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

Definitions 185The value of H 0 is usually written as H 0 = h 0 × 100 km s −1 Mpc −1 , where h 0parametrizes the existing experimental uncertainty. A conservative estimate forthis parameter is in the range 0.50 < h 0 < 0.65 (see [5] and references quotedtherein).It is not very convenient to normalize ρ gw to a quantity, ρ c , which isuncertain: this uncertainty would appear in all the subsequent formulae, althoughit has nothing to do with the uncertainties on the GW background. Therefore, werather characterize the stochastic GW background with the quantity h 2 0 gw( f ),which is independent of h 0 . All theoretical computations of a relic GW spectrumare actually computations of dρ gw /dln f and are independent of the uncertaintyon H 0 . Therefore, the result of these computations is expressed in terms of h 2 0 gw,rather than of 2 gw . Under the assumption that the stochastic background ofgravitational radiation is isotropic, unpolarized, stationary and Gaussian (see [4,7]for details), it is completely specified by its spectrum gw ( f ).To detect a stochastic GW background the optimal strategy consists inperforming a correlation between two (or more) detectors, since, as we willdiscuss below, the signal will be far too low to exceed the noise level in anyexisting or planned single detector (with the exception of the space interferometerLISA). The strategy has been discussed in [8–10], and is clearly reviewed in [4,7].Let us recall the main points of the analysis.The cross-correlation between the outputs s 1 (t) and s 2 (t) of two detectors isdefined asS =∫ T/2−T/2dt∫ T/2−T/2dt ′ s 1 (t)s 2 (t ′ )Q(t, t ′ ), (12.8)where T is the total integration time (e.g. one year) and Q is a filter function.The output of a single detector characterized by an intrinsic noise n(t) and onwhich acts a gravitational strain h(t) is of the form s(t) = n(t) + h(t). Thegravitational strain can be expressed in terms of the amplitudes h +,× of the wavein the following way [11]h(t) = F + h + (t) + F × h × (t), (12.9)where the detector pattern functions F +,× are introduced, which depend on thelocation, orientation and geometry of the detector and the direction of arrivalof the GW and its polarization (see section 12.3). These functions have valuesin the range 0 ≤ |F +,× | ≤ 1. The noise intrinsic to the detector is assumedstationary, Gaussian and statistically independent on the gravitational strain. As aconsequence of the assumed stationariety of both the stochastic GW backgroundand noise, the filter function turns out to be Q(t, t ′ ) = Q(t −t ′ ). Furthermore, thenoises in the two detectors are assumed uncorrelated, i.e. the ensemble average of2 This simple point has occasionally been missed in the literature, where one can find the statementthat, for small values of H 0 , gw is larger and therefore easier to detect. Of course, it is larger onlybecause it has been normalized using a smaller quantity.