  We study the pattern of oscillations in the primordial power spectrum in
axion monodromy inflation, accounting for drifts in the oscillation period that
can be important for comparing to cosmological data. In these models the
potential energy has a monomial form over a super-Planckian field range, with
superimposed modulations whose size is model-dependent. The amplitude and
frequency of the modulations are set by the expectation values of moduli
fields. We show that during the course of inflation, the diminishing energy
density can induce slow adjustments of the moduli, changing the modulations. We
provide templates capturing the effects of drifting moduli, as well as drifts
arising in effective field theory models based on softly broken discrete shift
symmetries, and we estimate the precision required to detect a drifting period.
A non-drifting template suffices over a wide range of parameters, but for the
highest frequencies of interest, or for sufficiently strong drift, it is
necessary to include parameters characterizing the change in frequency over the
e-folds visible in the CMB. We use these templates to perform a preliminary
search for drifting oscillations in a part of the parameter space in the Planck
nominal mission data.
