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Astron. Astrophys. 351, 811-814 (1999)

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3. Correlation of the Explorer and Nautilus detectors

The detectors used for the crosscorrelation experiment are Explorer (Astone et al. 1993) at CERN and Nautilus (Astone et al. 1997a) in Frascati: two aluminum cylinders with mass of 2200 kg, equipped with a capacitive resonant transducer. These detectors are parallel but, due to the distance ([FORMULA] km), the crosscorrelation should be corrected by a factor that is 6 (Michelson 1987, Vitale et al. 1997).

Their noise spectral densities have minima at the two resonances, [FORMULA] and [FORMULA] Hz, as shown in Fig. 1.

[FIGURE] Fig. 1. Averaged noise spectra of Explorer and Nautilus used for the cross-correlation experiment. The minima correspond to the resonances of the detectors.

We recall (Astone et al. 1997b) that the spectral density of a gravitational wave background which can be measured with signal to noise ratio equal to unity with a resonant detector is related to the detector characteristics by the formula


Here Q is the merit factor of the detector, M the mass, [FORMULA] the temperature, v the sound velocity in the material, [FORMULA] the resonance frequency, and [FORMULA] a frequency dependent term that reduces to unity for [FORMULA].

The bandwidth is usually smaller than 1 Hz, thus we can neglect the frequency dependences concerning both the g.w. background and the function [FORMULA] and use in the analysis Eq. (7), only considering the detector spectra frequency dependence. For this experiment we have tuned 2 the two detectors in order to have the same resonance frequency at one of the two resonant modes, [FORMULA]

We choose an overlapping bandwidth of [FORMULA], from 907.1508 to 907.2574 Hz: in this bandwidth the averaged Nautilus spectrum is constant at the level [FORMULA], and the Explorer spectrum varies a factor 2, from [FORMULA] to [FORMULA]

The Explorer data are sampled in a bandwidth of the order of 27.5 Hz from 900 to 927.5 Hz, with a sampling time of 18.18 ms; the Nautilus data in a bandwidth from 900 to 955.0, with a sampling time of 9.09 ms.

We performed the experiment, by tuning the detectors, only for a relatively short period of time, obtaining [FORMULA] hours of "good" data, on which the crosscorrelation was applied. The overlapped data cover a period of 12.57 hours from February, [FORMULA], 1997, 22 h, 18 m (day=35466.9298) to [FORMULA], 12 h, 11 m (day=35467.5916). The noise spectra of Explorer and Nautilus during this period are shown in Fig. 1.

We considered "good" the Explorer basic FFT's where the noise spectral density, averaged in the bandwidth of 0.1 Hz around the resonance frequency, was smaller than [FORMULA] thus vetoing [FORMULA] of the spectra. From these data alone, using Eq. (2), we obtained the limit


As regards Nautilus, we considered "good" the FFT's with noise below [FORMULA] thus vetoing less than the [FORMULA] of the spectra. From the Nautilus data alone, we obtained


We remark here that at the time we performed the experiment the Nautilus sensitivity was worse than usual (Astone et al. 1997a), roughly by a factor 10 (as this detector usually works at the same sensitivity of Explorer).

The above thresholds has been chosen to optimize the contribution of [FORMULA] in Eq. (6). The Explorer detector was the one which limited the overall bandwidth of the experiment, as clearly shown in Fig. 1.

The result of the crosscorrelation analysis of the data of the two detectors is shown in Fig. 2: the lower curve shows the modulus of the cross spectrum [FORMULA], compared to the square root of the product of the two spectra [FORMULA] 3. In the case of total correlation the two curves should coincide. In case of null correlation we expect the standard deviation be smaller than that obtained with only one detector by a factor [FORMULA], when integrating over the overlapping bandwidth [FORMULA] Hz. This factor represents the sensitivity improvement of the crosscorrelation experiment with respect to the use of only one detector, if they were "near" and had the same sensitivity.

[FIGURE] Fig. 2. The lower curve shows the result of the cross-correlation experiment (modulus of the cross spectrum). The upper curve shows the square root of the product of the spectra obtained with the two detectors. In case of total correlation of the two detectors the two curves should coincide. In the figure the factor [FORMULA] is [FORMULA] at each frequency.

The numerical results obtained by averaging over a 0.1 Hz bandwidth are:




By expressing the above in terms of [FORMULA], that is using Eq. (2) (with [FORMULA] in place of [FORMULA]) and taking into account the factor 6 due to the distance, we get


Comparing this result with those given by Eqs. (10) and (11), we notice that the gain obtained by crosscorrelating the two data sets is a factor [FORMULA] for Explorer and a factor [FORMULA] for Nautilus.

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© European Southern Observatory (ESO) 1999

Online publication: November 16, 1999