuations in a single field, the inflaton, generate approximately Gaussian primordial density perturbations. At present, the bispectrum and trispectrum of the density perturbations have not been observed and the probability distribution for these perturbations is consistent with Gaussianity. However, Planck satellite data will bring a new level of precision to bear on this issue, and it is possible that evidence for non-Gaussian effects in the primordial distribution will be discovered. One possibility is that a trispectrum will be observed without evidence for a nonzero bispectrum. It is not difficult for this to occur in inflationary models where quantum fluctuations in a field other than the inflaton contribute to the density perturbations. A natural question to ask is whether such an observation would rule out the standard scenarios. We explore this issue and find that it is possible to construct single-field models in which inflaton-generated primordial density perturbations have an observable trispectrum, but a bispectrum that is too small to be observed by the Planck satellite. However, an awkward fine-tuning seems to be unavoidable.