Inflation and primordial black holes in matter domination

In this thesis we will study a new interpretation of dark matter in terms of Primordial Black Holes (PBHs), i.e. black holes that are believed to form when large density perturbations, produced during the inflationary epoch, reenter the horizon and collapse because of their own gravitational force. Black holes produced in this way are referred to as "Primordial" since their production mechanism has a primordial origin during inflation. These large density fluctuations are produced if the inflationary potential possesses enough tuning freedom to feature a slow-roll plateau followed by a near inflection point that greatly enhances the power spectrum of scalar perturbations. We examine PBH formation during a radiation dominated (RD) epoch and during a matter dominated (MD) epoch driven by a gravitationally coupled scalar field (modulus) which decays before Big Bang Nucleosynthesis in order to preserve its successful predictions. We will require in both cases that the produced PBHs constitute 100% of dark matter today. In the MD case, we find that the mass of the scalar field affects the enhancement required in the curvature power spectrum: for large modulus masses, this enhancement turns out to be smaller than the one in the RD case, meaning that in the case of horizon reentry during moduli domination the potential requires less tuning to produce the same amount of PBHs. Therefore, we focus on PBH production during a MD epoch. We then introduce a model of string inflation called "Fibre Inflation", that works particularly well for our purposes since it naturally leads to a post-inflationary epoch of MD driven by an axion-like modulus. In this model the inflationary potential has enough tuning freedom to induce a period of ultra slow-roll that enhances the density perturbations at the required PBH scales: we employ this potential to obtain a numerical estimate of the scalar power spectrum and the other inflationary observables.