Perturbative and numerical methods for stochastic nonlinear oscillators

Interferometric gravitational wave detectors are devoted to pick up the effect induced on masses by gravitational waves. The variations of the length dividing two mirrors is measured through a laser interferometric technique. The Brownian motion of the masses related to the interferometer room temperature is a limit to the observation of astrophysical signals. It is referred to as thermal noise and it affects the sensitivity of both the projected and the future generation interferometers. In this paper, we investigate the relevance of small non-linear effects and point out their impact on the sensitivity curve of interferometric gravitational wave detectors (e.g. VIRGO, LIGO, GEO, etc.) through perturbative methods and numerical simulations. We find that in the first-order approximation the constants characterizing the power spectrum density (PSD) are renormalized but it retains its typical shape. This is due to the fact that the involved Feynman diagrams are of tadpole type. Higher-order approximations are required to give rise to up-conversion effects. This result is predicted by the perturbative approach and is in agreement with the numerical results obtained by studying the system's non-linear response by numerically simulating its dynamics.