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 (Heisenberg's uncertainty principle) exist only below a maximum mass M-max = 1.012h/root Gm vertical bar a(s)vertical bar where a(s) < 0 is the scattering length of the bosons and m is their mass [P. H. Chavanis, Phys. Rev. D 84, 043531 (2011)]. For M > M-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 and reducing the problem to the study of the motion of a particle in an effective potential. We find that the collapse time scales as (M/M-max - 1)(-1/4) for M -> M-max and as M-1/2 for M >> M-max. Other analytical results are given above and below the critical point corresponding to a saddle-node bifurcation. We apply our results to QCD axions with mass m = 10(-4) eV/c(2) and scattering length a(s) = -5.8 x 10(-53) m for which M-max = 6.5 x 10(-14) M-circle dot and R = 3.3 x 10(-4) R-circle dot. We confirm our previous claim that bosons with attractive self-interaction, such as QCD axions, may form low mass stars (axion stars or dark matter stars) but cannot form dark matter halos of relevant mass and size. These mini axion stars could be the constituents of dark matter. They can collapse into mini black holes of mass similar to 10(-14) M-circle dot 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 x 10(-20) eV/c(2) and scattering length a(s) = -8.29 x 10(-60) fm for which M-max = 0.39 x 10(6) M-circle dot and R = 33 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 similar to 10(6) M-circle dot (similar to those reported at the center of galaxies) in about one million years. We point out the limitations of the Gaussian ansatz to describe the late stages of the collapse dynamics. We also mention the possibility that, instead of forming a black hole, the collapse may be accompanied by a burst or relativistic axions (bosenova) leading to a cycle of collapses and explosions as observed for nongravitational Bose-Einstein condensates with attractive self-interaction.