4.2 Dynamics of binary boson stars

The dynamics of binary boson stars is sufficiently complicated that it generally requires numerical solutions. The necessary lack of symmetry and the resolution requirement dictated by the harmonic time dependence of the scalar field combine so that significant computational resources must be expended for such a study. However, boson stars serve as simple proxies for compact objects without the difficulties (shocks and surfaces) associated with perfect fluid stars, and, as such, binary BS systems have been studied in the two-body problem of general relativity. When sufficiently distant from each other, the precise structure of the star should be irrelevant as suggested by Damour’s “effacement theorem” [61Jump To The Next Citation Point].

First attempts at binary boson-star simulations assumed the Newtonian limit, since the SP system is simpler than the EKG one. Numerical evolutions of Newtonian binaries showed that in head-on collisions with small velocities, the stars merge forming a perturbed star [50]. With larger velocities, they demonstrate solitonic behavior by passing through each other, producing an interference pattern during the interaction but roughly retaining their original shapes afterwards [51]. In [50] it was also compute preliminary simulation of coalescing binaries, although the lack of resolution in these 3D simulations prevents of strong statements on these results. The head-on case was revisited in [26] with a 2D axisymmetric code. In particular, these evolutions show that final state will depend on the total energy of the system, that is, the addition of kinetic, gravitational and self-interaction energies. If the total energy is positive, the stars exhibit a solitonic behavior both for identical (see Figure 13View Image) as for non-identical stars. When the total energy is negative, the gravitational force is the main driver of the dynamics of the system. This case produces a true collision, forming a single object with large perturbations, which slowly decays by gravitational cooling, as displayed in Figure 14View Image.

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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 14View Image. Reprinted with permission from [26]; copyright by APS.
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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 13View Image. Reprinted with permission from [26]; copyright by APS.

The first simulations of boson stars with full general relativity were reported in [12], where the gravitational waves were computed for a head-on collision. The general behavior is similar to the one displayed for the Newtonian limit; the stars attract each other through gravitational interaction and then merge to produce a largely perturbed boson star. However, in this case the merger of the binary was promptly followed by collapse to a black hole, an outcome not possible when working within Newtonian gravity instead of general relativity. Unfortunately, very little detail was given on the dynamics.

Much more elucidating was work in axisymmetry [141], in which head-on collisions of identical boson stars were studied in the context of critical collapse (discussed in Section 6.1) with general relativity. Stars with identical masses of M = 0.47 ≈ 0.75 Mmax were chosen, and so it is not surprising that for small initial momenta the stars merged together to form an unstable single star (i.e., its mass was larger than the maximum allowed mass, Mmax). The unstable hypermassive star subsequently collapsed to a black hole. However, for large initial momentum the stars passed through each other, displaying a form of solitonic behavior since the individual identities were recovered after the interaction. The stars showed a particular interference pattern during the overlap, much like that displayed in Figures 1View Image and 13View Image.

Another study considered the very high speed, head-on collision of BSs [55Jump To The Next Citation Point]. Beginning with two identical boson stars boosted with Lorentz factors ranging as high as 4, the stars generally demonstrate solitonic behavior upon collision, as shown in the insets of Figure 18View Image. This work is further discussed in Section 6.2.

The interaction of non-identical boson stars was studied in [174Jump To The Next Citation Point] using a 3D Cartesian code to simulate head-on collisions of stars initially at rest. It was found that, for a given separation, the merger of two stars would produce an unstable star that collapses to a black hole if the initial individual mass were M ≥ 0.26 ≈ 0.4 Mmax. For smaller masses, the resulting star would avoid gravitational collapse and its features would strongly depend on the initial configuration. The parameterization of the initial data was written as a superposition of the single boson-star solution ϕ0(r), located at different positions r1 and r2

ϕ = (1)ϕ0(r1)eiωt + (2)ϕ0(r2)ei(𝜖ωt+δ). (82 )
Many different initial configurations are possible with this parameterization. The precise solution ϕ0 is unaffected by changing the direction of rotation (within in the complex plane isospace) via 𝜖 = ±1 or by a phase shift δ.

When 𝜖 = − 1, the Noether charge changes sign and the compact object is then known as an anti-boson star. Three particular binary cases were studied in detail: (i) identical boson stars (𝜖 = 1, δ = 0), (ii) the pair in phase opposition (𝜖 = 1, δ = π), and (iii) a boson–anti-boson pair (𝜖 = − 1, δ = 0). The trajectories of the centers of the stars are displayed in Figure 15View Image, together with a simple estimate of the expected trajectory assuming Newtonian gravity. The figure makes clear that the merger depends strongly on the kind of pair considered, that is, on the interaction between the scalar fields.

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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 [174Jump To The Next Citation Point]; copyright by APS.

A simple energy argument is made in [174Jump To The Next Citation Point] to understand the differing behavior. In the weak gravity limit when the stars are well separated, one can consider the local energy density between the two stars. In addition to the contribution due to each star separately, a remaining term Δ results from the interaction of the two stars and it is precisely this term that will depend on the parameters 𝜖 and δ. This term takes the simple form

Δ = Δ0 cos[(1 − 𝜖)ω t − δ], (83 )
where Δ0 is a positive definite quantity. One then observes that the identical pair will have an increased energy density Δ = + Δ0 resulting in a deeper (and more attractive) gravitational well between the stars. In contrast, the pair with opposite phases has a decreased energy density Δ = − Δ0 between them, resulting in a gravitational well less attractive than the area surrounding it resulting in repulsion. The boson–anti-boson pair has an interaction that is harmonic in time Δ = Δ0 cos(2ωt) and, therefore, sometimes positive and sometimes negative. However, if the time scale of interaction is not particularly fast, then the interaction averages to zero. Note that the boson–anti-boson pair trajectory is the closest to the simple Newtonian estimate. The qualitative behavior agrees very well with the numerical results.

The orbital case was later studied in [173Jump To The Next Citation Point]. This case is much more involved both from the computational point of view (i.e., there is less symmetry in the problem) and from the theoretical point of view, since for the final object to settle into a stationary, rotating boson star it must satisfy the additional quantization condition for the angular momentum of Eq. (71View Equation).

One simulation consisted of an identical pair each with individual mass M = 0.5, with small orbital angular momentum such that J ≤ N. In this case, the binary merges forming a rotating bar that oscillates for some time before ultimately splitting apart. This can be considered as a scattered interaction, which could not settle down to a stable boson star unless all the angular momentum was radiated.

In the case of boson–anti-boson pair, the total Noether charge is already trivial, and the final object resembles the structure of a rotating dipole. The pair in opposition of phase was not considered because of the repulsive effect from the interaction. The cases with very small angular momentum J ≪ N or with J ≤ N collapsed to a black hole soon after the merger. The trajectories for this latter case are displayed in Figure 16View Image, indicating that the internal structure of the star is irrelevant (as per the effacement theorem [61]) until that the scalar fields overlap.

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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 [173Jump To The Next Citation Point]; copyright by APS.

Other simulations of orbiting, identical binaries have been performed within the conformally flat approximation instead of full GR, which neglects gravitational waves (GW) [167Jump To The Next Citation Point]. Three different qualitative behaviors were found. For high angular momentum, the stars orbit for comparatively long times around each other. For intermediate values, the stars merged and formed a pulsating and rotating boson star. For low angular momentum, the merger produces a black hole. No evidence was found of the stars splitting apart after the merger.

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