4.1 Principles of the operation of resonant mass detectors

A typical “bar” detector consists of a cylinder of aluminum with a length ℓ ∼ 3 m, a very narrow resonant frequency between f ∼ 500 Hz and 1.5 kHz, and a mass M ∼ 1000 kg. A short gravitational wave burst with h ∼ 10− 21 will make the bar vibrate with an amplitude
− 21 δℓgw ∼ hℓ ∼ 10 m. (38 )
To measure this, one must fight against three main sources of noise.

  1. Thermal noise. The original Weber bar operated at room temperature, but the most advanced detectors today, Nautilus [53] and Auriga [58Jump To The Next Citation Point], are ultra-cryogenic, operating at T = 100 mK. At this temperature the root mean square (rms) amplitude of vibration is
    ( )1 ∕2 21∕2 --kT----- − 18 ⟨δ ℓ⟩th = 4π2M f2 ∼ 6 × 10 m. (39)
    This is far larger than the gravitational wave amplitude expected from astrophysical sources. But if the material has a high Q (say, 106) in its fundamental mode, then that changes its thermal amplitude of vibration in a random walk with very small steps, taking a time Q ∕f ∼ 1000 s to change by the full amount. However, a gravitational wave burst will cause a change in 1 ms. In 1 ms, thermal noise will have random-walked to an expected amplitude change 1∕2 1∕2 (1000 s∕1 ms) = Q times smaller, or (for these numbers)
    ( ) 2 1∕2 kT 1∕2 −21 ⟨δℓ ⟩th:1 ms = ---2----2- ∼ 6 × 10 m. (40) 4 π M f Q
    So ultra-cryogenic bars can approach the goal of detection near h = 10− 20 despite thermal noise.
  2. Sensor noise. A transducer converts the bar’s mechanical energy into electrical energy, and an amplifier increases the electrical signal to record it. If sensing of the vibration could be done perfectly, then the detector would be broadband: both thermal impulses and gravitational wave forces are mechanical forces, and the ratio of their induced vibrations would be the same at all frequencies for a given applied impulsive force.

    But sensing is not perfect: amplifiers introduce noise, and this makes small amplitudes harder to measure. The amplitudes of vibration are largest in the resonance band near f, so amplifier noise limits the detector sensitivity to gravitational wave frequencies near f. But if the noise is small, then the measurement bandwidth about f can be much larger than the resonant bandwidth f∕Q. Typical measurement bandwidths are 10 Hz, about 104 times larger than the resonant bandwidths, and 100 Hz is not out of the question [61].

  3. Quantum noise. The zero-point vibrations of a bar with a frequency of 1 kHz are
    ( )1∕2 ⟨δℓ2⟩1q∕u2ant = --ℏ---- ∼ 4 × 10− 21 m. (41) 2πM f
    This is comparable to the thermal limit over 1 ms. So, as detectors improve their thermal limits, they run into the quantum limit, which must be breached before a signal at 10–21 can be seen with such a detector.

    It is not impossible to do better than the quantum limit. The uncertainty principle only sets the limit above if a measurement tries to determine the excitation energy of the bar, or equivalently the phonon number. But one is not interested in the phonon number, except in so far as it allows one to determine the original gravitational wave amplitude. It is possible to define other observables that also respond to the gravitational wave and can be measured more accurately by squeezing their uncertainty at the expense of greater errors in their conjugate observable [111]. It is not yet clear whether squeezing will be viable for bar detectors, although squeezing is now an established technique in quantum optics and will soon be implemented in interferometric detectors (see below).

Reliable gravitational wave detection, whether with bars or with other detectors, requires coincidence observations, in which two or more detectors confirm each other’s findings. The principal bar detector projects around the world formed the International Gravitational Event Collaboration (IGEC) [204] to arrange for long-duration coordinated observations and joint data analysis. A report in 2003 of an analysis of a long period of coincident observing over three years found no evidence of significant events [52]. The ALLEGRO bar [28] at Louisiana State University made joint data-taking runs with the nearby LIGO interferometer, setting an upper limit on the stochastic gravitational-wave background at around 900 Hz of 2 h 100Ωgw (f) ≤ 0.53 [17]. More recently, because funding for many of the bar detector projects has become more restricted, only two groups continue to operate bars at present (end of 2008): the Rome [317] and Auriga [58] groups. The latest observational results from IGEC may be found in [56].

It is clear from the above discussion that bars have great difficulty achieving the sensitivity goal of 10–21. This limitation was apparent even in the 1970s, and that motivated a number of groups to explore the intrinsically wide-band technique of laser interferometry, leading to the projects described in Section 4.3.1 below. However, the excellent sensitivity of resonant detectors within their narrow bandwidths makes them suitable for specialized, high-frequency searches, including cross-correlation searches for stochastic backgrounds [120]. Therefore, novel and imaginative designs for resonant-mass detectors continue to be proposed. For example, it is possible to construct large spheres of a similar size (1 to 3 m diameter) to existing cylinders. This increases the mass of the detector and also improves its direction-sensing. One can in principle push to below 10–21 with spheres [118]. A spherical prototype called MiniGRAIL[260] has been operated in the Netherlands[182]. A similar prototype called the Schenberg detector[183] is being built in Brazil [23]. Nested cylinders or spheres, or masses designed to sense multiple modes of vibration may also provide a clever way to improve on bar sensitivities [88].

While these ideas have interesting potential, funding for them is at present (2008) very restricted, and the two remaining bar detectors are likely to be shut down in the near future, when the interferometers begin operating at sensitivities clearly better than 10–21.

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