A large $\mathrm{\mid }\eta \mathrm{\mid }$ approach to single field inflation

Single field models of inflation capable to produce primordial black holes usually require a significant departure from the standard, perturbative slow-roll regime. In fact, in many of these scenarios, the size of the slow-roll parameter $\mathrm{\mid }\eta \mathrm{\mid }$ becomes larger than one during a short phase of inflationary evolution. In order to develop an analytical control on these systems, we explore the limit of $\mathrm{\mid }\eta \mathrm{\mid }$ large, and promote $1\mathrm{/}\mathrm{\mid }\eta \mathrm{\mid }$ to a small quantity to be used for perturbative expansions. Formulas simplify, and we obtain analytic expressions for the two and three point functions of curvature fluctuations, which share some of the features found in realistic inflationary models generating primordial black holes. We study one-loop corrections in this framework: we discuss criteria for adsorbing ultraviolet divergences into the available parameters, leaving log-enhanced infrared contributions of controllable size.