Cosmological aspects of the heterotic string above the Hagedorn temperature

We consider the cosmology of the heterotic string above the grand-unification scale taking into account the thermodynamical phase structure of an ideal gas of heterotic string excitations above the Hagedorn temperature. We find that the Universe expands up to a critical volume $a_c^9$ above which the massive (Planckian) excitations should decay into massless excitations (nonequilibrium regime). The Universe then goes on expanding in a radiation-dominated phase with a scale factor a(t)∼β(t). Because the heterotic string is self-dual with respect to temperature, it is argued that there is an induced duality relation for the scale factor a(t).