Final fate of compact boson star mergers

Boson stars, self-gravitating objects made of a complex scalar field, have been proposed as simple models for very different scenarios, ranging from galaxy dark matter to black hole mimickers. Here we focus on a very compact type of boson stars to study binary mergers by varying different parameters, namely the phase shift, the direction of rotation, and the angular momentum. Our aim is to investigate the properties of the object resulting from the merger in these different scenarios by means of numerical evolutions. These simulations, performed by using a modification of the covariant conformal Z4 formalism of the Einstein equations that does not require the algebraic enforcing of any constraint, indicate that the final state after a head-on collision of low mass boson stars is another boson star. However, almost complete annihilation of the stars occurs during the merger of a boson-antiboson pair. The merger of orbiting boson stars form a rotating bar that quickly relaxes to a nonrotating boson star.

Figure 1

Robust stability. $L_2$ norm of $|\tilde{\gamma }-1|$ (top panel) and $|Z_x|$ (bottom panel) as a function of time—in crossing time units. Some modes increase for the CCZ4 system with ${\lambda }_0=0$ (blue dashed line), showing that this choice leads to a weakly pseudohyperbolic system. These modes (and all others) remain constant—a sign of the strong hyperbolicity of the system—in the other two cases; CCZ4 with ${\lambda }_0=1$ (black solid line) and CCZ4e (red dotted line), where the conformal constraints are algebraically enforced.

Figure 2

Gauge waves. $L_{\infty }$ norm of $|\tilde{\gamma }-1|$ (top panel) and $|\mathcal{H}|$ (bottom panel) as a function of time—in crossing time units. The BSSN system (red solid line) and CCZ4 with ${\kappa }_c=0$ (green dotted line) display an unbound growth in some constraints. Both CCZ4e (black solid line) and CCZ4 with ${\kappa }_c=1$ (blue dashed line) maintain the constraints under control at least for 100 crossing times.

Figure 3

Solitonic boson star. The top panel displays the metric components $\alpha (r)$ (blue solid line) and $\psi (r)$ (red solid line) for the typical solitonic boson stars with compactness $C\approx 0.118$ used here, compared to the Schwarzschild solution (dashed lines) in isotropic coordinates. The bottom panel shows the scalar field profile ${\varphi }_0(r)$, which is almost constant in the interior and decays rapidly at the surface of the star.

Figure 4

Solitonic boson star. Evolution of the real part of $\varphi$ at $r=0$. The solid red line illustrates the analytically expected value $\cos (\omega t)$ with $\omega =1.0666$. The blue circles show the numerical solution obtained with different evolution systems (i.e., BSSN and CCZ4), which cannot be distinguished by eye in this plot.

Figure 5

Solitonic boson star. The $L_2$ norm of $|\tilde{\gamma }-1|$ (top panel) and of $|\mathcal{H}|$ (bottom panel) as a function of time. The solution obtained with BSSN (red solid line) shows a small $|\tilde{\gamma }-1|$ constraint as a result of enforcing constraint in each integration time step, but the Hamiltionian constraint increases over time. The solutions obtained with CCZ4 are stable if we add the damping terms for the conformal constraints (black solid line)—otherwise there is a linear growth in $|\tilde{\gamma }-1|$ that will lead to an unstable evolution (blue dashed line).

Figure 6

Head-on binary collisions. ADM mass and Noether charge as a function on time for the different cases studied. The boson-boson binaries merging into a single one loses approximately 5% of their initial mass and Noether charge. In contrast, the boson-antiboson binaries annihilate during the merger, radiating most of the scalar field (and the corresponding mass). The total Noether charge for the B-aB cases is zero through all the evolution.

Figure 7

Head-on binary collisions. Time snapshots of the Noether charge in the plane $z=0$. Each row corresponds to the different $\mathrm{B}\text{−}\mathrm{B}(\theta )$ and $\mathrm{B}\text{−}\mathrm{aB}(\theta )$ cases studied here. The collision of the stars happens approximately at $t=28$. The result of the B-B is a single boson star except in the case of $\mathrm{B}\text{−}\mathrm{B}(\pi )$. The stars in the B-aB case annihilate each other during the merger.

Figure 8

Orbital binary collisions. ADM mass (top panel), angular momentum $J_z$ (middle panel), and Noether charge (bottom panel) as a function on time for the different tangential boost velocities. During the coalescence approximately 5% of the mass and Noether charge is lost, and almost all the angular momentum.

Figure 9

Orbital binary collisions. Noether charge in the plane $z=0$ for the different boost velocities $v=\{0,0.05,0.10,0.15\}$. The merger between solitonic boson stars happens approximately at $t\approx 30$ for the cases $v=\{0,0.05,0.10\}$ and at $t\approx 40$ for the quasicircular orbit case $v=0.15$. For the latter case, after the merger two blobs of scalar field take away a large fraction of the angular momentum from the system.

Figure 10

Orbital binary collisions. The $L_2$ norm of the physical (blue solid line), the conformal (red solid line), and the energy-momentum (green solid line) constraints as a function of time. With our choice of the damping parameters, all the constraints are perfectly under control during all the simulation.