  We analyze the non-equilibrium shape fluctuations of giant unilamellar
vesicles encapsulating motile bacteria. Owing to bacteria--membrane collisions,
we experimentally observe a significant increase in the magnitude of membrane
fluctuations at low wave numbers, compared to the well-known thermal
fluctuation spectrum. We interrogate these results by numerically simulating
membrane height fluctuations via a modified Langevin equation, which includes
bacteria--membrane contact forces. Taking advantage of the length and time
scale separation of these contact forces and thermal noise, we further
corroborate our results with an approximate theoretical solution to the
dynamical membrane equations. Our theory and simulations demonstrate excellent
agreement with non-equilibrium fluctuations observed in experiments. Moreover,
our theory reveals that the fluctuation--dissipation theorem is not broken by
the bacteria; rather, membrane fluctuations can be decomposed into thermal and
active components.
