Gravitino-overproduction problem in an inflationary universe

We show that the gravitino-overproduction problem is prevalent among inflation models in supergravity. An inflaton field $\ensuremath{\phi}$ generically acquires (effective) nonvanishing auxiliary field ${\mathcal{G}}_{\ensuremath{\Phi}}^{(\mathrm{eff})}$, if the K\"ahler potential is nonminimal. The inflaton field then decays into a pair of the gravitinos. We extensively study the cosmological constraints on ${\mathcal{G}}_{\ensuremath{\Phi}}^{(\mathrm{eff})}$ for a wide range of the gravitino mass. For many inflation models we explicitly estimate ${\mathcal{G}}_{\ensuremath{\Phi}}^{(\mathrm{eff})}$, and show that the gravitino-overproduction problem severely constrains the inflation models, unless such an interaction as $K=\ensuremath{\kappa}/2|\ensuremath{\phi}{|}^{2}{z}^{2}+\mathrm{H}.\mathrm{c}.$ is suppressed (here $z$ is the field responsible for the supersymmetry breaking). We find that many of them are already excluded or on the verge of, if $\ensuremath{\kappa}\ensuremath{\sim}O(1)$.