  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.
