Estimation of thermal noise in the mirrors of laser interferometric gravitational wave detectors: Two point correlation function

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Abstract
A general formula and a computational scheme for estimating the power spectrum of the displacement correlation function of suspended test masses such as those used in interferometric gravitational wave detectors are presented. Unlike previous mode-summation approaches, the fluctuation-dissipation theorem has been applied directly to the displacement correlation. The resulting formula expresses the correlation in terms of material damping parameters and mechanical Green’s functions, and provides an efficient and flexible method to compute thermally induced surface displacements of arbitrarily shaped anisotropic elastic bodies. The formula can be used for optimizing the shape and size of test masses in gravitational wave receivers. A simple one-dimensional example is included to clarify the relationship with the modal expansion approach and to illustrate the advantage of the Green’s function method. This paper presents the theoretical formulation; numerical evaluations of the formula will be presented elsewhere.