Gauge Couplings at High Temperature and the Relic Gravitino Abundance

In higher-dimensional supersymmetric theories gauge couplings of the effective four-dimensional theory are determined by expectation values of scalar fields. We find that at temperatures above a critical temperature $T_*$, which depends on the supersymmetry breaking mass scales, gauge couplings decrease like $T^{-\a}$, $\a > 1$. This has important cosmological consequences. In particular it leads to a relic gravitino density which becomes independent of the reheating temperature for $T_R > T_*$. For small gravitino masses, $m_{3/2} \ll m_{\gl}$, the mass density of stable gravitinos is essentially determined by the gluino mass. The observed value of cold dark matter, $\O_{\rm CDM}h^2 \sim 0.1$, is obtained for gluino masses $m_{\gl} = {\cal O}(1 {\rm TeV})$.