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 eta becomes larger than one during a short phase of inflationary evolution. In order to develop an analytical control on these systems, I explore the limit of eta large, and promote 1/eta to a small quantity to be used for perturbative expansions. Formulas simplify, and analytic expressions for the two and three point functions of curvature fluctuations are obtained. I will then discuss the behaviour of loop corrections to inflationary observables in this framework

Cosmological inflation predicts the existence of a stochastic background of gravitational waves (GW), whose features depend on the model of inflation under consideration. There exist well motivated frameworks leading to an enhancement of the primordial GW spectrum at frequency scales testable with GW experiments, with specific features as parity violation, anisotropies, and non-Gaussianity. I will explain the properties of such scenarios, and their distinctive predictions for what respect GW observables. I will then discuss perspectives for testing these predictions with future GW experiments.