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Single field models of inflation capable to produce primordial black holesusually require a significant departure from the standard, perturbativeslow-roll regime. In fact, in many of these scenarios, the size of theslow-roll parameter $|\eta|$ becomes larger than one during a short phase ofinflationary evolution. In order to develop an analytical control on thesesystems, we explore the limit of $|\eta|$ large, and promote $1/|\eta|$ to asmall quantity to be used for perturbative expansions. Formulas simplify, andwe obtain analytic expressions for the two and three point functions ofcurvature fluctuations, which share some of the features found in realisticinflationary models generating primordial black holes. We study one-loopcorrections in this framework: we discuss criteria for adsorbing ultravioletdivergences into the available parameters, leaving log-enhanced infraredcontributions of controllable size.

A large $|\eta|$ approach to single field inflation