  We study self-gravitating bosonic systems, candidates for dark-matter halos,
by carrying out a suite of direct numerical simulations (DNSs) designed to
investigate the formation of finitetemperature, compact objects in the
three-dimensional (3D) Fourier-truncated Gross-Pitaevskii-Poisson equation
(GPPE). This truncation allows us to explore the collapse and fluctuations of
compact objects, which form at both zero temperature and finite temperature. We
show that the statistically steady state of the GPPE, in the large-time limit
and for the system sizes we study, can also be obtained efficiently by tuning
the temperature in an auxiliary stochastic Ginzburg-Landau-Poisson equation
(SGLPE). We show that, over a wide range of model parameters, this system
undergoes a thermally driven first-order transition from a collapsed, compact,
Bose-Einstein condensate (BEC) to a tenuous Bose gas without condensation. By a
suitable choice of initial conditions in the GPPE, we also obtain a binary
condensate that comprises a pair of collapsed objects rotating around their
center of mass.
