Horizon crossing and inflation with large η

I examine the standard formalism of calculating curvature perturbations in inflation at horizon crossing, and derive a general relation which must be satisfied for the horizon-crossing formalism to be valid. This relation is satisfied for the usual cases of power-law and slow-roll inflation. I then consider a model for which the relation is strongly violated, and the curvature perturbation evolves rapidly on superhorizon scales. This model has Hubble slow-roll parameter η=3, but predicts a scale-invariant spectrum of density perturbations. I consider the case of hybrid inflation with large η, and show that such solutions do not solve the “η problem” in supergravity. These solutions correspond to field evolution which has not yet relaxed to the inflationary attractor solution, and may make possible new, more natural models on the string landscape.