D-Brane inflation in the non-relativistic and relativistic regimes

In this thesis we study inflation from a string theory point of view. The first part of the work is dedicated to reviewing how this mechanism can solve the main problems of the Standard Hot Big Bang (HBB) Cosmology and how it can be built in the context of General Relativity. The second part focuses on the application of the principles of string theory to inflation. In this framework an inflationary model can be viewed as an effective field theory in the super-gravity limit. In particular we concentrate on inflationary models in which the effective Lagrangian is obtained from the dimensional reduction of the Dirac-Born-Infeld (DBI) action, which describes the position and orientation of a $D7$-brane. This effective Lagrangian contains a non-trivial kinetic term. We study the effects of this non-trivial kinetic term on the speed of sound $c_s$, on the tensor-to-scalar ratio $r$ and on the spectral index $n_s$ in the case of monomial potentials, generalizing the work done by Ibanez and Pedro. We find that the system never enters the relativistic regime and therefore $c_s\simeq 1$. This justifies the truncation of the DBI action to second order derivatives. In the final chapter we see how the Swampland criteria, when applied to $DBI$ inflation, push the system into the relativistic regime and lead to a strong decrease of $c_s$.