Modified Gravity Theories and the Amplification of Inflationary Perturbations

Abstract

The superpotential technique allows the reconstruction of inflationary models. Assuming that the Hubble parameter is a function of the scalar field, we can reconstruct the scalar field potential. In this thesis, we are interested in using this technique to find the conditions on the reconstructed potential leading to the formation of Primordial Black Holes in the early universe. It has indeed been shown that PBHs can form if a growing mode of the Mukhanov-Sasaki equation is present, or if the scalar perturbation spectrum is blue-tilted. These two conditions place different constraints on the inflationary model. In particular, we applied the superpotential technique to a series of inflationary models. Compared to other reconstruction methods, this technique has been found to be very powerful in such a context. It leads to quite general predictions for the shape of the inflaton potential needed for the amplification. The case of inflation in the context of General Relativity was first investigated. Exploiting the conformal mapping and the Jordan-Einstein frame transformation, the analysis was extended to non-minimally coupled inflaton. The case of $f(R)$ theories was also studied in detail.