Cosmological Gravitino Problem in Gauge-Mediated Supersymmetry Breaking Models

We investigate the cosmological gravitino problem in gauge-mediated supersymmetry breaking models, where the gravitino becomes in general the lightest supersymmetric particle (LSP). In order to avoid the overclosure of the stable gravitino, the reheating temperature of inflation $T_R$ should be low enough. Furthermore, if the gravitino mass is larger than about 100 MeV, the decay of the next-to-LSP (NLSP) into the gravitino may modify disastrously the abundances of the light elements predicted by the big-bang nucleosynthesis (BBN). We consider the case in which the lighter stau is the NLSP and derive cosmological constraints from the BBN on the stau NLSP decay. We obtain a lower bound on the mass of stau $m_{\staul}$, which is more stringent than the current experimental limit $m_{\staul} > 90$ GeV for the gravitino mass region $m_{3/2} \gsim 5$ GeV. This lower bound, together with the overclosure constraint on the stable gravitino, gives an upper bound on $T_{R}$. We find that the reheating temperature can be as high as $10^9$--$10^{10}$ GeV for $m_{3/2} \simeq 5$--100 GeV