Thermal leptogenesis and gauge mediation

We show that a mini-thermal inflation occurs naturally in a class of gauge mediation models of supersymmetry (SUSY) breaking, provided that the reheating temperature T_R of the primary inflation is much higher than the SUSY-breaking scale, say T_R > 10^{10} GeV. The reheating process of the thermal inflation produces an amount of entropy, which dilutes the number density of relic gravitinos. This dilution renders the gravitino to be the dark matter in the present universe. The abundance of the gravitinos is independent of the reheating temperature T_R, once the gravtinos are thermally produced after the reheating of the primary inflation. We find that the thermal leptogenesis takes place at T_L\simeq 10^{12-14} GeV for m_{3/2}\simeq 100 keV - 10 MeV without any gravitino problem.