Numerical framework for simulating quantum spacetime fluctuations with prescribed spectra

Abstract

Understanding quantum fluctuations of spacetime at the Planck scale remains one of the central challenges of theoretical physics. This study introduces a stochastic framework to model such fluctuations, aiming to test whether a phenomenological approach can reproduce expected statistical signatures of quantum geometry. The metric field was represented as a one-dimensional Gaussian process with a prescribed power spectrum, and its Fourier modes were evolved through an Ornstein–Uhlenbeck process to enforce stationarity. Numerical simulations were carried out on a discretized domain with periodic boundary conditions, and statistical analyses were performed on power spectra, spatial correlations, temporal autocorrelations, and field distributions. The results showed that the empirical power spectrum reproduced the target distribution across more than two decades in wavenumber, with a clear suppression of high-frequency modes due to ultraviolet damping. The spatial correlation function indicated a coherence scale of approximately 150–200 Planck lengths, beyond which fluctuations decorrelate, making spacetime effectively smooth at larger scales. Temporal autocorrelations decayed exponentially with a relaxation time of about 200 Planck units, demonstrating that spacetime fluctuations possess finite memory. The field amplitudes followed a Gaussian distribution, supporting the assumption of central-limit behavior in the linear regime. Stationary field snapshots confirmed equilibrium behavior throughout the simulation. Overall, the study establishes a reproducible and computationally efficient framework for simulating Planck-scale metric fluctuations. The findings highlight short correlation lengths, finite coherence times, and Gaussian statistics as key features of quantum spacetime, providing a bridge between phenomenological modeling and fundamental theory.