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.