New insights into the formation and growth of boson stars in dark matter halos

This work studies the formation and growth of boson stars and their surrounding miniclusters by gravitational condensation using nonlinear dynamical numerical methods. Fully dynamical attractive and repulsive self-interactions are also considered for the first time. In the case of pure gravity, we numerically prove that the growth of boson stars inside halos slows down and saturates as has been previously conjectured, and detail its conditions. Self-interactions are included using the Gross-Pitaevskii-Poisson equations. We find that in the case of strong attractive self-interactions the boson stars can become unstable and collapse, in agreement with previous stationary computations. At even stronger coupling, the condensate fragments. Repulsive self-interactions, as expected, promote boson star formation, and lead to solutions with larger radii.

Figure 1

Snapshots of the density field from one simulation with $\tilde{N}=1005.3$, $\tilde{L}=18$. (a) Projected density at the initial time. (b) Projected density at $\tilde{t}=10$, which shows that minicluster is forming in the box. (c) Projected density at $\tilde{t}=30$. (d) Projected density at $\tilde{t}=200$. A single dense object is visible at the center of the minicluster.

Figure 2

Density profiles of the minicluster at different times (colored dots) compared with solitonic profiles (solid lines) as given by Eq. (c10) with the same central densities.

Figure 3

The mean stacked mass of boson stars evolution (solid lines) for different box sizes $\tilde{L}=25$, 20, 18, 15 and total mass $\tilde{N}=691$, 754, 817, 880, 942, 1005, 1131. The data from simulation with the same box size $\tilde{L}$ but different total mass $\tilde{N}$ are divided into 500 time bins. The shaded regions show the $1-\sigma$ intervals. The time and mass of boson stars are normalized by the condensation time, ${\tau }_{\text{gravity}}$ and the total mass, ${\tilde{N}}_{691}^{1/3}$, where ${\tilde{N}}_{691}=\tilde{N}/691$. Note that here ${\tau }_{\text{gravity}}$ is computed using Eq. (11) for the initial configuration, i.e. $R=L$, $v=v_0$, and $n=N/L^3$, to avoid ambiguities in the definitions of halo radius and density.

Figure 4

Snapshots of the density field from one simulation with $\tilde{N}=1005.3$, $\tilde{L}=18$, $\tilde{g}=-0.007$. (a) Projected density at the initial time. (b) Projected density at $\tilde{t}=10$, which shows that minicluster is forming in the box. (c) Projected density at $\tilde{t}=30$. (d) Projected density at $\tilde{t}=48$. Compared with the case without self-interactions, the boson star formed at the center is denser.

Figure 5

Density profiles of the minicluster at different times (colored dots) from simulations of the GPP equations with $\tilde{g}=-0.007$. Solitonic profiles given by Eq. (c11) and Eq. (c10) are plotted in solid and dashed lines, respectively.

Figure 6

Maximum density growth with respect to time from simulations assuming different self-interaction couplings: $\tilde{g}=0$ (without self-interactions), $\tilde{g}=-0.000005$, $\tilde{g}=-0.001$, $\tilde{g}=-0.007$, $\tilde{g}=-0.01$, and $\tilde{g}=-0.012$. The box size $\tilde{L}=18$ and the total mass $\tilde{N}=1005.3$. The horizontal dashed lines mark the densities which satisfy $\tilde{\rho }{\tilde{g}}^2=0.52$, when the boson star will collapse.

Figure 7

Snapshots of the density field at $\tilde{t}=200$ from simulations in a box of size $\tilde{L}=18$ and total mass $\tilde{N}=1005.3$ assuming different $\tilde{g}$. (a) $\tilde{g}=0.01$ (b) $\tilde{g}=0.1$ (c) $\tilde{g}=1.0$ (d) $\tilde{g}=10.0$.

Figure 8

Density profiles of the miniclusters from simulations (colored dots), compared with solitonic profiles (solid lines) as given by Eq. (c10) and Thomas-Fermi approximation [57] [Eq. (c16)] with the same central densities.

Figure 9

Maximum density growth with respect to time from simulations assuming different $\tilde{g}$. The box size $\tilde{L}=18$ and the total mass $\tilde{N}=1005.3$.

Figure 10

Mass growth of boson stars with respect to time from simulations assuming different $\tilde{g}$. The box size $\tilde{L}=18$ and the total mass $\tilde{N}=1005.3$. Note that the case with $g=10$ is not shown because no boson star is formed in the box.

Figure 11

Snapshots of the density field from simulations of the GPP equations in a box of size $\tilde{L}=18$, and total mass $\tilde{N}=1005.3$. The self-interaction coupling constant $\tilde{g}=-0.04$. Note that due to resolution limit, we cannot resolve the central region of the densest object, so we cutoff the projected density in the plot at $\approx 1500$. (a) $\tilde{t}=0$ (b) $\tilde{t}=10$ (c) $\tilde{t}=13$ (d) $\tilde{t}=13.7$.

Figure 12

Snapshots of the density field from simulations of the GPP equations (left column) and GP equations (right column). We pick $\tilde{g}=-1$ (first row) and $\tilde{g}=-80$ (second row). The box size $\tilde{L}=18$, and the total mass $\tilde{N}=1005.3$. (a) $\tilde{t}=1.0$, $\tilde{g}=-1$ (b) $\tilde{t}=1.0$, $\tilde{g}=-1$ (c) $\tilde{t}=0.2$, $\tilde{g}=-80$ (d) $\tilde{t}=0.2$, $\tilde{g}=-80$.

Figure 13

Relative errors of the total energy with respect to time for different time step sizes. To show the fourth-order convergence of the algorithm in time, we have multiplied the relative errors by a factor of $2^4$ and $4^4$ for the cases with time step sizes $4\pi \mathrm{\Delta }\tilde{t}$ and $2\pi \mathrm{\Delta }\tilde{t}$, respectively.

Figure 14

Maximum density growth with respect to time for spatial resolutions.

Figure 15

Maximum density with respect to time for different spatial resolutions. The bosons have attractive self-interaction, i.e. $\tilde{g}=-0.007$.

Figure 16

Total mass of boson star with attractive self-interactions ($g<0$) as a function of the central density.

Figure 17

Core radius of boson star with attractive self-interactions ($g<0$) as a function of the central density. Circles: numeric results. Solid line: fitting function, Eq. (c12).

Figure 18

Parameter $\alpha$ in the fitting formula Eq. (c11) for the case with attractive self-interactions ($g<0$). Circles: numeric results. Solid line: fitting function, Eq. (c13).

Figure 19

Parameter $\beta$ in the fitting formula Eq. (c11) for the case with attractive interactions ($g<0$) as a function of the central density. Solid line: fitting function, Eq. (c14).

Figure 20

Core radius of boson star with repulsive self-interactions ($g>0$) as a function of the central density. Circles: numeric results. Solid line: fitting function, Eq. (c15).

Figure 21

Parameter $\alpha$ in the fitting formula Eq. (c11) for the case with repulsive self-interactions ($g>0$) as a function of the central density. Circles: numeric results. Solid line: fitting function, Eq. (c13).

Figure 22

Mean fractional deviation from the fitting formula ${\tilde{M}}_{*,\mathrm{sat}}\propto {\tilde{N}}^{\alpha }$. The colored lines show the results from simulations grouped by box size, while the black line shows the results combing all simulations.