(Small) Resonant non-Gaussianities: Signatures of a Discrete Shift
  Symmetry in the Effective Field Theory of Inflation
  We apply the Effective Field Theory of Inflation to study the case where the
continuous shift symmetry of the Goldstone boson \pi is softly broken to a
discrete subgroup. This case includes and generalizes recently proposed String
Theory inspired models of Inflation based on Axion Monodromy. The models we
study have the property that the 2-point function oscillates as a function of
the wavenumber, leading to oscillations in the CMB power spectrum. The
non-linear realization of time diffeomorphisms induces some self-interactions
for the Goldstone boson that lead to a peculiar non-Gaussianity whose shape
oscillates as a function of the wavenumber. We find that in the regime of
validity of the effective theory, the oscillatory signal contained in the
n-point correlation functions, with n>2, is smaller than the one contained in
the 2-point function, implying that the signature of oscillations, if ever
detected, will be easier to find first in the 2-point function, and only then
in the higher order correlation functions. Still the signal contained in
higher-order correlation functions, that we study here in generality, could be
detected at a subleading level, providing a very compelling consistency check
for an approximate discrete shift symmetry being realized during inflation.
