Space-Dependent Step Features: Transient Breakdown of Slow-roll,   Homogeneity and Isotropy During Inflation
  A step feature in the inflaton potential can model a transient breakdown of slow-roll inflation. Here we generalize the step feature to include space-dependence, allowing it also to model a breakdown of homogeneity and isotropy. The space-dependent inflaton potential generates a classical curvature perturbation mode characterized by the wavenumber of the step inhomogeneity. For inhomogeneities small compared with the horizon at the step, space-dependence has a small effect on the curvature perturbation. Therefore the smoothly oscillating quantum power spectrum predicted by the homogeneous step is robust with respect to sub-horizon space-dependence. For inhomogeneities equal to or greater than the horizon at the step, the space-dependent classical mode can dominate, producing a curvature perturbation in which modes of wavenumber determined by the step inhomogeneity are superimposed on the oscillating power spectrum. Generation of a space-dependent step feature may therefore provide a mechanism to introduce primordial anisotropy into the curvature perturbation. Space-dependence also modifies the quantum fluctuations, in particular via resonance-like features coming from mode coupling to amplified superhorizon modes. However these effects are small relative to the classical modes. 
Rose Lerner and John McDonald