Thermal leptogenesis and gauge mediation

We show that a minithermal 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 $T_R>{10}^{10}\mathrm{GeV}.$ 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 gravitinos are thermally produced after the reheating of the primary inflation. We find that the thermal leptogenesis takes place at $T_L\simeq {10}^{12}–{10}^{14}\mathrm{GeV}$ for $m_{3/2}\simeq 100\mathrm{keV}–10\mathrm{MeV}$ without any gravitino problem.