Cosmic inflation is responsible for the primordial density fluctuations which are the seeds of today's large-scale structures and which are observed through the temperature fluctuations of the Cosmic Microwave Background. In this thesis, we explore a particular model of inflation with modified kinetics : Dirac-Born-Infeld (DBI) inflation inspired by string theory. In this scenario, inflation is generated by the motion of a probe brane and particles can be created during inflation when trapped branes are present along the inflationary valley. We will show that such a coupling between the inflaton and matter fields leads to interesting features in the primordial power spectrum, similar to features à la Starobinsky sourced by the potential or scalar-tensor features sourced by a suddenly-varying inflaton-matter coupling function in k-inflation. We will insist on the possible observational consequences of our theories.