  We analyze the shape and amplitude of oscillatory features in the primordial
power spectrum and non-Gaussianity induced by periodic production of heavy
degrees of freedom coupled to the inflaton $\phi$. We find that non-adiabatic
production of particles can contribute effects which are detectable or
constrainable using cosmological data even if their time-dependent masses are
always heavier than the scale $\dot \phi^{1/2}$, much larger than the Hubble
scale. This provides a new role for UV completion, consistent with the criteria
from effective field theory for when heavy fields cannot be integrated out.
This analysis is motivated in part by the structure of axion monodromy, and
leads to an additional oscillatory signature in a subset of its parameter
space. At the level of a quantum field theory model that we analyze in detail,
the effect arises consistently with radiative stability for an interesting
window of couplings up to of order $\lesssim 1$. The amplitude of the
bispectrum and higher-point functions can be larger than that for Resonant
Non-Gaussianity, and its signal/noise may be comparable to that of the
corresponding oscillations in the power spectrum (and even somewhat larger
within a controlled regime of parameters). Its shape is distinct from
previously analyzed templates, but was partly motivated by the oscillatory
equilateral searches performed recently by the {\it Planck} collaboration. We
also make some general comments about the challenges involved in making a
systematic study of primordial non-Gaussianity.
