  A large class of quintessential inflationary models, recently proposed by
Peebles and Vilenkin, leads to post-inflationary phases whose effective
equation of state is stiffer than radiation. The expected gravitational waves
logarithmic energy spectra are tilted towards high frequencies and
characterized by two parameters: the inflationary curvature scale at which the
transition to the stiff phase occurs and the number of (non conformally
coupled) scalar degrees of freedom whose decay into fermions triggers the onset
of a gravitational reheating of the Universe. Depending upon the parameters of
the model and upon the different inflationary dynamics (prior to the onset of
the stiff evolution) the relic gravitons energy density can be much more
sizeable than in standard inflationary models, for frequencies larger than 1
Hz. We estimate the required sensitivity for detection of the predicted
spectral amplitude and show that the allowed region of our parameter space
leads to a signal smaller (by one 1.5 orders of magnitude) than the advanced
LIGO sensitivity at a frequency of 0.1 KHz. The maximal signal, in our context,
is expected in the GHz region where the energy density of relic gravitons in
critical units (i.e. $h_0^2 \Omega_{GW}$) is of the order of $10^{-6}$, roughly
eight orders of magnitude larger than in ordinary inflationary models. Smaller
detectors (not necessarily interferometers) can be relevant for detection
purposes in the GHz frequency window. We suggest/speculate that future
measurements through microwave cavities can offer interesting perspectives.
