  We investigate gravitational analogue models to describe slowly rotating
objects (e.g., dark-matter halos, or boson stars) in terms of Bose-Einstein
condensates, trapped in their own gravitational potentials. We begin with a
modified Gross-Pitaevskii equation, and show that the resulting background
equations of motion are stable, as long as the rotational component is treated
as a small perturbation. The dynamics of the fluctuations of the velocity
potential are effectively governed by the Klein-Gordon equation of a "Eulerian
metric," where we derive the latter by the use of a relativistic Lagrangian
extrapolation. Superradiant scattering on such objects is studied. We derive
conditions for its occurence and estimate its strength. Our investigations
might give an observational handle to phenomenologically constrain
Bose-Einstein condensates.
