Gravitational collapse of Bose-Einstein condensate dark matter halos
  We study the mechanisms of the gravitational collapse of the Bose-Einstein
condensate dark matter halos, described by the zero temperature time-dependent
nonlinear Schr\"odinger equation (the Gross-Pitaevskii equation), with
repulsive inter-particle interactions. By using a variational approach, and by
choosing an appropriate trial wave function, we reformulate the
Gross-Pitaevskii equation with spherical symmetry as Newton's equation of
motion for a particle in an effective potential, which is determined by the
zero point kinetic energy, the gravitational energy, and the particles
interaction energy, respectively. The velocity of the condensate is
proportional to the radial distance, with a time dependent proportionality
function. The equation of motion of the collapsing dark matter condensate is
studied by using both analytical and numerical methods. The collapse of the
condensate ends with the formation of a stable configuration, corresponding to
the minimum of the effective potential. The radius and the mass of the
resulting dark matter object are obtained, as well as the collapse time of the
condensate. The numerical values of these global astrophysical quantities,
characterizing condensed dark matter systems, strongly depend on the two
parameters describing the condensate, the mass of the dark matter particle, and
of the scattering length, respectively. The stability of the condensate under
small perturbations is also studied, and the oscillations frequency of the halo
is obtained. Hence these results show that the gravitational collapse of the
condensed dark matter halos can lead to the formation of stable astrophysical
systems with both galactic and stellar sizes.
