Horizon crossing and inflation with largeη

Abstract

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 $\ensuremath{\eta}=3$, but predicts a scale-invariant spectrum of density perturbations. I consider the case of hybrid inflation with large $\ensuremath{\eta}$, and show that such solutions do not solve the ``$\ensuremath{\eta}$ 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.