Running from Features: Optimized Evaluation of Inflationary Power
  Spectra
  In models like axion monodromy, temporal features during inflation which are
not associated with its ending can produce scalar, and to a lesser extent,
tensor power spectra where deviations from scale-free power law spectra can be
as large as the deviations from scale invariance itself. Here the standard
slow-roll approach breaks down since its parameters evolve on an efolding scale
$\Delta N$ much smaller than the efolds to the end of inflation. Using the
generalized slow-roll approach, we show that the expansion of observables in a
hierarchy of potential or Hubble evolution parameters comes from a Taylor
expansion of the features around an evaluation point that can be optimized.
Optimization of the leading-order expression provides a sufficiently accurate
approximation for current data as long as the power spectrum can be described
over the well-observed few efolds by the local tilt and running. Standard
second-order approaches, often used in the literature, ironically are worse
than leading-order approaches due to inconsistent evaluation of observables. We
develop a new optimized next-order approach which predicts observables to
$10^{-3}$ even for $\Delta N\sim 1$ where all parameters in the infinite
hierarchy are of comparable magnitude. For models with $\Delta N \ll 1$, the
generalized slow-roll approach provides integral expressions that are accurate
to second order in the deviation from scale invariance. Their evaluation in the
monodromy model provides highly accurate explicit relations between the running
oscillation amplitude, frequency and phase in the curvature spectrum and
parameters of the potential.
