  We study the realization of slow-roll inflation in $\mathcal N = 1$
supergravities where inflation is the result of the evolution of a single
chiral field. When there is only one flat direction in field space, it is
possible to derive a single-field effective field theory parametrized by the
sound speed $c_s$ at which curvature perturbations propagate during inflation.
The value of $c_s$ is determined by the rate of bend of the inflationary path
resulting from the shape of the $F$-term potential. We show that $c_s$ must
respect an inequality that involves the curvature tensor of the Kahler manifold
underlying supergravity, and the ratio $M/H$ between the mass $M$ of
fluctuations ortogonal to the inflationary path, and the Hubble expansion rate
$H$. This inequality provides a powerful link between observational constraints
on primordial non-Gaussianity and information about the $\mathcal N = 1$
supergravity responsible for inflation. In particular, the inequality does not
allow for suppressed values of $c_s$ (values smaller than $c_s \sim 0.4$)
unless (a) the ratio $M/H$ is of order 1 or smaller, and (b) the fluctuations
of mass $M$ affect the propagation of curvature perturbations by inducing on
them a nonlinear dispersion relation during horizon crossing. Therefore, if
large non-Gaussianity is observed, supergravity models of inflation would be
severely constrained.

High Energy Physics - Theory;Cosmology and Nongalactic Astrophysics;High Energy Physics - Phenomenology