Inflation and flat directions in modular invariant superstring effective theories

The potential during inflation must be very flat in, at least, the direction of the inflaton. In renormalizable global supersymmetry, flat directions are ubiquitous, but they are not preserved in a generic supergravity theory. It is known that at least some of them are preserved in no-scale supergravity, and simple generalizations of it. We here study a more realistic generalization, based on string-derived supergravity, using the linear supermultiplet formalism for the dilaton. We consider a general class of hybrid inflation models, where a Fayet-Illiopoulos $D$ term drives some fields to large values. The potential is dominated by the $F$ term, but flatness is preserved in some directions. This allows inflation, with the dilaton stabilized in its domain of attraction, and some moduli stabilized at their vacuum values. Another modulus may be the inflaton.