Detecting Collective Excitations in Self-Gravitating Bose-Einstein Condensates via Faraday Waves

We propose Faraday waves as a probe for collective excitations in self-gravitating Bose-Einstein condensates (SGBECs). Using a semi-classical approach based on linear stability analysis of the Gross-Pitaevskii-Newton equations, we derive a damped Mathieu equation governing parametric instabilities. Our analysis reveals well-separated regions of parametric resonance and Jeans instability in parameter space, with distinct growth rate characteristics: Jeans instability decreases monotonically to zero at the critical wavenumber $k_J$, while parametric resonance exhibits non-monotonic behavior with a clear maximum. These findings provide explicit experimental guidelines for accessing the parametric resonance regime. Numerical simulations demonstrate the transition from Faraday wave formation to Jeans collapse as gravitational strength increases, validating our theoretical framework.