2. Stochastic g.w. search
The sensitivity of a g.w. antenna is usually given in terms of its strain noise spectral density or spectral amplitude (unit of ). Using it is easy to infer the detector sensitivity for various classes of signals, as bursts, periodic signals and stochastic g.w. (Astone et al. 1997b).
As regards stochastic g.w., the dimensionless function of the frequency (Brustein et al. 1995)
is related to the detector sensitivity, by the formula (Astone et al. 1996):
where is the Hubble constant. Then we have
Using one detector the measurement of its noise spectrum only provides an upper limit for the g.w. stochastic background spectrum. This limit can be considerably improved, or even an estimation of the spectrum can be attempted by crosscorrelating the output signals of two (or more) antennas (Michelson 1987, Astone et al. 1997b). Let us consider two "near" and "aligned" antennas 1 with spectral densities and . The crosscorrelation function only depends on the common excitation of the detectors, as due to the g.w. stochastic background spectrum acting on both of them, and is not affected by the noises acting independently on the two detectors.
As the analysis of the data is usually performed in the frequency domain, we consider the cross spectrum that is the Fourier transform of , where all the signals are properly normalized to represent the input strain of the detectors. The cross spectrum is a complex quantity which is identically zero, for each frequency, in the case of no correlation between the two detectors, therefore ideally providing unlimited sensitivity for any common excitation. The actual sensitivity, however, is limited because the estimate obtained over a finite observation time has a statistical error. It can be shown (Bendat & Piersol 1966) that the standard deviation of each sample of the spectrum is
where is the total measuring time and is the frequency step in the spectrum.
If the g.w. background spectrum is expected (Brustein et al. 1995) to be approximately constant over a few hertz or a few tens of hertz, the statistical error can be reduced by estimating its intensity over spectral intervals larger than the spectral step . In this case the uncertainty of the estimate, obtained from Eq. (5),
represents the overall sensitivity of the experiment.
The above expression shows that the spectral interval has to be carefully chosen: as large as possible to increase the statistics, but small enough to avoid including spectral samples of larger value, outside the flat regions of the two noise spectra.
where with are the Fourier transforms of the data of each detector, properly normalized, and * indicates the complex coniugate.
The optimal sensitivity is obtained (Michelson 1987, Flanagan 1993, Vitale et al. 1997) when the detectors, besides being aligned, are at a distance , where is a function of the frequency of the wave, , roughly 50 km at 1 kHz.
If the distance is there is a decrease in the efficiency of the detection, as the crosscorrelation falls down due to the phase shift between the waves acting on the two detectors. We have, in general:
where is the overlapping reduction function, discussed by Flanagan (1993). The quantity is a function of the frequency of the wave, of the location of the detectors and of their relative orientation, which is equal to the unity for "near" and "aligned" detectors.
We note, for sake of completeness, that Eq. (7) is the optimal detection strategy only if the integration bandwidth is so small that we can neglect the frequency variations of , of the detector noise spectrum and of the signal . A larger bandwidth requires (Michelson 1987, Flanagan 1993, Vitale et al. 1997) to apply to the data a weight function which takes into account all the frequency dependences. This is not the case in the present analysis.
2.1. Use of the frequency domain data base
The analysis of the data is done in practice using a frequency domain data base, where each basic FFT is completely characterized by recording information on the status of the experimental apparatus and on the quality of the data. This allows to take into account the data taking interruptions and the non-stationarity of the noise.
For a crosscorrelation experiment the length of the basic FFT is not crucial, as we need to integrate over the overlapping bandwidths of the experiments we want to correlate. We have used the length optimized for the pulsar search, that is hours.
The analysis procedure is based on the use of Eq. (7) and it is described in (Astone 1997). We note here that in the organization of the data base for crosscorrelation analysis common criteria should be used for the different detectors (the lengths of the basic FFTs, the rule for choosing the inizial time of the first spectrum of each new run, and possibly the sampling times)
© European Southern Observatory (ESO) 1999
Online publication: November 16, 1999