Nonminimally coupled topological-defect boson stars: Static solutions

We consider spherically symmetric static composite structures consisting of a boson star and a global monopole, minimally or nonminimally coupled to the general relativistic gravitational field. In the nonminimally coupled case, Marunovic and Murkovic [Classical Quantum Gravity 31, 045010 (2014)] have shown that these objects, so-called topological-defect boson stars, can be sufficiently gravitationally compact so as to potentially mimic black holes. Here, we present the results of an extensive numerical parameter space survey which reveals additional new and unexpected phenomenology in the model. In particular, focusing on families of topological-defect boson stars which are parameterized by the central amplitude of the boson field, we find configurations for both the minimally and nonminimally coupled cases that contain one or more shells of bosonic matter located far from the origin. In parameter space, each shell spontaneously appears as one tunes through some critical central amplitude of the boson field. In some cases the shells apparently materialize at spatial infinity: in these instances their areal radii are observed to obey a universal scaling law in the vicinity of the critical amplitude. We derive this law from the equations of motion and the asymptotic behavior of the fields.

Figure 1

Asymptotic mass, $M_{\infty }$, as a function of boson star central amplitude, $\psi (0)$, for ground state boson stars with no quartic self-interaction potential (so-called mini-boson stars). Stars located to the left of the first turning point are stable against small perturbation while stars located to the right are unstable [4]. Using our terminology, the set of mini-boson stars is a family consisting of a single branch, since the mass is everywhere $C^1$.

Figure 2

Asymptotic mass, $M_{\infty }$, as a function of boson star central amplitude, $\psi (0)$, for a hypothetical family of solutions with three branches. The branches of the family are separated by vertical lines, the positions of which correspond to values of $\psi (0)$ where $d{M}_{\infty }/d\psi (0)$ is ill defined.

Figure 3

Convergence of independent residuals for a solution near the limit of our code’s ability to resolve solutions. This limit occurs when features are present at very large distances from the origin. Here we plot the scaled residuals of the metric function $a$ evaluated on grids of 8192, 4096 and 2048 points using a second-order finite difference scheme for the IR evaluator. With the scaling given in the figure, overlap of curves implies second-order convergence. As described in the subsequent sections, the large spikes near the middle and right of the graph are caused by the presence of shells of matter far from the origin. However, even in the vicinity of these shells, convergence is sufficiently precise that it is difficult to distinguish the separate scaled residuals.

Figure 4

Radial component of the metric, $a(r)$, as a function of areal radius, $r$, for representative solutions from families (b), (d) and (e). Here the meaning of the global monopole’s solid deficit angle is obvious: rather than approaching flat space as $r\to \infty$, we approach a space-time which is the four-dimensional analog of a cone.

Figure 5

Time component of the metric, $\alpha (r)$, as a function of areal radius, $r$, for the same solutions plotted in Fig. 4. When the energy contribution of the global monopole is strong, observers at infinity see time at the center of symmetry as flowing faster rather than slower as is the case for ordinary compact stars.

Figure 6

Global monopole field, $\varphi (r)$, as a function of areal radius, $r$, for the previously plotted solutions. Relative to the boson star profile (Fig. 7), where the effect of coupling to the monopole is clear, for the majority of the parameter space the global monopole field is not significantly distorted by the presence of the boson star. In the presence of large nonminimal couplings, however, the field can become significantly distorted near the origin, which contributes to the compactness of the stars [13].

Figure 7

Boson star field, $\psi (r)$, as a function of areal radius, $r$, for the previously plotted solutions. Here, we can see that the solutions from families (d) and (e) are not monotonically decreasing, instead exhibiting successive shells of matter. Excluding the central peak, the solutions from families (d) and (e) consist of seven and three shells, respectively.

Figure 8

Mass function, $M(r)$, as a function of areal radius, $r$, for the previously plotted solutions. It can be seen from inspection that the majority of the bosonic mass is contained within the matter shells rather than near the origin. In the minimally coupled case, the mass contributions from the monopole and boson star are roughly equal and opposite, while in the nonminimal case the global monopole can contribute a positive effective mass [13, 27].

Figure 9

Charge function, $N(r)$, as a function of areal radius, $r$, for the previously plotted solutions. When the number of shells is relatively small and well separated, each matter shell is seen to contribute roughly the same quantity of bosonic matter.

Figure 10

Progression of boson star profile, $\psi$, about a critical central amplitude for family (d) as a function of central amplitude, $\psi (0)$. Approaching the critical central amplitude (${\psi }_i^c\approx 1.255843\times {10}^{-4}$) from below (black line), there are no shells far from the origin. After crossing the critical central amplitude (red line), there is a shell of bosonic matter located far from the origin. As the central amplitude is further increased (blue line), the asymptotic shell migrates inwards.

Figure 11

Progression of global monopole field, $\varphi$, about a critical central amplitude (${\psi }_i^c\approx 1.255843\times {10}^{-4}$) for family (d) for the same solutions shown in Fig. 10. The global monopole field is not significantly affected by the presence of the asymptotic shells and exhibits no significant changes near the critical point.

Figure 12

Asymptotic mass as a function of central amplitude for family (c). Unlike the other families, family (c) does not exhibit critical central amplitudes and consists of only a single branch. This is likely due to the size of the global monopole self-interaction (${\lambda }_{\mathrm{GM}}=1.00$) which greatly reduces the length scale of the monopole (in the case of the minimally coupled monopole, the transformation ${\lambda }_{\mathrm{GM}}\to {\kappa }^2{\lambda }_{\mathrm{GM}},r\to r/\kappa ,t\to t/\kappa$ generates a new solution from an existing one). As such, the space-time achieves its asymptotic solid angle deficit on a length scale small compared to the size of the boson star.

Figure 13

Asymptotic mass as a function of central amplitude for family (d). The bottom plot shows an expanded view of the top one, highlighting the structure. Note the turning point showcased in the subplot and marked with the vertical dashed line. This corresponds to a matter shell that was originally progressing inwards (as a function of boson star central amplitude), progressed to some minimal distance from the origin (corresponding to the turning point) and then reversed direction and progressed outwards.

Figure 14

Progression of mass function, $M(x)$, about a critical central amplitude (${\psi }_i^c\approx 1.255843\times {10}^{-4}$) for solutions from family (d) as a function of central amplitude. Here we have plotted the same solutions shown in Fig. 10. As one progresses across the critical point, a new shell of matter appears far from the origin (red dashed line) and then moves inward (blue dot-dashed line). Note that the inner and asymptotic shells contain approximately the same amount of bosonic matter and that as we cross the critical central amplitude, the asymptotic mass changes discontinuously. Also note the use of the compactified spatial coordinate, $x$, here and in many of the plots below.

Figure 15

Progression of criticality function, $\delta (x)$, about a critical central amplitude (${\psi }_i^c\approx 1.255843\times {10}^{-4}$) for solutions from family (d) as a function of central amplitude. As before, we have plotted the same solutions shown in Figs. 10 and 11; however, we have plotted versus $x$ to more clearly showcase the turning points of the criticality function. Note that as we cross the critical central amplitude, $\delta (x)$, never dips below 0 asymptotically.

Figure 16

Asymptotic mass as a function of central amplitude for family (e). The subplot shows an expanded view of the upper plot highlighting the structure. The nonminimal global monopole coupling appears to smooth out the transitions for at least a subsection of the parameter space. However, note the discontinuity between the final and penultimate branches of the uppermost subplot, which is not an artifact of the resolution of the plot. At the critical central amplitude, the mass approaches $\approx 1.5$ on the left and $\approx -1.3997$ on the right.

Figure 17

Asymptotic mass as a function of central amplitude for family (f) with a minimally coupled global monopole and nonminimally coupled boson star. The subplot shows expanded views of the uppermost plot, highlighting structure which is insufficiently resolved in the first plot. Unlike Figs. 16 and 18, there is no smoothing between the solution branches.

Figure 18

Asymptotic mass as a function of central amplitude for family (g). As with family (e), the branches exhibit significant smoothing. Correspondingly, the smoothing behavior appears to be an effect of the nonminimal global monopole coupling rather than nonminimal boson star coupling.

Figure 19

Progression of boson star profile, $\psi$, about a critical central amplitude for family (e) as a function of central amplitude. Approaching the critical central amplitude (${\psi }_i^c\approx 4.04229\times {10}^{-2}$) from below (black line), there are no shells far from the origin. As the critical central amplitude is crossed (red line), a matter shell appears some finite distance from the origin. As the central amplitude is further increased (blue line), the shell increases in mass and begins to migrate inwards. In contrast to the behavior of the minimally coupled case (Fig. 10), the shells of matter appear or disappear at some finite distance from the origin.

Figure 20

Progression of mass function, $M(x)$, about a critical central amplitude (${\psi }_i^c\approx 4.04229\times {10}^{-2}$) for family (e) as a function of central amplitude. Here we have plotted the same solutions shown in Fig 19. In contrast to the apparent behavior of the minimally coupled case where the critical points can be determined by eye, in the nonminimally coupled case the shells of matter disappear at some finite distance from the origin and the asymptotic mass is continuous across the critical central amplitude. In general, when the shells of matter appear or vanish at a finite distance from the origin, the criticality function is of limited use in determining the value of the critical central amplitudes.

Figure 21

Areal radius of outermost shell ($r_s$) as a function of $|\psi (0)-{\psi }_i^c|$ for selected branches of family (d). Within a given family, the areal radius of the outermost shell follows the scaling law (58) with very similar exponents, $p\approx 1$. It is possible that the variations in the computed exponents, relative to $p=1$, would disappear in the limit $r_s\to \infty$, $|\psi (0)-{\psi }_i^c|\to 0$, with the metric functions approaching their asymptotic values. However, our code is incapable of exploring this regime.

Figure 22

Areal radius of outermost shell ($r_s$) as a function of $|\psi (0)-{\psi }_i^c|$ for selected branches of families (a), (b) and (d). Given the variation in the parameters, the scaling exponent $p$ is remarkably consistent across families ($p\approx 1$). As in the case of a single family (see Fig. 21), it is possible that these small variations would disappear in the asymptotic limit.

Figure 23

Areal radius of outermost shell ($r_s$) as a function of $|\psi (0)-{\psi }_i^c|$ for selected branches of families (e) and (f). Here we plot the penultimate branch of family (e) as it is the only one which exhibits a mass gap (see Fig. 16). It is observed that both the minimally coupled and nonminimally coupled cases exhibit approximately the same scaling exponent, $p\approx 1$.

Figure 24

Profile of the boson star profile for a solution from family (b) (see Table 1). It can be seen that the double precision shooting method ($\epsilon \approx 10^{-16}$) does not localize $\omega$ sufficiently to integrate the solution to the asymptotic regime. Here we compare the true solution (black line) to the bounding solutions generated via the shooting method and observe that the integration with double precision fails before all relevant features are resolved.