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

92 Resonant detectors for gravitational waves and their bandwidthspectrum due to the thermal noise isS f = 2ω 0Q mkT e (8.2)where T e is the equivalent temperature which includes the effect of the backactionfrom the electronic amplifier.By referring the noise to the displacement of the bar ends, we obtain thepower spectrum of the displacement due to Brownian noise:S B ξ= S fm 2 1(ω 2 − ω 2 0 )2 + ω2 ω 2 0Q 2 . (8.3)From this we can calculate the mean square displacement¯ ξ 2 = kT emω 2 0(8.4)that can also be obtained, as is well known, from the equipartition of the energy.To this noise we must add the wide-band noise due to the electronic amplifier(the contribution to the narrow-band noise due to the amplifier has already beenincluded in T e ).For the sake of simplicity we consider an electromechanical transducer thatconverts the displacement of the detector in a voltage signalV = αξ (8.5)with the transducer constant α (typically of the order of 10 7 Vm −1 ). Thus, theelectronic wide-band power spectrum, S 0 , is expressed in units of V 2 Hz −1 andthe overall noise power spectrum referred to the bar end is given byS n ξ = 2kT eω 0mQ1+ S 0(ω 2 − ω0 2)2 + ω2 ω02 α 2 . (8.6)Q 2We now calculate the signal due to a gravitational wave with amplitude h andwith optimum polarization impinging perpendicularly to the bar axis. The bardisplacement corresponds [6] to the action of a forcef = 2 mLḧ. (8.7)π 2The bar end spectral displacement due to a flat spectrum of GW (as for a deltaexcitation)is similar to that due to the action of the Brownian force. Therefore, ifonly the Brownian noise were present, we would have a nearly infinite bandwidth,