(ed.). Gravitational waves (IOP, 2001)(422s).

206 Generalities on the stochastic GW backgroundMany-detector correlationThe M-detector correlation signal S is the obvious generalization ofequation (12.8) for a two-detector:∫ T/2∫ T/2S = dt 1 ... dt M s 1 (t 1 )s 2 (t 2 )...s M (t M )Q(t 1 ,...,t M ).−T/2 −T/2Let us remark that, as a consequence of the fact that each detector output signalis assumed to be a random Gaussian variable having zero mean value, 〈S〉 isdifferent from zero only for an even number of detectors (M = 2N).Under the same hypothesis assumed in section 12.2.1 for the treament of thetwo-detector case, the authors of [7] show that the SNR for the optimally-filtered2N-detector correlation is given bySNR 4 ≈ ∑ {...}(SNR (12) SNR (34) ...SNR (2N,2N−1) ) 4 (12.56)where the sum run over all the possible permutations of the sequence{(ij), (kl),...,(pq)} with (i < j, k < l,...,p < q). In the case of afrequency-independent background, with fixed detection and false alarm rates,the minimum detectable value of gw from data obtained via a 2N-detectorcorrelation experiment, is given by(1) 2 gw (δ, α)[= C(δ, α) ∑ ()12 ] 1/N{...} gw (12) (δ, α) gw (34) (δ,α)... (2N−1,2N)gw (δ, α)(12.57)withC(δ, α) ={ √ 2[erfc −1 (2α) − erfc −1 (2δ)]} 2(N−1)/N .In table 12.10 are reported the minimum values of h 2 0 gw obtained fromequation (12.57) for the four-interferometer correlations. It is important to notethat, as clearly shown from the comparison with tables 12.8 and 12.9, theseminimum values are always greater than those for optimal combination of datafrom multiple detector pairs.By summarizing the results of this section we can say that the improvementin sensitivity obtained using more detectors is not large compared to the case ofa single pair correlation (see table 12.4). This is due to the fact that correlating2, 4,...,2N detectors or combining in a optimal way data from multiple detectorpairs, one does not change the general dependence that gw ∼ T −1/2tot