Boson stars with nontrivial topology

We construct boson star solutions in the presence of a phantom field, allowing for a nontrivial topology of the solutions. The wormholes residing at the core of the configurations lead to a number of qualitative changes of the boson star solutions. In particular, the typical spiraling dependence of the mass and the particle number on the frequency of the boson stars is lost. Instead, the boson stars with nontrivial topology approach a singular configuration in the limit of vanishing frequency. Depending on the value of the coupling constant, the wormhole geometry changes from a single throat configuration to a double throat configuration, featuring a belly in between the two throats. Depending on the mass of the boson field and its self-interaction, the mass and the size of these objects cover many orders of magnitude, making them amenable to various astrophysical observations. A stability analysis reveals that the unstable mode of the Ellis wormhole is retained in the presence of the bosonic matter. However, the negative eigenvalue can get very close to zero, by tuning the parameters of the self-interaction potential appropriately.

Figure 1

Probe limit: (a) The boson field function $\varphi$ versus the compactified coordinate $\mathrm{atan}\eta$ for throat size $r_0=1$ and several values of the frequency $\omega$. (b) The value of the boson field function at the throat ${\varphi }_0$ versus the frequency $\omega$ for several values of the throat size, $r_0=1$, 2, and 3. The $Q$-ball limit corresponds to $r_0=0$. The dotted vertical lines represent the minimal and the maximal values of the frequency ${\omega }_{\min }$ and ${\omega }_{\max }$. The black dot represents the limiting value of ${\varphi }_0$. (c) The mass $M$ for the same sets of solutions. (d) The particle number $Q$ for the same sets of solutions. (e) The mass $M$ versus the particle number $Q$ for the same sets of solutions. The dotted line corresponds to the mass of $Q$ free bosons. (f) The mass $M$ and the particle number $Q$ for frequency $\omega =1$ versus the throat size $r_0$.

Figure 2

The metric functions $A$ (a) and $N$ (b), the mass function $m$ (c) and the boson field function $\varphi$ (d) versus the compactified coordinate $\mathrm{atan}(\eta /r_0)$ for $\alpha =0.1$.

Figure 3

$\alpha$ dependence: The values of the metric function $A_0$ (a) and the boson function ${\varphi }_0$ (b) at the throat (or equator), the mass $M$ (c), the scaled mass $M/\alpha$ (d) and the particle number $Q$ (e) versus the boson frequency $\omega$. The scaled mass $M/\alpha$ versus the particle number $Q$, where the inset magnifies the cusp structure (f). In (b), (d), (e) and (f) the solid black curves represent the probe limit. The probe limit of (a) is trivial, $A_0=1$. The thin vertical line indicates ${\omega }_{\max }$.

Figure 4

Single or double throat: The ratio of the critical value ${\alpha }_{\text{cr}}$ to $\alpha$ (a) and the surface gravity $\kappa$ (b) versus the boson frequency $\omega$ for several values of $\alpha$.

Figure 5

Throat geometry: The metric function $R$ versus the radial coordinate $\eta$ for a sequence of solutions with various values of the frequency $\omega$ and coupling constant $\alpha =0.05$ (a) and $\alpha =5$ (b).

Figure 6

Throat geometry: Three-dimensional view of the isometric embedding for $\alpha =5$, $\omega =0.18$ (a) and $\alpha =0.05$, $\omega =0.24$ (b).

Figure 7

Limiting behavior of the solutions for $\omega \to 0$: The logarithm of the metric function $A$ versus the compactified coordinate $\mathrm{atan}\eta$ (a), a zoom of $\ln A$ versus $\ln \eta$ with the (approximate) limiting behavior close to $\eta =0$ (b), the metric function $N$ versus $\mathrm{atan}\eta$ (c), a zoom of the boson field function $\varphi$ versus $\eta$ with the (approximate) limiting behavior close to $\eta =0$ (d), and the Kretschmann scalar $K$ versus $\ln \eta$ (e) for several values of $A_0$ and $\alpha =0.1$. The function $1/\ln A_0$, where $A_0=A(0)$, versus the boson frequency $\omega$ for $\alpha =0.1$ (f).

Figure 8

The mass $M$ in solar masses $M_{\odot }$ versus the radius $R_{\star }$ in kilometers for families of solutions with $\alpha =0.01$ and 1, where the insets magnify the vicinities of the limiting solution. The corresponding Schwarzschild black hole curve is also indicated (a). The mass $M$ in solar masses $M_{\odot }$ versus the energy density of the boson field ${ϵ}_0^{(\mathrm{\Phi })}=-T_t^{t(\mathrm{\Phi })}$ at $\eta =0$, with ${ϵ}_0^{(\mathrm{\Phi })}/c^2$ in units of ${10}^{14}\mathrm{g}/{\mathrm{cm}}^3$ for the same set of solutions (b).

Figure 9

Instability: The eigenvalue ${\sigma }^2$ versus the value of the metric function $A_0=A(0)$ at the throat (or equator) for several values of $\alpha$, where the inset magnifies the region of the eigenvalues close to zero. The fat dot denotes the eigenvalue of the Ellis wormhole (a). The maximal value of the eigenvalue ${\sigma }_{\max }^2$ versus the coupling constant $\alpha$ for $\alpha \to 0$ (b).