Gravitino overproduction in inflaton decay

Most of the inflation models end up with non-vanishing vacuum expectation values of the inflaton fields ?   in the true vacuum, which induce, in general, non-vanishing auxiliary field G?$G_{?}$ for the inflaton potential in supergravity. We show that the presence of nonzero G?$G_{?}$ gives rise to inflaton decay into a pair of the gravitinos and are thereby severely constrained by cosmology especially if the gravitino is unstable and its mass is in a range of O(100) GeV–O(10) TeV$O(100)\text{GeV}\text{–}O(10)\text{TeV}$. For several inflation models, we explicitly calculate the values of G?$G_{?}$ and find that most of them are excluded or on the verge of being excluded for the gravitino mass in that range. We conclude that an inflation model with vanishing G?$G_{?}$, typically realized in a chaotic inflation, is favored in a sense that it naturally avoids the potential gravitino overproduction problem.