Suspension-thermal noise in spring-antispring systems for future gravitational-wave detectors

Spring-antispring systems have been investigated in the context of low-frequency seismic isolation in high-precision optical experiments. These systems provide the possibility to tune the fundamental resonance frequency to, in principle, arbitrarily low values, and at the same time maintain a compact design. It was argued though that thermal noise in spring-antispring systems would not be as small as one may naively expect from lowering the fundamental resonance frequency. In this paper, we present calculations of suspension-thermal noise for spring-antispring systems potentially relevant in future gravitational-wave detectors, i.e. the beam-balance tiltmeter, and the Roberts linkage. We find a concise expression of the suspension-thermal noise spectrum, which assumes a form very similar to the well-known expression for a simple pendulum. For systems such as the Roberts linkage foreseen as passive seismic isolation, we find that while they can provide strong seismic isolation due to a very low fundamental resonance frequency, their thermal noise is determined by the dimension of the system and is insensitive to fine-tunings of the geometry that can strongly influence the resonance frequency. By analogy, i.e. formal similarity of the equations of motion, this is true for all horizontal mechanical isolation systems with spring-antispring dynamics. This imposes strict requirements on mechanical spring-antispring systems for seismic isolation in potential future low-frequency gravitational-wave detectors as we discuss for the four main concepts, atom-interferometric, superconducting, torsion-bars, and conventional laser interferometer, and generally suggests that thermal noise needs to be evaluated carefully for high-precision experiments implementing spring-antispring dynamics.