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arXiv:1604.05904 · Physical review. D/Physical review. D.

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

Pierre-Henri Chavanis

Published Oct 20, 2016• 112 citations
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Abstract

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 ${M}_{\mathrm{max}}=1.012\ensuremath{\hbar}/\sqrt{Gm|{a}_{s}|}$ 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 $M>{M}_{\mathrm{max}}$ 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 $(M/{M}_{\mathrm{max}}\ensuremath{-}1{)}^{\ensuremath{-}1/4}$ for $M\ensuremath{\rightarrow}{M}_{\mathrm{max}}^{+}$ and as ${M}^{\ensuremath{-}1/2}$ for $M\ensuremath{\gg}{M}_{\mathrm{max}}$. 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 $m={10}^{\ensuremath{-}4}\text{ }\text{ }\mathrm{eV}/{c}^{2}$ and scattering length ${a}_{s}=\ensuremath{-}5.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}53}\text{ }\text{ }\mathrm{m}$ for which ${M}_{\mathrm{max}}=6.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}{M}_{\ensuremath{\bigodot}}$ and $R=3.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{R}_{\ensuremath{\bigodot}}$. 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 $\ensuremath{\sim}{10}^{\ensuremath{-}14}{M}_{\ensuremath{\bigodot}}$ 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 $m=1.93\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}\text{ }\text{ }\mathrm{eV}/{c}^{2}$ and scattering length ${a}_{s}=\ensuremath{-}8.29\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}60}\text{ }\text{ }\mathrm{fm}$ for which ${M}_{\mathrm{max}}=0.39\ifmmode\times\else\texttimes\fi{}1{0}^{6}{M}_{\ensuremath{\bigodot}}$ and $R=33\text{ }\text{ }\mathrm{pc}$. 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 $\ensuremath{\sim}1{0}^{6}{M}_{\ensuremath{\bigodot}}$ (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.

DOI
10.1103/physrevd.94.083007

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We study the stability of a Bose-Einstein condensate of harmonically trapped atoms with negative scattering length, specifically 7Li.{}^{7}\mathrm{Li}.7Li. Our method is to solve the time-dependent nonlinear Schr\"odinger equation numerically. For an isolated condensate, with no gain or loss, we find that the system is stable (apart from quantum tunneling) if the particle number N is less than a critical number Nc.{N}_{c}.Nc​. For N>Nc,N>{N}_{c},N>Nc​, the system collapses to high-density clumps in a region near the center of the trap. The time for the onset of collapse is on the order of one trap period. Within numerical uncertainty, the results are consistent with the formation of a ``black hole'' of infinite density fluctuations, as predicted by Ueda and Huang [Phys. Rev. A (to be published)]. We numerically obtain ${N}_{c}\ensuremath{\approx}1251.$ We then include gain-loss mechanisms, i.e., the gain of atoms from a surrounding ``thermal cloud,'' and the loss due to two- and three-body collisions. The number N now oscillates in a steady state, with a period of about 145 trap periods. We obtain ${N}_{c}\ensuremath{\approx}1260$ as the maximum value in the oscillations.
We consider the dynamics of a gas of free bosons within a semiclassical Fokker-Planck equation for which we give a physical justification. In this context, we find a striking similarity between the Bose-Einstein condensation in the canonical ensemble, and the gravitational collapse of a gas of classical self-gravitating Brownian particles. The paper is mainly devoted to the complete study of the Bose-Einstein "collapse" within this model. We find that at the Bose-Einstein condensation temperature Tc, the chemical potential mu(t) vanishes exponentially with a universal rate that we compute exactly. Below Tc, we show analytically that square root mu(t) vanishes linearly in a finite time t coll. After t coll, the mass of the condensate grows linearly with time and saturates exponentially to its equilibrium value for large time. We also give analytical results for the density scaling functions, for the corrections to scaling, and for the exponential relaxation time. Finally, we find that the equilibration time (above Tc) and the collapse time T coll(below Tc) both behave like -T -3 c ln|T-Tc|, near Tc.
We consider a Bose-Einstein condensate (BEC) with attractive two-body interactions in a cigar-shaped trap, initially prepared in its ground state for a given negative scattering length, which is quenched to a larger absolute value of the scattering length. Using the mean-field approximation, we compute numerically, for an experimentally relevant range of aspect ratios and initial strengths of the coupling, two critical values of quench. One corresponds to the weakest attraction strength, the quench to which causes the system to collapse before completing even a single return from the narrow configuration (pericenter) in its breathing cycle. The other is a similar critical point for the occurrence of collapse before completing two returns. In the latter case, we also compute the limiting value, as we keep increasing the strength of the postquench attraction towards its critical value, of the time interval between the first two pericenters. We also use a Gaussian variational model to estimate the critical quenched attraction strength below which the system is stable against the collapse for long times. These time intervals and critical attraction strengths, apart from being fundamental properties of nonlinear dynamics of self-attractive BECs, may provide clues to the design of upcoming experiments that are trying to create robust BEC breathers.
In this paper, we study the Bose-Einstein condensation of a scalar field with an attractive self-interaction, with or without gravitational interactions. We confirm through full dynamical simulation that the condensation timescale due to self-interaction is inversely proportional to the square of the number density nnn and the self-coupling constant $g:\ensuremath{\tau}\ensuremath{\propto}{n}^{\ensuremath{-}2}{g}^{\ensuremath{-}2}$. We also investigate the condensation timescale when self-interaction and gravity are both important by solving the Gross-Pitaevskii-Poisson equations, and find that the condensation time scales according to an additive model for the cross section. We discuss the relevance of our results to theoretical models of boson star formation by condensation.
We consider dynamics of Bose-Einstein condensates with long-range attractive interaction proportional to 1/rb1/{r}^{b}1/rb and arbitrary angular dependence. It is shown exactly that collapse of a Bose-Einstein condensate without contact interactions is possible only for $b\ensuremath{\geqslant}2$. The case b=2b=2b=2 is critical and requires the number of particles to exceed a critical value to allow collapse. The critical collapse in that case is a strong one, trapping into the collapsing region a finite number of particles. The case b>2b>2b>2 is supercritical with an expected weak collapse which traps a rapidly decreasing number of particles during approach to collapse. For b<2b<2b<2 a singularity at r=0r=0r=0 is not strong enough to allow collapse but an attractive 1/rb1/{r}^{b}1/rb interaction admits stable self-trapping (a stable three-dimensional soliton solution) even in the absence of an external trapping potential.