We recall here that upper limits for the stochastic background, in the frequency range of kHz, have been set using bar detectors at room temperature in Glasgow[2,3] ( ), interferometers[4] ( ) (Garching-Glasgow), the small antenna Altair[5], and the two resonant g.w. detectors Explorer and Nautilus[6].

The result obtained with Explorer and Nautilus is very interesting as it is the first time that two cryogenic resonant detectors have been used to do this measurement. As this is presently the best upper limit in the frequency range of 1 kHz, we recall here briefly the result. The detector noise spectral densities have minima at the two resonances, around 904 and 921 Hz, and are shown in Fig.. The analysis has been done over a bandwidth of order of 0.05 Hz , . In this bandwidth the averaged Nautilus spectrum is constant at the level , and the Explorer spectrum varies by a factor 2, in the range

The overlapped data cover a period of from February, 7th, 1997, 22 h, 18 m (day=35466.9298) to the 8th, 12 h, 11 m (day=35467.5916).

The result of the cross-correlation analysis of the data of the two detectors
is shown in Fig.2:
the lower curve shows the modulus of the cross spectrum
, compared to
the square root of the product of the two spectra
.
In the case of total correlation the two curves should coincide
^{a}.

In case of null correlation we expect the standard deviation to be smaller than that obtained with only one detector by a factor , when integrating over the bandwidth . This factor represents the sensitivity improvement of the cross-correlation experiment with respect to the use of only one detector, if they were ``near" and had the same sensitivity.

The result obtained by averaging over a 0.1 Hz bandwidth is:

By expressing the above in terms of
, that is using Eq. (with
in place of ) and taking into account the
factor , that is the worsening due to the distance between Explorer and
Nautilus, we get

We note that by extending the period of correlation to one year, we can obtain, with the detectors in operation now, an upper limit less than unity. This would be already very interesting for the various theoretical scenarios of the gravitational wave background.