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

186 Generalities on the stochastic GW backgroundtheir Fourier components satisfies〈ñ ∗ i ( f )ñ j ( f ′ )〉=δ( f − f ′ 1)δ ij 2S n (i) (| f |), (12.10)where the function S n (| f |), with dimensions Hz −1 , is known as the noise powerspectrum 3 . The factor 1 2is conventionally inserted in the definition so that the totalnoise power is obtained integrating S n ( f ) over the physical range 0 ≤ f < ∞,rather than from −∞ to ∞.As discussed in [4, 7–10], for any given form of the signal, i.e. for any givenfunctional form of h 2 0 gw( f ), it is possible to find explicitly the filter functionQ(t) which maximizes the signal-to-noise ratio (SNR). It can be shown [7] thatunder the above-mentioned assumptions, the Fourier transform of this optimalfilter and the corresponding value of the optimal SNR turn out to be, respectively:γ(| f |) gw (| f |)Q( f ) = λ| f | 3 S n (1) (| f |)S n (2) (| f |) , (12.11)[ ( )9H 4 ∫ ∞0SNR =8π 4 F 2 γ 2 (| f |) 2 ]gw (| f |) 1/4T d f0 f 6 S n (1) (| f |)S n (2) , (12.12)(| f |)where F is a normalization factor less than one depending only upon the geometryof the detectors, and λ is a real overall normalization constant that, assuming forthe spectrum a power-law gw ( f ) = β f β (with β = constant), is fixed bythe condition 〈S〉 = β T . The function γ(f ) appearing in both formulae isthe overlap reduction function introduced in [10], which takes into account thedifference in location and orientation of the two detectors. At this stage let usonly remark that γ(f ) is maximum and equal to one in the case of two detectorswith the same location and orientation. The detailed expression for F and γ(f )will be discussed in section 12.3. Finally, let us note that in equation (12.12) wehave taken into account the fact that what has been called S in equation (12.8)is quadratic in the signals and, with usual definitions, it contributes to the SNRsquared. This differs from the convention used in [4, 7].In principle the expression for the SNR, equation (12.12), is all that weneed in order to discuss the possibility of detection of a given GW background.However, it is useful, for order of magnitude estimates and for intuitiveunderstanding, to express the SNR in terms of a characteristic amplitude of thestochastic GW-background and of a characteristic noise level, although, as wewill see, the latter is a quantity that describes the noise only approximately, incontrast to equation (12.12) which is exact. We will introduce these quantities inthe next two subsections.3 Unfortunately there is not much agreement about notations in the literature. The noise powerspectrum, that we denote by S n ( f ) following, for example, [10], is called P( f ) in [4]. Other authorsuse the notation S h ( f ), which we instead reserve for the power spectrum of the signal. To make thingsworse, S n is sometimes defined with or without the factor 1 2 in equation (12.10).