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

200 Generalities on the stochastic GW backgroundTable 12.4. Minimum values of h 2 0 gw for cross-correlation measurements betweenVIRGO and the major interferometers for one year of observation, and different valuesof the pair (δ, α).Correlation (0.95, 0.05) (0.95, 0.10) (0.90, 0.10)VIRGO ⋆ LIGO-LA 5.3 × 10 −6 4.7 × 10 −6 4.2 × 10 −6VIRGO ⋆ LIGO-WA 6.4 × 10 −6 5.7 × 10 −6 5.0 × 10 −6VIRGO ⋆ GEO-600 7.2 × 10 −6 6.4 × 10 −6 5.6 × 10 −6VIRGO ⋆ TAMA-300 1.2 × 10 −4 1.1 × 10 −4 9.5 × 10 −5SNR verifies the following boundwhere the complementary error functionSNR 2 ≥ √ 2[erfc −1 (2α) − erfc −1 (2δ)]erfc(x) ∼ √ 2 ∫ ∞dz e −z2πhas been introduced. Therefore, in the case of a stochastic GW background havinga constant frequency spectrum gw ( f ) = gw , from equation (12.12) one has:x gw ≥ 4π 23H 2 0[ ∫1 ∞F √ d fT0γ 2 ( f )f 6 S (1)n ( f )S (2)n ( f )] −1/2[erfc −1 (2α) − erfc −1 (2δ)].(12.50)Two interferometersThe minimum values of h 2 0 gw for cross-correlation measurements betweendifferent interferometer pairs (F = 2/5 in equation (12.50)) in the case T = 10 7 s(4 months), α = 0.05, and δ = 0.95, are summarized in table II of [7]. Intable 12.4 we report the minimum values of h 2 0 gw for the correlation of VIRGOwith the major interferometers for one year of observation and different values ofthe detection and false alarm rates 4 .We now consider the sensitivity that could be obtained at VIRGO if theplanned interferometer were correlated with a second identical interferometerlocated at a few tens of kilometres from the first, and with the same orientation(the case A at the end of section 12.3.1).A rough estimate of the sensitivity can be given using the same line ofreasoning developed in section 12.2.3. From equation (12.49) we see that we4 Let us note that α and δ need not sum to one.