P010010-00-R - LIGO - California Institute of Technology

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degrees of freedom, from which the test mass is hung as the lower mass of a quadruple
pendulum. This design is expected to move the seismic wall to ≈ 10 Hz.
1.3.2 Thermal Noise
The second fundamental noise source is due to the fact that the masses are at finite
temperature. Finite temperature dictates that the atoms which comprise the masses,
as well as the wires which suspend the masses, vibrate. Vibrations of the test mass
atoms cause the surfaces of the mirrors to vibrate, generating a signal. The vibrations
of the wires which support the test mass shake the mirror, similarly to seismic noise.
Thermal noise which affects the surface of the test mass itself is referred to as “internal
thermal noise,” while the thermal noise from the suspension wires is called “pendulum
thermal noise.”
Internal Thermal Noise
The “Fluctuation-Dissipation Theorem” describes the physical mechanism for the
existence of thermal noise.[15] The theorem states that for an object in equilibrium
with the environment, the path through which energy can dissipate out of the object,
hence cool the object, is the same path through which the environmental energy
can enter into the object, thus driving the vibrations. The traditional way to think
about the internal thermal noise of the test mass is to recognize that, for every
mode of an object, equipartition gives kBT/2 of energy. The energy is frequency
independent, so it drives the mode across the entire bandwidth of its response. The
larger the mechanical Q of the mode, the more of this energy is concentrated around
the resonant frequency, while the noise level at all other frequencies is lower. The
total noise of the mass is calculated by characterizing the modes of the test mass, and
summing their contributions. The power spectrum of the fluctuations of the surface
of the test mass due to a single mode is given by [16]
Sxn(f) = 4kBT
�
ω
αnm 2πf
2 nφn(f)
((2πf) 2 − ω2 n) 2 + ω4 nφn(f) 2
�
(1.3)