Thermally excited vibrations of the mirrors of laser interferometer gravitational-wave detectors

The effect of thermally excited mirror vibrations on length measurements using a laser interferometer gravitational-wave detector is calculated, and the number of vibrational modes which must be included to predict the Brownian motion of the mirror surface relative to its center of mass is estimated. These calculations account for both the full three-dimensional shape of the vibrational modes of the mirrors and the spatial shape of the optical modes of the interferometer. A convergence pattern for the number of modes which must be included in a complete thermal noise estimate is found; all vibrational modes with acoustic wavelengths greater than the laser beam spot size must be considered. When the full geometries of the mirror and the optical mode are taken into account, the thermal noise power in the gravitational-wave frequency bandwidth for mirror and laser parameters relevant to the LIGO (Laser Interferometer Gravitational-Wave Observatory) interferometer is approximately a factor of 6 larger than the noise obtained in earlier estimates which typically considered only the lowest few modes. If mirror vibrational thermal noise were the limiting noise source for the detection of some astrophysical sources, then this difference in noise power would result in a factor of 15 difference in estimated determination rates for these sources.