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

36 Gravitational-wave detectorsThis is far larger than the gravitational wave amplitude. In order to detectgravitational waves against this noise, bars are constructed to have a veryhigh Q, of order 10 6 or better.The reason that bars need a high Q is different from the reason thatinterferometers also strive for high-Q systems. To see how Q helps bars,we recall that Q is defined as Q = f · τ where f is the resonant frequencyof the mode and τ is the decay time of the oscillations. If Q is large, thenthe decay time is long. If the decay time is long, then the amplitude ofoscillation changes very slowly in thermal equilibrium. Essentially, the bar’smode of vibration changes its amplitude by a random walk with very smallsteps, taking time Q/ f ∼ 1000 s to change by the full amount. On the otherhand, a gravitational wave burst will cause an amplitude change in time ofthe order 1 ms, during which the thermal noise will have random walked toan expected amplitude change that is Q 1 2 = ( 1000 1ms s ) 1 2 times smaller. In thiscase(〈δl 2 〉 1 )1kT 22th:1ms=∼ 6 × 10 −214π 2 Mf 2 m.QThus, thermal noise only affects a measurement to the extent that it changesthe amplitude of vibration during the time of the gravitational-wave burst.This change is similar to that produced by a gravitational wave of amplitude6 × 10 −21 . It follows that, if thermal noise were the only noise source,bars would be operating at around 10 −20 today. Bar groups expect in fact toreach this level during the next few years, as they reduce the other competingsources of noise. Notice that the effect of thermal noise has nothing todo with the frequency of the disturbance, so it is not the reason that barsobserve near their resonant frequency. In fact, both thermal impulses andgravitational-wave forces are mechanical forces on the bar, and the ratio oftheir induced vibrations is the same at all frequencies for a given appliedimpulsive force.• Sensor noise. Because the oscillations of the bar are very small, bars requirea transducer to convert the mechanical energy of vibration into electricalenergy, and an amplifier that increases the electrical signal to a level whereit can be recorded. If the amplifier were perfect, then the detector wouldin fact be broadband: it would amplify the smaller off-resonant responsesjust as well as the on-resonance ones. Conversely, real bars are narrow-bandbecause of sensor noise, not because of their mechanical resonance.Unfortunately sensing is not perfect: amplifiers introduce noise and thismakes small amplitudes harder to measure. The amplitudes of vibration arelargest in the resonance band near the resonant frequency f 0 , so amplifiernoise limits the detector sensitivity to frequencies near f 0 . Now, the signal(a typical gravitational-wave burst) has a duration time τ w ∼ 1 ms, so theamplifier’s bandwidth should be at least 1τ w in order for it to be able torecord a signal every τ w . In other words, bars require amplifiers with very