Collapse of a self-gravitating Bose-Einstein condensate with attractive self-interaction

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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.