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 13) 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 14.

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 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, ). 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 1 and 13.

Another study considered the very high speed, head-on collision of BSs [55]. 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 18. This work is further discussed in Section 6.2.

The interaction of non-identical boson stars was studied in [174] 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 . 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 , located at different positions and

Many different initial configurations are possible with this parameterization. The precise solution is unaffected by changing the direction of rotation (within in the complex plane isospace) via or by a phase shift .When , 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 (, ), (ii) the pair in phase opposition (, ), and (iii) a boson–anti-boson pair (, ). The trajectories of the centers of the stars are displayed in Figure 15, 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.

A simple energy argument is made in [174] 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 results from the interaction of the two stars and it is precisely this term that will depend on the parameters and . This term takes the simple form

where is a positive definite quantity. One then observes that the identical pair will have an increased energy density resulting in a deeper (and more attractive) gravitational well between the stars. In contrast, the pair with opposite phases has a decreased energy density 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 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 [173]. 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. (71).

One simulation consisted of an identical pair each with individual mass , with small orbital angular momentum such that . 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 or with collapsed to a black hole soon after the merger. The trajectories for this latter case are displayed in Figure 16, indicating that the internal structure of the star is irrelevant (as per the effacement theorem [61]) until that the scalar fields overlap.

Other simulations of orbiting, identical binaries have been performed within the conformally flat approximation instead of full GR, which neglects gravitational waves (GW) [167]. 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.

Living Rev. Relativity 15, (2012), 6
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