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^-α, α>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<<m_g~, the mass density of stable gravitinos is essentially determined by the gluino mass. The observed value of cold dark matter, Ω_CDMh²~0.1, is obtained for gluino masses m_g~=O(1 TeV).