Stability and physical properties of spherical excited scalar boson stars

We study the time evolution of spherical, excited—with $n$ radial nodes—scalar boson stars in general relativity minimally coupled to a complex massive scalar field with quartic self-interactions. We report that these stars, with up to $n=10$, can be made dynamically stable, up to timescales of $t\sim \frac{10^4}{c\mu }$, where $\mu$ is the inverse Compton wavelength of the scalar particle, for sufficiently large values of the self-interactions coupling constant $\lambda$, which depends on $n$. We observe that the compactness of these solutions is rather insensitive to $n$, for large $\lambda$ and fixed frequency. Generically, along the branches where stability was studied, these excited boson stars are not compact enough to allow for innermost stable circular orbits or light rings. Finally, we discuss the angular velocity of particles along timelike circular orbits, suggesting an application, for solutions in the Newtonian limit, to galactic rotation curves.

Figure 1

ADM mass versus the frequency for boson star solutions with fixed $\mathrm{\Lambda }$ (left) and fixed $n$ (right).

Figure 2

Energy density of excited boson stars (omitting the $\theta$ coordinate) for $\mathrm{\Lambda }=500$, $\omega =0.92$, where $R_{\text{areal}}=\sqrt{x^2+y^2}$.

Figure 3

Compactness as a function of $\mathrm{\Lambda }$ and $n$ for $\omega =0.90$ (left) and $\omega =0.92$ (right).

Figure 4

Effective radius, $R_{99}$, as a function of $\mathrm{\Lambda }$ and $n$ for $\omega =0.90$.

Figure 5

Radial profile for different times (top left), minimum value of the lapse (top right), violation of the Hamiltonian constrain for certain times (bottom left) and the L2 norm of the Hamiltonian constraint ($||\mathcal{H}|{|}_2$) (bottom right) for the boson star model $n=2$, $\mathrm{\Lambda }=75$ and $\omega =0.92$.

Figure 6

Radial profile for different times and radial profile of candidate final state (top left), minimum value of the lapse (top right), the energy of the star at different spatial hypersurfaces (bottom left) and violations of the Hamiltonian constraint for certain times and the L2 norm of the Hamiltonian constraint (bottom right) for the boson star model $n=2$, $\mathrm{\Lambda }=125$ and $\omega =0.90$.

Figure 7

Radial profile for different times (top left), minimum value of the lapse (top right), the energy of the star at different spatial hypersurfaces (bottom left) and violations of the Hamiltonian constraint for certain times and the L2 norm of the Hamiltonian constraint (bottom right) for the boson star model $n=2$, $\mathrm{\Lambda }=150$ and $\omega =0.92$.

Figure 8

Radial profiles $|{\mathrm{\Phi }}_{n,\mathrm{\Lambda },\omega }|$ for the threshold of stability for several $n$.

Figure 9

The ${\mathrm{\Lambda }}_{\text{threshold}}$ as a function of $n$.

Figure 10

Effective potential for $n=0,\mathrm{\Lambda }=3000,\omega =0.90$, the most compact star studied.

Figure 11

Angular velocity as a function of $R_{\text{orbit}}$ for $n=6$, $\mathrm{\Lambda }=800$, $\omega =0.92$. The vertical lines show the location of the nodes.

Figure 12

Angular velocity as a function of $R_{\text{orbit}}$ for $n=6$, $\omega =0.92$ for several $\mathrm{\Lambda }$ (left). Angular velocity as a function of $R_{\text{orbit}}$ for $\mathrm{\Lambda }=300$, $\omega =0.92$ for several $n$ (right). Same qualitative behavior for $\omega =0.90$.

Figure 13

Rotational velocity as a function of $R_{\text{orbit}}$ for $n=10$, $\omega =0.92$ for several $\mathrm{\Lambda }$ (left). Angular velocity as a function of $R_{\text{orbit}}$ for $\mathrm{\Lambda }=500$, $\omega =0.92$ for several $n$ (right). Same qualitative behavior for $\omega =0.90$.

Figure 14

Absolute value of the Hamiltonian constraint at $t=1200$ for different resolutions for a boson star with $n=2$, $\omega =0.92$, $\mathrm{\Lambda }=250$.

Figure 15

Evolution of the total mass/energy for different resolutions.

Figure 16

Difference between total mass at $t=1200$ and total initial mass (left) and rescaled functions to third-order convergence (right).