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.Comment: v2 minor revisions; 39 pages, 5 figure
