A Stochastic-Geodesic-Noise Search for the Einstein Telescope: A Phenomenological Geodesic-Diffusion Noise Model

We present a phenomenological search framework for geodesic diffusion noise in the Einstein Telescope (ET). The hypothesized signal is a stochastic path-length fluctuation that produces a strain power spectral density (PSD) ∝ f-γ, with γ=2 (random-walk) as the primary case and other indices as alternatives. The analysis is a nested likelihood-ratio test under a complex-Wishart approximation for Welch cross-spectral estimators, with spline-constrained instrument PSDs and a low-rank environmental coherence model. We emphasize that this is not a quantum-gravity derivation: it is an effective stochastic model tested against data and existing experimental limits. Significance is calibrated via a parametric bootstrap that is explicitly validated with time-domain simulations. We provide a hardened reference implementation and a commissioning plan, and we show how published experimental bounds map onto the model amplitude.