Inflationary perturbations near horizon crossing

We study the behaviour of inflationary density perturbations in the vicinity of horizon crossing, using numerical evolution of the relevant mode equations. We explore two specific scenarios. In one, inflation is temporarily ended because a portion of the potential is too steep to support inflation. We find that perturbations on super-horizon scales can be modified, usually leading to a large amplification, because of entropy perturbations in the scalar field. This leads to a broad feature in the power spectrum, and the slow-roll and Stewart--Lyth approximations, which assume the perturbations reach an asymptotic regime well outside the horizon, can fail by many orders of magnitude in this regime. In the second scenario we consider perturbations generated right at the end of inflation, which re-enter shortly after inflation ends --- such perturbations can be relevant for primordial black hole formation.