Gravitational waves 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 waves (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 one and it 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 backreaction of the GW leaving the horizon during the de Sitter phase and re-entering 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 ($H_{dS}\laq 10^{-6}~M_{P}$) in order to a have a sufficiently long stiff phase reaching an energy scale of the order of the $1~{\rm TeV}$ and even lower if we want the stiff phase to touch the hadronic era (corresponding to $T_{had}\sim 140~{\rm MeV}$). By looking more positively at our exercise we see that, if an inflationary phase is followed by a stiff phase, there exist an appealing possibility of "graviton reheating" whose effective temperature can be generally quite low.