Jeans mass-radius relation of self-gravitating Bose-Einstein condensates
  and typical parameters of the dark matter particle
  We study the Jeans mass-radius relation of Bose-Einstein condensate dark
matter in Newtonian gravity. We show at a general level that it is similar to
the mass-radius relation of Bose-Einstein condensate dark matter halos [P.H.
Chavanis, Phys. Rev. D {\bf 84}, 043531 (2011)]. Bosons with a repulsive
self-interaction generically evolve from the Thomas-Fermi regime to the
noninteracting regime as the Universe expands. In the Thomas-Fermi regime, the
Jeans radius remains approximately constant while the Jeans mass decreases. In
the noninteracting regime, the Jeans radius increases while the Jeans mass
decreases. Bosons with an attractive self-interaction generically evolve from
the nongravitational regime to the noninteracting regime as the Universe
expands. In the nongravitational regime, the Jeans radius and the Jeans mass
increase. In the noninteracting regime, the Jeans radius increases while the
Jeans mass decreases. The transition occurs at a maximum Jeans mass which is
similar to the maximum mass of Bose-Einstein condensate dark matter halos with
an attractive self-interaction. We use the mass-radius relation of dark matter
halos and the observational evidence of a ``minimum halo'' (with typical radius
$R\sim 1\, {\rm kpc}$ and typical mass $M\sim 10^8\, M_{\odot}$) to constrain
the mass $m$ and the scattering length $a_s$ of the dark matter particle.
