Figure 1

Single boson star. Phase oscillation of the real part of the scalar field at the center of symmetry up to $t=50$ for the resolution $\Delta x=0.25$ (similar behavior was followed up to $t=500$). The solid line indicates the analytically expected value, the crosses show the values read off from the numerical solution within the static coordinates, and the circles show the same by using the harmonic coordinates.Reuse & Permissions

Figure 2

Single boson star. $L_{\infty }$ of $g_{xx}$ versus time for the solution obtained employing the static coordinate condition. The figure displays the result of the evolution for three different base resolutions $\Delta x=\{0.375,0.25,0.156\}$ (in addition, each numerical solution is obtained with three levels of refinement). The values obtained converge to a constant value as expected, and the maximum variation observed to $t=50$ is $\simeq 0.0073\%$ for the coarsest base resolution while for the highest one it is just $\simeq 0.0015\%$.Reuse & Permissions

Figure 3

Single boson star. $L_{\infty }$ of $g_{xx}$ versus time for the solution obtained employing the harmonic-coordinate condition. The figure displays the result of the evolution for three different base resolutions $\Delta x=\{0.25,0.375,0.50\}$ (in addition, each numerical solution is obtained with three levels of refinement). After a small transient behavior, the values obtained converge to a constant value. The maximum variation observed to $t=500$ is $\simeq 0.72\%$ for the coarsest base resolution while $\simeq 0.11\%$ for the highest one.Reuse & Permissions

Figure 4

Single boson star. Convergence of the $L_2$ norm of the $Z$-constraint residual [defined by Eq. (30)] versus time by using harmonic coordinates, for the same resolutions as in Fig. 3. As the resolution is increased the constraint violation is clearly reduced.Reuse & Permissions

Figure 5

Boson/boson pair. $L_{\infty }$ norm of the Hamiltonian constraint residual versus the initial (coordinate) separation of the boson stars. As the distance is increased the violation decreases as expected since each star is defined so as to satisfy the constraints. We choose a value of $D=50$ which lies close to the asymptotic behavior and is still manageable in terms of the boundaries.Reuse & Permissions

Figure 6

Boson/boson pair. Geometry behavior for different central values of the scalar field that defines the stars. As the stars merge, the behavior of geometric variables changes drastically for different values. Stars with initial central densities $>0.005$ seem to give rise to a black hole as illustrated by their considerable growth; those below this value yield much smoother behavior. This is illustrated in the figure which displays the maximum of the component of the metric $g_{xx}$ versus time for different central amplitudes of the scalar field $\{{\varphi }_0=0.02,0.015,0.01,0.05\}$, respectively. These give rise to stars with masses $\{M=0.47,0.42,0.36,0.26\}$, respectively.Reuse & Permissions

Figure 7

Boson/boson pair. $L_{\infty }(g_{xx})$ as a function of time for three different resolutions and their differences. We can observe clearly that the merger occurs at approximately $t\simeq 550$. The solution is qualitatively convergent during the first part of the merger until $t\simeq 800$ when boundary effects negatively affect the convergence.Reuse & Permissions

Figure 8

Boson/boson pair. 2D $z=0$ cuts of the Noether density $J^0$ at different times. As the stars come closer and merge, the maximum value of $J^0$ grows significantly, followed by a quadrupolar oscillation.Reuse & Permissions

Figure 9

Boson/boson pair. Contours corresponding to the value $J^0=2\times {10}^{-5}$ at the $z=0$ plane. From top to bottom the contours are shown for times $\{0,160,240,300\}$, $\{340,380,420,460\}$,$\{500,540,580,640\}$, $\{680,760,840,900\}$, and $\{980,1060,1140,1200\}$ indicated by solid lines, solid lines with crosses, solid lines with squares, and solid lines with triangles, respectively, in each plot. As the stars get closer, the initially spherical contours deform until merging, as illustrated by a cusp on the topmost figure. Afterwards, $J^0$ exhibits essentially quadrupolar-type oscillations.Reuse & Permissions

Figure 10

Boson/boson pair. Coefficients corresponding to the $l=2$ modes of $r{\Psi }_4$ as a function of time, extracted at $r_{\mathrm{ext}}=200$. After some initial transient due to spurious radiation in the initial data, the signal is clearly visible corresponding to the merger of the stars and later decaying to smaller values. At late times, $t>800$, contamination with boundary effects obscures the extracted signals.Reuse & Permissions

Figure 11

Boson/boson pair. The coefficient $C_{2,+2}$ for different extraction radii. This figure illustrates both the outgoing waves due to the dynamics of the spacetime and the incoming waves that have traveled to the boundary and bounced off it. Additionally, some remnant dependence on $r$ is visible indicating that the extraction is still performed not sufficiently far from the sources. Nevertheless, the structure of the outgoing waves is clearly visible.Reuse & Permissions

Figure 12

Boson/boson pair. Radiated energy as a function of time. The top plot indicates the energy radiated in all modes (dashed line) together with that radiated solely in $l=2$ modes. Clearly, until boundary effects begin to affect the results, practically all energy is radiated in the $l=2$ modes. The bottom plot displays the difference between the $l=2$ radiated energies at three different resolutions, indicating convergent behavior until $t\simeq 950$.Reuse & Permissions

Figure 13

Boson/boson in opposition of phase pair. 2D $z=0$ cuts of the Noether density $J^0$ at different times Although the stars come closer, they never seem to merge.Reuse & Permissions

Figure 14

Boson/boson in opposition of phase pair. Contours with value $J^0=2\times {10}^{-5}$ on the $z=0$ plane. From top to bottom the contours are shown for times $\{180,360,540,720\}$ and $\{720,900,1080,1200\}$ indicated by solid lines, solid lines with crosses, solid lines with squares, and solid lines with triangles, respectively, in each plot. The stars initially get closer (top frame) but afterwards they move apart from each other (bottom frame); this process continues on as the stars are trapped in a common gravitational well.Reuse & Permissions

Figure 15

Boson/antiboson pair. The $C_{2,m}$ coefficients of the $r{\Psi }_4$ decomposition (top frame) and the emitted radiation versus time (bottom frame).Reuse & Permissions

Figure 16

Boson/antiboson pair. 2D $z=0$ cuts of the Noether density $J^0$ at different times. Although the stars come closer and merge, the bosonic part is distinguishable from the antibosonic one by means of the different sign of the Noether density.Reuse & Permissions

Figure 17

Boson/antiboson pair. Contours with value $J^0=2\times {10}^{-5}$ on the $z=0$ plane. From top to bottom contours are shown for times $\{180,360,540,600,660\}$ and $\{660,780,840,900\}$ indicated by solid lines, solid lines with crosses, solid lines with squares, and solid lines with triangles, respectively, in each plot. Only the right star has such a contour initially as the left has a negative initial value for $J^0$. As the stars come closer to each other, the “tunneling” behavior is illustrated by the positive value appearing on the left side (top frame). As time progresses, this process essentially reiterates itself from the left side to the right side (bottom frame). The dynamics that follows repeats this side-to-side “motion” in the Noether density with the stars trapped in a common gravitational well.Reuse & Permissions

Figure 18

The proper distance from the center of the boson star to the center of the computational domain as a function of time for the different cases and the Newtonian approximation. The position of the boson is identified with the maximum of the energy density.Reuse & Permissions

Figure 19

The maximum of the energy density $\rho$ is plotted as a function of time for the three different cases.Reuse & Permissions