Ultraviolet aspects of Peccei--Quinn Inflation

The possibility of observing a value of the tensor-to-scalar ratio $r$ of the order $r\sim 10^{-3}$ would project us into a new era for the cosmology of the early Universe. Such an observation would lead to confirmation of inflation and a measurement of the energy scale at which this phase occurred, revolutionizing the idea of the early stages of the evolution of the Universe. This scale would be beyond the reach of any possible terrestrial experiment, exploring a region of energies of the order of $10^{12}-10^{13} \rm{GeV} $. Since a theory of quantum gravity is missing, this relegates inflation models to effective models that are reliable only within a certain range of energies. Hence, the question of whether the predictions of these models are reliable is crucial. In other words, is it always possible to ignore the tower of higher-dimensional operators present? This thesis aims to answer this question by focusing on the model called Peccei-Quinn inflation. This model offers the possibility of explaining inflation, dark matter and providing a solution to the strong CP problem. It also predicts a value of $r \sim 10^{-3 }$ and makes this model falsifiable in the future. In addition, this thesis addresses a crucial aspect related to the production of dark matter through axions. It involves modeling the evolution after inflation, which is a crucial point. The findings significantly alter what was previously known about the Peccei-Quinn model. In the thesis we prove that Peccei–Quinn inflation is extremely sensitive to higher-dimensional operators, undermining its validity as an effective field theory. Further combined with the discussion on the axion quality required for solving the strong CP problem, we examine the validity of this scenario. We also show that after Peccei–Quinn inflation, resonant amplifications of the field fluctuations are inevitably triggered, casting serious doubts on the typical assumption of a homogeneous evolution. In conclusion, this thesis asks and tries to answer some profound questions regarding theoretical models that are in the sights of future groundbreaking observations in cosmology that will potentially provide a deeper understanding of the fundamental properties of our Universe.