Inflation over a local maximum of a potential

We calculate the power spectrum of curvature perturbations when the inflaton field is rolling over the top of a local maximum of a potential. We show that the evolution of the field can be decomposed into a late-time attractor, which is identified as the slow roll solution, plus a rapidly decaying non-slow roll solution, corresponding to the field rolling “up the hill” to the maximum of the potential. The exponentially decaying transient solution can map to an observationally relevant range of scales because the Universe is also expanding exponentially. We consider the two branches separately and we find that they are related through a simple transformation of the slow roll parameter η and they predict identical power spectra. We generalize this approach to the case where the inflaton field is described by both branches simultaneously and find that the mode equation can be solved exactly at all times. Even though the slow roll parameter η is evolving rapidly during the transition from the transient solution to the late-time attractor solution, the resultant power spectrum is an exact power-law spectrum. Such solutions may be useful for model building on the string landscape.