Method for electromechanical modeling of Johnson noise in Advanced LIGO

We develop a complete framework for modeling general electromechanical systems in the quasi-electrostatic regime. The equations are applicable to a broad range of electrostatic problems and offer the advantage of being theoretically tractable for scaling arguments. Additionally, we show how the formalism can be used together with finite element simulations to obtain estimates for non-stationary effects such as charge accumulation in insulators. As a demonstration, we combined the formalism with measurements from Advanced LIGO to give an updated estimate for the Johnson noise coupling to the gravitational-wave channel. The induced signal was determined to be 10 times lower than the instrument's design sensitivity in the detection band and scaling as f ⁻². (paper)