Thermal noise

4.2 Thermal noise

Thermal noise associated with the mirror masses and the last stage of their suspensions is likely to be the most significant noise source at the low frequency end of the operating range of long baseline gravitational wave detectors [85

]. Above the operating range there are the internal resonances of the test masses. The thermal noise in the operating range comes from the tails of these resonant modes. For any simple harmonic oscillator such as a mass hung on a spring or hung as a pendulum the spectral density of thermal motion of the mass can be expressed as [85

]

where $kB$ is Boltmann’s constant, $T$ is the temperature, $m$ is the mass and $ϕ(ω )$ is the loss angle or loss factor of the oscillator of angular resonant frequency $ω0$ . This loss factor is the phase lag angle between the displacement of the mass and any force applied to the mass at a frequency well below $ω0$ . In the case of a mass on a spring the loss factor is a measure of the mechanical loss associated with the material of the spring. For a pendulum, most of the energy is stored in the lossless gravitational field. Thus the loss factor is lower than that of the material which is used for the wires or fibres used to suspend the pendulum. Indeed following Saulson [85

] it can be shown that for a pendulum of mass $m$ , suspended on four wires or fibres of length $l$ , the loss factor of the pendulum is related to the loss factor of the material by

where $I$ is the moment of the cross section of each wire, and $T$ is the tension in each wire whose material has a Young’s modulus $E$ . In general for most materials it appears that the intrinsic loss factor is essentially independent of frequency over the range of interest for gravitational wave detectors (although care has to be taken with some materials in that a form of damping known as thermo-elastic damping can become important for wires of small cross-section [69

] and for some bulk crystalline materials [10

]). In order to estimate the internal thermal noise of a test mass, each resonant mode of the mass can be regarded as a harmonic oscillator. When the detector operating range is well below the resonances of the masses, following Saulson [85 ], the effective spectral density of thermal displacement of the front face of each mass can be expressed as:

In this formula $m$ is the mass of the test mass, $ω$ is an angular frequency in the operating range of the detector, $ω0$ is the resonant angular frequency of the fundamental mode, $ϕmat(ω )$ is the intrinsic material loss, and $β$ is a correction factor to include the effect of summation of the motion over the higher order modes of the test mass (taking into account the effect of a finite optical beam size and correction for the effective masses of the modes). Typically, as calculated by Gillespie and Raab [35 ], $β$ is a number less than 10. A different and more general treatment of internal thermal noise using evaluation of the relevant mechanical impedance has been carried out by Bondu et al. [8 ]. This was based on work of Yuri Levin [55 ] and gives good agreement with the results of Gillespie and Raab.

In order to keep thermal noise as low as possible the mechanical loss factors of the masses and pendulum resonances should be as low as possible. Further the test masses must have a shape such that the frequencies of the internal resonances are kept as high as possible, must be large enough to accommodate the laser beam spot without excess diffraction losses, and must be massive enough to keep the fluctuations due to radiation pressure at an acceptable level. Test masses range in mass from 6 kg for GEO 600 to 30 kg for the first proposed upgrade to LIGO. To approach the best levels of sensitivity discussed earlier the loss factors of the test masses must be $≃$ 3 × 10^–8 or lower, and the loss factor of the pendulum resonances should be smaller than 10^–10. Discussions relevant to this are given in [78 , 81 ]. Obtaining these values puts significant constraints on the choice of material for the test masses and their suspending fibres. One viable solution which should allow detector sensitivities to approach the level desired for upgraded detectors is to use fused silica masses hung by fused silica fibres [12 , 80 ], as the intrinsic loss factors in samples of synthetic fused silica have been measured at around 3 × 10^–8 [61 , 93 ]. Still, the use of other materials such as sapphire is being seriously considered for future detectors as mentioned in Section 6 [11 , 50 , 78 ]. The technique of hydroxy-catalysis bonding provides a method of jointing oxide materials in a suitably low loss way to allow ‘monolithic’ suspension systems to be constructed [79 ]. A picture of such a prototype fused silica mass suspended by two fused silica fibres, which has been constructed in Glasgow and is being tested at the University of Perugia, is shown in Figure 5 .

Figure 5: Prototype ‘monolithic’ fused silica test mass suspension. The mass (3 kg) here is of 12.5 cm diameter. Note final suspension will use four fibres.