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

The physics of resonant mass detectors 3510 −1410 −16h (Hz −1/2 )10 −1810 −2010 −2210 −2410 1 10 2 10 3 10 4frequency (Hz)Figure 3.4. Noise curves of the initial LIGO (left) and VIRGO (right) detectors. TheVIRGO curve is in strain per root Hz, as the GEO curves earlier. The LIGO curve iscalibrated in metres per root Hz, so to convert to a limit on h one multiplies by the squareroot of the bandwidth and divides by the length of the detector arm, 4000 m.Let us suppose we have a typical bar with length L ∼ 1 m. (In the future,spheres may go up to 3 m.) Depending on the length of the bar and its material,the resonant frequency will be f ∼ 500 Hz to 1.5 kHz and mass M ∼ 1000 kg.A short burst gravitational wave h will make the bar vibrate with an amplitudeδl gw ∼ hl ∼ 10 −21 m.Unlike the interferometers, whose response is simply given by this equation,the bars respond in a complicated way depending on all their internal forces.However, if the duration of the wave is short, the amplitude will be of the sameorder as that given here. If the wave has long duration and is near the bars resonantfrequency, then the signal can build up to much larger amplitudes. Normally, bardetector searches have been targeted at short-duration signals.The main sources of noise that compete with this very small amplitude are:• Thermal noise. This is the most serious source of noise. Interferometers canlive with room-temperature thermal noise because their larger size makestheir response to a gravitational wave larger, and because they observe atfrequencies far from the resonant frequency, where the noise amplitude islargest. However, bars observe at the resonant frequency and have a veryshort length, so they must reduce thermal noise by going to low temperatures.The best ultra-cryogenic bars today operate at about T = 100 mK, wherethe rms amplitude of vibration is found by setting the kinetic energy of thenormal mode, M(δ ˙l) 2 /2, equal to kT/2, the equipartition thermal energy ofa single degree of freedom. This gives then〈δl 2 〉 1 2th=( kT4π 2 Mf 2 )12∼ 6 × 10 −18 m.