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 $T_{\ast }$, which depends on the supersymmetry breaking mass scales, gauge couplings decrease like $T^{-\text{a}}$, $\text{a}>1$. This has important cosmological consequences. In particular it leads to a relic gravitino density which becomes independent of the reheating temperature for $T_R>T_{\ast }$. For small gravitino masses, $m_{3\mathrm{/}2}\ll m_{\text{gl}}$, the mass density of stable gravitinos is essentially determined by the gluino mass. The observed value of cold dark matter, ${\text{O}}_{\mathrm{C}\mathrm{D}\mathrm{M}}{h}^2\sim 0.1$, is obtained for gluino masses $m_{\gl} = {\cal O}(1 {\rm TeV})$.