具有吸引自相互作用的自引力玻色-爱因斯坦凝聚体的坍缩
Collapse of a self-gravitating Bose-Einstein condensate with attractive self-interaction
摘要：
研究了具有吸引自相互作用的自引力玻色-爱因斯坦凝聚体的坍缩。平衡态中，引力和自相互作用引起的吸引力被量子压力抵消，仅存在于最大质量$M_{\rm max}=1.012\hbar/\sqrt{Gm|a_s|}$以下，其中$M_{\rm max}=1.012\hbar/\sqrt{Gm|a_s|}$是玻色子的散射长度，$m$是它们的质量。对于$M>M_{\rm max}$，系统预计会崩溃并形成黑洞。我们通过对波函数作高斯近似来研究坍缩动力学。我们发现对于$M\rightarrow M_{\rm max}^+$，塌陷时间尺度为$(M/M_{\rm max}-1)^{-1/4}$，对于$M\gg M_{\rm max}$，塌陷时间尺度为$M^{-1/2}$。我们将我们的结果应用到标准轴子的质量$m=10^{-4}\, {\rm eV}/c^2$和散射长度$a_s=-5.8\times 10^{-53}\, {\rm m}$，其中$M_{\rm max}=6.5\times 10^{-14}M_{\odot}$和$R=3.3\times 10^{-4}\, R_{\odot}$。我们证实了我们以前的说法，玻色子与吸引力的自我相互作用，如标准的轴子，可能会形成低质量的恒星，但不能形成暗物质晕的相关质量和大小。这些迷你轴子星可能是暗物质的组成部分。它们可以在几个小时内坍缩成质量为$\sim 10^{-14}\, M_{\odot}$的迷你黑洞。在这种情况下，暗物质晕将由微型黑洞组成。我们也将我们的结果应用到超轻轴子的质量$m=1.93\times 10^{-20}\, {\rm eV}/c^2$和散射长度$a_s=-8.29\times 10^{-60}\, {\rm fm}$的$M_{\rm max}=0.39\times 10^6\, M_{\odot}$和$R=33\, {\rm pc}$。这些超轻的轴子可以聚集成暗物质晕。具有吸引力的自相互作用的轴子暗物质晕可以在大约一百万年内坍缩成质量为$\sim 10^{6}\, M_{\odot}$的超大质量黑洞。
英文摘要：
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.
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