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

98 Resonant detectors for gravitational waves and their bandwidthcan combine n Fourier spectra in one unique spectrum taking into considerationalso the phase of the signal. The final spectrum has the same characteristics ofthe spectrum over the entire period t m and equation (8.36) still applies. (b) Theexact frequency is unknown. In this case when we combine the n spectra we loseinformation on the phase. The result is that the final combined spectrum over theentire period has a larger variance and the left part of equation (8.36) has to bechanged to√2N(ω 0 )h = √tm △t . (8.37)We come now to the measurement of the GW stochastic background [11,12,16]. Using one detector, the measurement of the noise spectrum corresponding toequation (8.11) only provides an upper limit for the GW stochastic backgroundspectrum, since the noise is not so well known that we can subtract it. Theestimation of the GW stochastic background spectrum can be considerablyimproved by employing two (or more) antennae, whose output signals are crosscorrelated. Let us consider two antennae, that may in general be different, withtransfer functions T 1 and T 2 , and displacements ξ 1 and ξ 2 : the displacement crosscorrelationfunction∫R ξ1 ξ 2(τ) = ξ 1 (t + τ)ξ 2 (t) dt (8.38)only depends on the common excitation of the detectors, due to the GW stochasticbackground spectrum S gw acting on both of them, and is not affected by the noisesacting independently on the two detectors.The Fourier transform of equation (8.38) is the displacement cross spectrum.This spectrum, multiplied by T 1 T 2 (4L 2 /π 4 ), is an estimate of the gravitationalbackground S gw . The estimate, obtained over a finite observation time t m , has astatistical error. It can be shown [11] that√Nh1 (ω)N h2 (ω)δS gw (ω) = S gw (ω) = √ (8.39)tm δ fwhere t m is the total measuring time and δ f is the frequency step in the powerspectrum. From equation (8.11) we have the obvious result that, for resonantdetectors, the error is smaller at the resonances. If the resonances of the twodetectors coincide the error is even smaller. In practice, it is better to have twodetectors with the same resonance and bandwidth. If one bandwidth is smaller, theminimum error occurs in a frequency region overlapping the smallest bandwidth.From the measured S gw one can calculate the value of the energy density ofthe stochastic GW referred to as the critical density (the energy density neededfor a closed universe). We have = 4π 2 f 23 H 2 S gw( f ) (8.40)where H is the Hubble constant.