Collapse of a self-gravitating Bose-Einstein condensate with attractive
  self-interaction
  We study the collapse of a self-gravitating Bose-Einstein condensate with
attractive self-interaction. Equilibrium states in which the gravitational
attraction and the attraction due to the self-interaction are counterbalanced
by the quantum pressure exist only below a maximum mass $M_{\rm
max}=1.012\hbar/\sqrt{Gm|a_s|}$ where $a_s<0$ is the scattering length of the
bosons and $m$ is their mass. For $M>M_{\rm max}$ the system is expected to
collapse and form a black hole. We study the collapse dynamics by making a
Gaussian ansatz for the wave function. We find that the collapse time scales as
$(M/M_{\rm max}-1)^{-1/4}$ for $M\rightarrow M_{\rm max}^+$ and as $M^{-1/2}$
for $M\gg M_{\rm max}$. We apply our results to standard axions with mass
$m=10^{-4}\, {\rm eV}/c^2$ and scattering length $a_s=-5.8\times 10^{-53}\,
{\rm m}$ for which $M_{\rm max}=6.5\times 10^{-14}M_{\odot}$ and $R=3.3\times
10^{-4}\, R_{\odot}$. We confirm our previous claim that bosons with attractive
self-interaction, such as standard axions, may form low mass stars but cannot
form dark matter halos of relevant mass and size. These mini axions stars could
be the constituents of dark matter. They can collapse into mini black holes of
mass $\sim 10^{-14}\, M_{\odot}$ in a few hours. In that case, dark matter
halos would be made of mini black holes. We also apply our results to
ultralight axions with mass $m=1.93\times 10^{-20}\, {\rm eV}/c^2$ and
scattering length $a_s=-8.29\times 10^{-60}\, {\rm fm}$ for which $M_{\rm
max}=0.39\times 10^6\, M_{\odot}$ and $R=33\, {\rm pc}$. These ultralight
axions could cluster into dark matter halos. Axionic dark matter halos with
attractive self-interaction can collapse into supermassive black holes of mass
$\sim 10^{6}\, M_{\odot}$ in about one million years.
