Dynamical Boson Stars

Abstract

The idea of stable, localized bundles of energy has strong appeal as a model for particles. In the 1950s, John Wheeler envisioned such bundles as smooth configurations of electromagnetic energy that he called geons, but none were found. Instead, particle-like solutions were found in the late 1960s with the addition of a scalar field, and these were given the name boson stars. Since then, boson stars find use in a wide variety of models as sources of dark matter, as black hole mimickers, in simple models of binary systems, and as a tool in finding black holes in higher dimensions with only a single Killing vector. We discuss important varieties of boson stars, their dynamic properties, and some of their uses, concentrating on recent efforts.

Figure 1

Demonstration of the solitonic nature of the (mini-)boson star. Shown are snapshots of the magnitude squared of the complex scalar field for a head-on collision of two identical mini-boson stars. The interacting stars display an interference pattern as they pass through each other, recovering their individual identities after the collision. However, note that the BSs have a larger amplitude after their interaction and so are not true solitons. The collision can therefore be considered inelastic. Reprinted with permission from [49]. See also [141] (e.g., Figure 5.12).

Figure 2

Profiles characterizing static, spherically symmetric boson stars with a few different values of the central scalar field (top left). Reprinted with permission from [141].

Figure 3

Left: The mass of the boson star as a function of the central value of the scalar field in adimensional units . Right: Maximum mass as a function of Λ (squares) and the asymptotic Λ → ∞ relation of Eq. (52) (solid curve). Reprinted with permission from [57]; copyright by APS.

Figure 4

The compactness of a stable boson star (black solid line) as a function of the adimensional self-interaction parameter Λ ≡ λ/(4π Gm²). The compactness is shown for the most massive stable star (the most compact BS is unstable). This compactness asymptotes for Λ → ∞ to the value indicated by the red, dashed line. Also shown for comparison is the compactness of a Schwarzschild BH (green dot-dashed line), and the maximum compactness of a non-spinning neutron star (blue dotted line). Reprinted with permission from [8]; copyright by IOP.

Figure 5

The mass (solid) and the number of particles (dashed) versus central scalar value for charged boson stars with four values of as defined in Section 3.3. The mostly-vertical lines crossing the four plots indicate the solution for each case with the maximum mass (solid) and maximum particle number (dashed). Reprinted with permission from [123]; copyright by Elsevier.

Figure 6

Top: Total mass (in units of ) and fundamental frequency of an oscillaton as a function of the central value of the scalar field ϕ₁(r = 0). The maximum mass is . Bottom: Plot of the total mass versus the radius at which g_rr achieves its maximum. Reprinted with permission from [5]; copyright by IOP.

Figure 7

The scalar field in cylindrical coordinates ϕ(ρ, z) for two rotating boson-star solutions: (left) k = 1 and (right) k = 2. The two solutions have roughly comparable amplitudes in scalar field. Note the toroidal shape. Reprinted with permission from [141].

Figure 8

Left: The maximum of the central value of each of the two scalar fields constituting the multi-state BS for the fraction η = 3, where η ≡ N⁽²⁾/N⁽¹⁾ defines the relative “amount” of each state. Right: The frequencies associated with each of the two states of the multi-state BS. At t = 2000, there is a flip in which the excited state (black solid) decays and the scalar field in the ground state (red dashed) becomes excited. Discussed in Section 3.7. Reprinted with permission from [25]; copyright by APS.

Figure 9

Oscillation frequencies of various boson stars are plotted against their mass. Also shown are the oscillation frequencies of unstable BSs obtained from the fully nonlinear evolution of the dynamical system. Unstable BSs are observed maintaining a constant frequency as they approach a stable star configuration. Reprinted with permission from [196]; copyright by APS.

Figure 10

The instability time scale of an excited boson star (the first excitation) to one of three end states: (i) decay to the ground state, (ii) collapse to a black hole, or (iii) dispersal. Reprinted with permission from [15]; copyright by APS.

Figure 11

Very long evolutions of a perturbed, slightly sub-critical, boson star with differing outer boundaries. The central magnitude of the scalar field is shown. At early times (t < 250 and the middle frame), the boson star demonstrates near-critical behavior with small-amplitude oscillations about an unstable solution. For late times (t > 250), the solution appears converged for the largest two outer boundaries and suggests that sub-critical boson stars are not dispersing. Instead, they execute large amplitude oscillations about low-density boson stars. Reprinted with permission from [142].

Figure 12

The evolution of r² ρ (where ρ is the energy density of the complex scalar field) with massive field (left) and massless (right). In the massive case, much of the scalar field collapses and a perturbed boson star is formed at the center, settling down by gravitational cooling. In the massless case, the scalar field bounces through the origin and then disperses without forming any compact object. Reprinted with permission from [198]; copyright by APS.

Figure 13

Collision of identical boson stars with large kinetic energy in the Newtonian limit. The total energy (i.e., the addition of kinetic, gravitational and self-interaction) is positive and the collision displays solitonic behavior. Contrast this with the gravity-dominated collision displayed in Figure 14. Reprinted with permission from [26]; copyright by APS.

Figure 14

Collision of identical boson stars with small kinetic energy in the Newtonian limit. The total energy is dominated by the gravitational energy and is therefore negative. The collision leads to the formation of a single, gravitationally bound object, oscillating with large perturbations. This contrasts with the large kinetic energy case (and therefore positive total energy) displayed in Figure 13. Reprinted with permission from [26]; copyright by APS.

Figure 15

The position of the center of one BS in a head-on binary as a function of time for (i) [B-B] identical BSs, (ii) [B-poB] opposite phase pair, and (iii) [B-aB] a boson-anti-boson pair. A simple argument is made, which qualitatively matches these numerical results, as discussed in Section 4.2. Also shown is the expected trajectory from a simple Newtonian two-body estimate. Reprinted with permission from [174]; copyright by APS.

Figure 16

The position of the center of one BS within an orbiting binary as a function of time for the two cases: (i) [B-B] identical BSs and (ii) [B-poB] opposite phase pair. Notice that the orbits are essentially identical at early times (and large separations), but that they start to deviate from each other on closer approach. This is consistent with the internal structure of each member of the binary being irrelevant at large separations. Reprinted with permission from [173]; copyright by APS.

Figure 17

Evolution of a boson star (solid line) perturbed by a shell of scalar field (dashed line). Shown is the mass density ∂M/∂r for each contribution. By t ≈ 100 the real scalar field pulse has departed the central region and perturbed the boson star into an unstable, compact configuration. Contrast the t = 0 frame with that of t = 97.5 and note the increase in compaction. This unstable BS survives until t ≈ 500 only because the initial perturbation has been tuned to one part in 10¹⁵ and indicates Type I critical behavior. Reprinted with permission from [142].

Figure 18

Evolutions of the head-on collisions of identical boson stars boosted toward each other with initial Lorentz factors γ as indicated. Time flows downward within each column and the top edge displays the axis of symmetry. The color-scale indicates the value of |ϕ|. In the middle frames one sees the interference pattern characteristic of high kinetic energy BS collisions (as mentioned in Figure 1). In the last column on the right, the collision produces a BH with apparent horizon indicated by the black oval in the third frame. Reprinted with permission from [55]; copyright by APS.