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
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
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.
http://www.livingreviews.org/lrr-2012-6 |
Living Rev. Relativity 15, (2012), 6
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