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