  The sensitivity achievable by a pair of VIRGO detectors to stochastic and
isotropic gravitational wave backgrounds of cosmological origin is discussed in
view of the development of a second VIRGO interferometer. We describe a
semi-analytical technique allowing to compute the signal-to-noise ratio for
(monotonic or non-monotonic) logarithmic energy spectra of relic gravitons of
arbitrary slope. We apply our results to the case of two correlated and
coaligned VIRGO detectors and we compute their achievable sensitivities. The
maximization of the overlap reduction function is discussed. We focus our
attention on a class of models whose expected sensitivity is more promising,
namely the case of string cosmological gravitons. We perform our calculations
both for the case of minimal string cosmological scenario and in the case of a
non-minimal scenario where a long dilaton dominated phase is present prior to
the onset of the ordinary radiation dominated phase. In this framework, we
study possible improvements of the achievable sensitivities by selective
reduction of the thermal contributions (pendulum and pendulum's internal modes)
to the noise power spectra of the detectors. Since a reduction of the shot
noise does not increase significantly the expected sensitivity of a VIRGO pair
(in spite of the relative spatial location of the two detectors) our findings
support the experimental efforts directed towards a substantial reduction of
thermal noise.
