Gravitational Bose-Einstein Condensation in the Kinetic Regime

We study Bose-Einstein condensation and the formation of Bose stars in virialized dark matter halos and miniclusters by universal gravitational interactions. We prove that this phenomenon does occur and it is described by a kinetic equation. We give an expression for the condensation time. Our results suggest that Bose stars may form kinetically in mainstream dark matter models such as invisible QCD axions and fuzzy dark matter.

Figure 1

Formation of a Bose star from a random field with initial distribution $|{\tilde{\psi }}_{\tilde{\mathit{p}}}{|}^2\propto e^{-{\tilde{\mathit{p}}}^2}$ and total mass $\tilde{N}=50$ in the box $0\le \tilde{x},\tilde{y},\tilde{z}<125$. These values correspond to the center of the axion minicluster with $M_c\sim {10}^{-13}{M}_{\odot }$ and $\mathrm{\Phi }\sim 2.7$ in the discussion at the end of this Letter. (a),(b) Sections $\tilde{z}=\text{const}$ of the solution $|\tilde{\psi }(\tilde{t},\tilde{\mathit{x}})|$ at (a) $\tilde{t}=0$ and (b) $\tilde{t}>{\tilde{\tau }}_{gr}\approx 1.08\times {10}^6$. (c) Radial profile $|\tilde{\psi }(\tilde{r})|$ of the object in (b) (points) compared to the Bose star ${\tilde{\psi }}_s(\tilde{r})$ with ${\tilde{\omega }}_s\approx -0.7$ (line). (d) Maximum of $|\tilde{\psi }(\tilde{\mathit{x}})|$ over the box as a function of time. (e) Spectra (3) at times of (a) and (b) and at the eve of Bose star nucleation, $\tilde{t}=1.05\times {10}^6\sim {\tilde{\tau }}_{gr}$. (f) The spectrum at $t\sim {\tau }_{gr}$ (dashed line) versus the solution of Eq. (5) (circles) and thermal law $\tilde{F}\propto {\tilde{\omega }}^{-1/2}$ (dots).

Figure 2

(a) Time to Bose star formation in the cases of Gaussian (empty circle) and $\delta$-peaked (empty square) initial distributions. The $\delta$ graphs are shifted downwards (${\tau }_{gr}\to {\tau }_{gr}/10$) for visualization purposes. Lines depict fits by Eq. (4). (b) The same for isolated miniclusters. (c),(d) Slices $\tilde{z}=\text{const}$ of the solution $|\tilde{\psi }(\tilde{t},\tilde{\mathit{x}})|$ describing the formation of a Bose star in the center of a minicluster; $\tilde{N}=290$, $\tilde{L}\approx 63$.