### 4.3 Thermal noise

Thermal noise associated with the mirror masses and the last stage of their suspensions is the most
significant noise source at the low frequency end of the operating range of initial long baseline gravitational
wave detectors [276]. Advanced detector configurations are also expected to be limited by thermal noise at
their most sensitive frequency band [211, 238, 168, 121]. 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 [276]
where is Boltzmann’s constant, is the temperature, is the mass and is the loss
angle or loss factor of the oscillator of angular resonant frequency . 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 . 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 [276] it can be shown that for a pendulum of mass , suspended on four wires or fibres
of length , the loss factor of the pendulum is related to the loss factor of the material by
where is the moment of the cross-section of each wire, and is the tension in each wire, whose
material has a Young’s modulus . 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 [246] and for some bulk crystalline materials [102]). 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 [276], the effective spectral density of thermal displacement of the front
face of each mass can be expressed as the summation of the motion of the various mechanical
resonances of the mirror as also discussed by Gillespie and Raab [153]. However, this intuitive
approach to calculating the thermally-driven motion is only valid when the mechanical loss is
distributed homogeneously and, therefore, not valid for real test-mass mirrors. The mechanical loss is
known to be inhomogeneous due to, for example, the localisation of structural defects and stress
within the bulk material, and the mechanical loss associated with the polished surfaces is higher
than the levels typically associated with bulk effects. Therefore, Levin suggested using a direct
application of the fluctuation-dissipation theorem to the optically-sensed position of the mirror
substrate surface [211]. This technique imposes a notional pressure (of the same spatial profile
as the intensity of the sensing laser beam) to the front face of the substrate and calculates
the resulting power dissipated in the substrate on its elastic deformation under the applied
pressure. Using such an approach we find that can then be described by the relation
where is the peak amplitude of the notional oscillatory force and is the power dissipated in the
mirror described as,
where and are the strain and mechanical loss located at specific positions within the volume
. This formalisation highlights the importance of where mechanical dissipation is located with respect to
the sensing laser beam. In particular, the thermal noise associated with the multi-layer dielectric mirror
coatings, required for high reflectivity, will in fact limit the sensitivity of second-generation
gravitational-wave detectors at their most sensitive frequency band, despite these coatings typically
being only 4.5 µm in thickness [168]. Identifying coating materials with lower mechanical
loss, and trying to understand the sources of mechanical loss in existing coating materials, is a
major R&D effort targeted at enhancements to advanced detectors and for third generation
instruments [224].
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 currently range in mass from 6 kg for GEO600 to 40 kg for Advanced 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}.

Obtaining these values puts significant constraints on the choice of material for the test masses and their
suspending fibres. GEO600 utilises very-low–loss silica suspensions, a technology, which should allow
detector sensitivities to approach the level desired for second generation instruments [105, 266, 268], since
the intrinsic loss factors in samples of synthetic fused silica have been measured down to around
5 × 10^{–9} [68]. Still, the use of other materials such as sapphire is being seriously considered for future
detectors [104, 195, 266] such as in LCGT [234, 247].

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 [267]. A recent discussion on
the level of mechanical loss and the associated thermal noise in advanced detectors resulting
from hydroxy-catalysis bonds is given by Cunningham et al. [123]. Images of the GEO600
monolithic mirror suspension and of the prototype Advanced LIGO mirror suspension are shown in
Figure 8.