Gravitational waves from primordial black holes in string inflation

We consider a model of inflation, called Fibre Inflation, derived from string theory in the framework of type IIB flux compactification and we study the production of primordial black holes due to the gravitational collapse of the curvature perturbations. They are a good candidate for dark matter and if their mass was between 10^-17 and 10^-13 solar masses then their contribution to the total dark matter abundance would be between 10% and 100%. The inflaton potential has enough tuning freedom to allow for a flat plateau at large field values, corresponding to the usual slow roll behaviour, and an inflection point near the minimum that enhances the scalar perturbations. It is the latter feature that greatly increases the density anisotropies from the usual δρ/ρ ~ 10^-5, typical of the CMB radiation, to δρ/ρ ~ 10^-1 due to a peak in the primordial power spectrum that allows for the formation of black holes. The latter is computed solving the Mukhanov-Sasaki equation numerically using the Hubble slow roll parameters extracted from the solutions to the Friedmann equations and the inflaton equation of motion. The formation of black holes is followed by the production of a stochastic background of secondary gravitational waves: in this thesis we compute their amplitude and compare it with current observational bounds and the sensitivities of earth- and space-based interferometers. Our result is within reach of experiments like DECIGO and BBO, hence in the future it will be possible to have a comparison between our theoretical results and observations.