  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 non-zero 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.
