Horizon crossing and inflation with large η
William H. Kinney
TL;DR
This work questions the universal applicability of the horizon-crossing formalism for inflationary curvature perturbations, deriving exact conservation conditions that must hold for its validity and exploring regimes where it fails. By solving exactly solvable cases—including de Sitter, power-law, and ultra-slow-roll with $η=3$—the authors show that the curvature perturbation can evolve on superhorizon scales, requiring evaluation of the spectrum at the end of inflation rather than at horizon crossing in certain cases. They extend the analysis to non-slow-roll hybrid inflation, deriving $n-1 = 2 r_+$ with $r_± = \tfrac{3}{2}[1 \mp \sqrt{1 - \tfrac{4}{9} α}]$ and demonstrating that horizon crossing breaks down for large $η$, though transient solutions can still be observationally viable. The results imply that non-slow-roll dynamics enlarge the inflationary parameter space but do not solve the $η$-problem in string/supergravity contexts, while providing a framework for new model-building opportunities on the string landscape.
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 $η= 3$, 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.