A naturally large four-point function in single field inflation

Non-Gaussianities of the primordial density perturbations have emerged as a very powerful possible signal to test the dynamics that drove the period of inflation. While in general the most sensitive observable is the three-point function in this paper we show that there are technically natural inflationary models where the leading source of non-Gaussianity is the four-point function. Using the recently developed Effective Field Theory of Inflation, we are able to show that it is possible to impose an approximate parity symmetry and an approximate continuos shift symmetry on the inflaton fluctuations that allow, when the dispersion relation if of the form omega similar to c(s)k, for a unique quartic operator, while approximately forbidding all the cubic ones. The resulting shape for the four-point function is unique. In the models where the dispersion relation is of the form omega similar to k(2)/M a similar construction can be carried out and additional shapes are possible.