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 TR 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 mτ1, which is more stringent than the current experimental limit mτ1>90 GeV for the gravitino mass region m3/2≳5 GeV. This lower bound, together with the overclosure constraint on the stable gravitino, gives an upper bound on TR. We find that the reheating temperature can be as high as 109–1010 GeV for m3/2≃5–100 GeV.