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

Sensitivity and bandwidth of resonant detectors 93in terms of the signal-to-noise ratio (SNR). For a GW excitation with powerspectrum S h (ω), the spectrum of the corresponding bar end displacement isWe can then write the SNR asS ξ = 4L2 ω 4 S hπ 4 1(ω 2 − ω 2 0 )2 + ω2 ω 2 0Q 2 . (8.8)SNR = S ξS n ξ= 4L2 ω 4 S h1π 4 S f1 + Ɣ(Q 2 (1 − ω2m 2ω 2 0) 2 + ω2 )ω02(8.9)where the quantity Ɣ is defined by [7]Ɣ = S 0β 1α 2 ¯ξ 2 ∼ T n(8.10)β QT eT n is the noise temperature of the electronic amplifier and β indicates the fractionof energy which is transferred from the bar to the transducer. It can readily beseen that Ɣ ≪ 1.The GW spectrum that can be detected with SNR = 1 is:⎛ ⎛ ( ) ⎞⎞S h (ω) = π 2 kT e ω03 21⎝1MQv 2 ω 3 + Ɣ ⎝Q 2 1 − ω2ωω02 + ω2 ⎠⎠ ω02 (8.11)where v is the sound velocity in the bar material (v = 5400 m s −1 in aluminum).For ω = ω 0 we obtain the highest sensitivityS h (ω 0 ) = π 2 kT e 1MQv 2 (8.12)ω 0having considered Ɣ ≪ 1.Another useful quantity often used is the spectral amplitude˜h = √ S h . (8.13)We remark that the best spectral sensitivity, obtained at the resonancefrequency of the detector, only depends, according to equation (8.12), on thetemperature T , on mass M and on the quality factor Q of the detector, providedT = T e . Note that this condition is rather different from that required for optimumpulse sensitivity (see later). The bandwidth of the detector is found by imposingthat S h (ω) is equal to twice the value of S h (ω 0 ). We obtain, in terms of thefrequency f = ω/2π, that△ f = f 0Q1√Ɣ. (8.14)