Slow and Safe Gravitinos
  It has been argued that supergravity models of inflation with vanishing sound
speeds, $c_s$, lead to an unbounded growth in the production rate of
gravitinos. We consider several models of inflation to delineate the conditions
for which $c_s = 0$. In models with unconstrained superfields, we argue that
the mixing of the goldstino and inflatino in a time-varying background prevents
the uncontrolled production of the longitudinal modes. This conclusion is
unchanged if there is a nilpotent field associated with supersymmetry breaking
with constraint ${\bf S^2} =0$, i.e. sgoldstino-less models. Models with a
second orthogonal constraint, ${\bf S(\Phi-\bar{\Phi})} =0$, where $\bf{\Phi}$
is the inflaton superfield, which eliminates the inflatino, may suffer from the
over-production of gravitinos. However, we point out that these models may be
problematic if this constraint originates from a UV Lagrangian, as this may
require using higher derivative operators. These models may also exhibit other
pathologies such as $c_s > 1$, which are absent in theories with the single
constraint or unconstrained fields.
