Stringy bounces and gradient instabilities
  Bouncing solutions are obtained from a generally covariant action
characterized by a potential which is a nonlocal functional of the dilaton
field at two separated space-time points. Gradient instabilities are shown to
arise in this context but they are argued to be nongeneric. After performing a
gauge-invariant and frame-invariant derivation of the evolution equations of
the fluctuations, a heuristic criterion for the avoidance of pathological
instabilities is proposed and corroborated by a number of explicit examples
that turn out to be compatible with a quasi-flat spectrum of curvature
inhomogeneities for typical wavelengths larger than the Hubble radius.
