We study non-thermal gravitino production in the minimal supergravity inflation. In this minimal model utilizing orthogonal nilpotent superfields, the particle spectrum includes only graviton, gravitino, inflaton, and goldstino. We find that a substantial fraction of the cosmic energy density can be transferred to the longitudinal gravitino due to non-trivial change of its sound speed. This implies either a breakdown of the effective theory after inflation or a serious gravitino problem.
Hasegawa, F., Mukaida, K., Nakayama, K., Terada, T., & Yamada, Y. (2017). Gravitino problem in minimal supergravity inflation. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 767, 392–397. https://doi.org/10.1016/j.physletb.2017.02.030