Relativistic nontopological soliton stars in a U(1) gauge Higgs model

We study spherically symmetric nontopological soliton (NTS) stars numerically in the coupled system of a complex scalar field, a U(1) gauge field, a complex Higgs scalar field, and Einstein gravity, where the symmetry is broken spontaneously. The gravitational mass of NTS stars is limited by a maximum mass for a fixed breaking scale, and the maximum mass increases steeply as the breaking scale decreases. In the case of the breaking scale is much less than the Planck scale, the maximum mass of NTS stars becomes the astrophysical scale, and such a star is relativistically compact so that it has the innermost stable circular orbit. The first author contributed with a part of the numerical calculations. The second contributed with planning and conducting the research, and the third contributed with all numerical calculations and finding new properties of the system.

Figure 1

Field configurations of numerical solutions in the Planck breaking scale case $\eta /M_{\mathrm{P}}=1$. We show three subcases of the parameter: (a) $\mathrm{\Omega }=1.183$ (left column), (b) $\mathrm{\Omega }=1.178$ (middle column), and (c) $\mathrm{\Omega }=1.008$ (right column). The scalar fields $u$, $f$, and the gauge field $\alpha$ are plotted in the upper panels, and the metric components $\sigma$ and $m$ are plotted in the lower panels.

Figure 2

The same ones as Fig. 1 in the lower breaking scale, $\eta /M_{\mathrm{P}}={10}^{-3}$. We show for (d) $\mathrm{\Omega }=1.183$ (left column), (e) $\mathrm{\Omega }=1.178$ (middle column), and (f) $\mathrm{\Omega }=0.783$ (right column).

Figure 3

The energy density $\epsilon$, the radial pressure $p_r$, and the tangential pressure $p_{\theta }$, where ${\epsilon }_{\max }$ is the maximum value of $\epsilon$. The upper panels correspond to the Planck breaking-scale case, $\eta /M_{\mathrm{P}}=1$, and the lower does the lower breaking-scale case, $\eta /M_{\mathrm{P}}={10}^{-3}$.

Figure 4

The charge densities ${\rho }_{\psi }$, ${\rho }_{\varphi }$, and the total charge density ${\rho }_{\text{total}}={\rho }_{\psi }+{\rho }_{\varphi }$ are plotted, where these charge densities are normalized by the maximum value of ${\rho }_{\psi }$. The upper panels correspond to the Planck breaking-scale case, $\eta /M_{\mathrm{P}}=1$, and the lower does the lower breaking-scale case, $\eta /M_{\mathrm{P}}={10}^{-3}$.

Figure 5

The gravitational mass of NTS stars as a function of $\mathrm{\Omega }$ for various breaking scales $\eta /M_{\mathrm{P}}$. The vertical axis is taken for $M_G/\eta$ and the horizontal axis for $\log ({\mathrm{\Omega }}_{\max }-\mathrm{\Omega })$. The (red) broken curve denotes the mass of NTS solutions, dust balls [19, 20], decouple to gravity, i.e., $\eta /M_{\mathrm{P}}\to 0$. The points (a)–(f) correspond to the solutions shown in Figs. 1, 2, 3, 4.

Figure 6

The gravitational mass of NTS stars for the lower breaking scales $\eta /M_{\mathrm{P}}={10}^{-3}$. In this case the NTS star solutions are classified into three types: miniboson stars, matter-interacting NTS stars, and gravitating NTS stars.

Figure 7

The gravitational mass $M_G$ of the NTS stars as a function of surface radius $R$ for various breaking scales. On the vertical axis $M_G$ is normalized by the Planck mass $M_{\mathrm{P}}$, and in the horizontal axis $R$ is normalized by the Planck length $R_{\mathrm{P}}≔\sqrt{G}$.

Figure 8

The compactness of NTS stars as a function of the gravitational mass $M_G$ for various breaking scales.

Figure 9

The mass ratio $M_G/M_{\mathrm{free}}$ as a function of $M_G$ for various breaking scales. The asterisk mark represents the maximum NTS solution for each breaking scale.

Figure 10

The mass of the maximum NTS stars (left panel), and the surface radius (right panel) for each breaking scale $\eta /M_{\mathrm{P}}$ are plotted as dots. The plots are fitted by double power laws, respectively.

Figure 11

The compactness of the maximum NTS stars for various breaking scales.