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

204 Generalities on the stochastic GW backgroundabout one order of magnitude [20].While resonant bars have been taking data for years, spherical detectors areat the moment still at the stage of theoretical studies (although prototypes mightbe built in the near future), but could reach extremely interesting sensitivities. Inparticular, the correlation between VIRGO and one sphere with a diameter of 3 m,made of Al 5056 (M = 38 ton), gives sensitivities that are, respectively, h 2 0 gw ∼2 × 10 −5 , if the sphere is located in the AURIGA site, and h 2 0 gw ∼ 4 × 10 −5 ifthe sphere is at the NAUTILUS site [19]. Instead, the correlation of two spheres ofthis type but located at the same site could reach a sensitivity h 2 0 gw ∼ 4 × 10 −7[19]. All these figures improve using a denser material or increasing the spherediameter, but it might be difficult to build a heavier sphere. Another verypromising possibility is given by hollow spheres [16]. The theoretical studiesof [16] suggest for the correlation of two 40-ton colocated hollow spheres, madeof Al 5056, a sensitivity of h 2 0 gw ∼ 6 × 10 −8 at f = 218 Hz (one year ofobservation and SNR = 1).12.4.3 More than two detectorsWhen the outputs of M > 2 detectors are available, the information aboutthe magnitude of the stochastic GW background can be extracted in two ways:combining the measurements from each detector pair or directly correlating theoutputs of the detectors. Both these techniques have been extensively treatedin [7], and here we shall only review the key results obtained in this analysis.Multiple detector pairsWe indicate withS (ij)1, S (ij)2,...,S n (ij)ij, (i, j = 1,...,M)the n ij different measurements, each of length T , of the optimally-filtered crosscorrelationsignal S (ij) between the ith and jth detectors (see equation (12.8)).Under the following hypothesis:• T ≫ the light travel time between any pair of detectors;• the optimal filter functions (see equation (12.8)) for each detector pair arenormalized in a way that〈S (ij) 〉= 1n ij∑n ijS (ij)k= β T, ∀(i, j)k=1for a stochastic background having a power-law spectrum gw ( f ) = β f β ;• the noise is large, i.e. the covariance matrix built from the cross-correlationsignals taken during the same time interval T is approximately diagonalC (ij)(kl) =〈S (ij) S (kl) 〉−〈S (ij) 〉〈S (kl) 〉≃δ ik δ kl (σ (ij) ) 2