Fluctuation-dissipation theorem for thermo-refractive noise

TL;DR

This paper presents a straightforward method to calculate thermal fluctuation spectra of temperature-dependent quantities, demonstrated by accurately computing thermo-refractive noise in LIGO with advantages for complex geometries.

Contribution

It introduces a simple, computationally efficient prescription for spectral calculations of thermal fluctuations applicable to irregular geometries and volume-integrated quantities.

Findings

Accurately computes thermo-refractive noise spectrum in LIGO.

Method agrees with previous detailed calculations.

Offers computational advantages for complex geometries.

Abstract

We introduce a simple prescription for calculating the spectra of thermal fluctuations of temperature-dependent quantities of the form $\hat{\delta T}(t)=\int d^3\vec{r} \delta T(\vec{r},t) q(\vec{r})$. Here $T(\vec{r}, t)$ is the local temperature at location $\vec{r}$ and time $t$, and $q(\vec{r})$ is an arbitrary function. As an example of a possible application, we compute the spectrum of thermo-refractive coating noise in LIGO, and find a complete agreement with the previous calculation of Braginsky, Gorodetsky and Vyatchanin. Our method has computational advantage, especially for non-regular or non-symmetric geometries, and for the cases where $q(\vec{r})$ is non-negligible in a significant fraction of the total volume.