  Gravitino production in the primordial Universe is investigated into details.
After briefly reviewing inflation, supersymmetry and supergravity, we first
study the scattering of massive W bosons in the thermal bath of particles,
during the period of reheating. It is found that the process generates in the
cross section terms which eventually lead to unitarity breaking above a certain
scale. This happens by virtue of the supergravity vertex. We show that the
longitudinal polarizations of the on-shell W become strongly interacting in the
high energy limit, and that the inclusion of diagrams with off-shell scalars of
the MSSM does not cancel the divergences. Next, we consider the dynamics and
the decay into gravitinos of a scalar field S, which starts oscillating in its
potential at the end of inflation. We embed S in a model of gauge mediation
with metastable vacua, where the hidden sector is of the O'Raifeartaigh type.
By demanding that the gravitinos thus produced provide with the observed Cold
Dark Matter density, we modify previous results in the literature, and find
that it is easy to account for gravitino Dark Matter with an arbitrarily low
reheating temperature.
