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 $e$-folding scale $\mathrm{\Delta }N$ much smaller than the $e$-folds 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 $e$-folds 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 $\mathrm{\Delta }N\sim 1$ where all parameters in the infinite hierarchy are of comparable magnitude. For models with $\mathrm{\Delta }N\ll 1$, 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.

• Received 20 March 2015

DOI:https://doi.org/10.1103/PhysRevD.92.043501