Cosmological implications of quantum corrections and higher-derivative extension

We discuss the challenges for the early universe cosmology from quantum corrections, and in particular higher-derivative terms, in the gravitational and inflaton sectors of the models. The work is divided in two parts. In the first one we review the already wellknown issues due to quantum corrections to the inflaton potential, in particular focusing on chaotic/slow-roll single-field models. We will point out some issues concerning the proposed mechanisms to cope with the corrections, and also argue how the presence of higher-derivative corrections could be problematic for those mechanisms. In the second part we will more directly focus on higher-derivative corrections. We will show how, in order to discuss a number of high-energy phenomena relevant to inflation (such as its actual onset) one has to deal with energy scales where the derivative expansion breaks down, presenting problems such as quantum vacuum instability and ghosts. To discuss such phenomena in the convenient framework of the effective theory, one must then abandon the derivative expansion and resort to the full nonlocal formulation of the theory, which is in fact equivalent to re-integrating back the relevant physics, but with the benefit of using a more compact single-field formalism. Finally, we will briefly discuss possible advantages offered by the presence of higher derivatives and a nonlocal theory to build better controlled UV models of inflation.