Coupling the Inflationary Sector to Matter
  We describe the coupling of matter fields to an inflationary sector of
supergravity, the inflaton $\Phi$ and a stabilizer $S$, in models where the
Kahler potential has a flat inflaton direction. Such models include, in
particular, advanced versions of the hyperbolic $\alpha$-attractor models with
a flat inflaton direction Kahler potential, providing a good fit to the
observational data. If the superpotential is at least quadratic in the matter
fields $U^{i}$, with restricted couplings to the inflaton sector, we prove that
under certain conditions: i) The presence of the matter fields does not affect
a successful inflationary evolution. ii) There are no tachyons in the matter
sector during and after inflation. iii) The matter masses squared are higher
than $3H^2$ during inflation. The simplest class of theories satisfying all
required conditions is provided by models with a flat direction Kahler
potential, and with the inflaton $\Phi$ and a stabilizer $S$ belonging to a
hidden sector, so that matter fields have no direct coupling to the
inflationary sector in the Kahler potential and in the superpotential.
