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

190 Generalities on the stochastic GW backgroundIf, in the integral giving the optimal SNR, equation (12.12) orequation (12.29), we consider only a range of frequencies f such that theintegrand is approximately constant, we can writeSNR ≃[2F 2 T f γ 2 ( f )S 2 h ( f )S 2 n ( f ) ] 1/4=[F 2 T f γ 2 ( f )h 4 c ( f )2 f 2 S 2 n ( f ) ] 1/4. (12.31)The right-hand side of equation (12.31) is proportional to h c ( f ), and we cantherefore define h n ( f ) equating the right-hand side of equation (12.31) toh c ( f )/h n ( f ), so that[ ]1 fSn ( f ) 1/2h n ( f ) =( 1 2 T f . (12.32))1/4 F|γ(f )|From the derivation of equation (12.32) we can understand the limitationsimplicit in the use of h n ( f ). It gives a measure of the noise level onlyunder the approximation that leads from equation (12.29), which is exact, toequation (12.31). This means that f must be small compared to the scaleon which the integrand in equation (12.29) changes, so that γ(f )S h ( f )/S n ( f )must be approximately constant. In a large bandwidth this is non-trivial, andof course depends also on the form of the signal; for instance, if h 2 0 gw is flat,then S h ( f ) ∼ 1/f 3 . For accurate estimates of the SNR there is no substitutefor a numerical integration of equation (12.12) or equation (12.29), unless thefrequency range f in which we are interested is sufficiently small. However, fororder of magnitude estimates, equation (12.27) for h c ( f ) and equation (12.32) forh n ( f ) are simpler to use, and they have the advantage of clearly separating thephysical effect, which is described by h c ( f ), from the properties of the detectors,that enter only in h n ( f ).Equation (12.32) also shows very clearly the advantage of correlating twodetectors compared with the use of a single detector. With a single detector, theminimum observable signal, at SNR = 1, is given by the condition S h ( f ) ≥S n ( f ). This means, from equation (12.21), a minimum detectable value for h c ( f )given byh 1dmin ( f ) = (2 fS n( f )) 1/2 , (12.33)where the superscript 1d reminds us that this quantity refers to a single detector.From equation (12.32), by indicating with ¯h 1dminthe minimum detectable signalfor a detector which noise level equals the geometric average of the noise levelsof two detectors in coincidence, the minimum detectable signal for this detectorpair is:h 2dmin ( f ) = 1 ¯h 1dmin ( f )(2T f ) 1/4 [F|γ(f )|] 1/2≃ 1.1 × 10 −2 ( 1Hzf) 1/4 ( ) 1 year 1/4 ¯h 1dTmin ( f ) . (12.34)[F|γ(f )|] 1/2