Gravitational wave constraints on post-inflationary phases stiffer than radiation

We point out that the existence of post-inflationary phases stiffer than radiation leads to the production of stochastic gravitational wave (GW) backgrounds whose logarithmic energy spectra (in critical units) are typically ``blue'' at high frequencies. The maximal spectral slope (for present frequencies larger than 10^-16 Hz) is of order 1 and is related to the maximal sound velocity of the stiff plasma governing the evolution of the geometry. The duration of the stiff phase is crucially determined by the back reaction of the GW leaving the horizon during the de Sitter phase and reentering during the stiff phase. Therefore, the maximal (inflationary) curvature scale has to be fine-tuned to a value smaller than the limits set by the large scale measurements (H₃ₒ10^-6 M₏) in order to have a sufficiently long stiff phase reaching an energy scale of the order of 1 TeV and even lower if we want the stiff phase to touch the hadronic era (corresponding to T₇₀₃140 MeV). By looking more positively at our exercise we see that, if an inflationary phase is followed by a stiff phase, there exists the appealing possibility of ``graviton reheating'' whose effective temperature can be generally quite low.