Piercing of a boson star by a black hole

New light fundamental fields are natural candidates for all or a fraction of dark matter. Self-gravitating structures of such fields might be common objects in the universe, and could comprise even galactic halos. These structures would interact gravitationally with black holes, a process of the utmost importance since it dictates their lifetime, the black hole motion, and possible gravitational radiation emission. Here, we study the dynamics of a black hole piercing through a much larger fully relativistic boson star, made of a complex minimally coupled massive scalar without self-interactions. As the black hole pierces through the bosonic structure, it is slowed down by accretion and dynamical friction, giving rise to gravitational-wave emission. Since we are interested in studying the interaction with large and heavy scalar structures, we consider mass ratios up to $q\sim 10$ and length ratios $\mathcal{L}\sim 62$. Somewhat surprisingly, for all our simulations, the black hole accretes more than 95% of the boson star material, even if an initially small black hole collides with large velocity. This is a consequence of an extreme “tidal capture” process, which binds the black hole and the boson star together, for these mass ratios. We find evidence of a “gravitational atom” left behind as a product of the process.

Figure 1

Configuration describing an isolated BS with mass $M=0.53$ in isolation, corresponding to a value of the scalar field at the center ${\varphi }_0=0.02,{\alpha }_0=0.873,\omega =0.936$. Top: scalar field and metric components as a function of the radial coordinate. Bottom: Hamiltonian constraint along the $R$ axis. The red curve has been multiplied by 16, the expected factor for fourth-order convergence. We use mesh refinement, and $\mathrm{\Delta }$ represents $\mathrm{d}x,\mathrm{d}y$, and $\mathrm{d}z$ of the coarsest level.

Figure 2

Snapshots of evolution, depicting the scalar field absolute value $|\mathrm{\Phi }|$ for the initial data IB in Table 1. From left to right, the snapshots are taken at instants $t=0.0$, 300.8, 380.8, 390.4 in the top row, and at $t=396.8$, 406.4, 416.0, 448.0 in the bottom row. Pink lines depict contours of constant lapse function $\alpha =0.2$, which are a good approximation for the location of the horizon, further indicated by arrows. Notice how the BS is tidally distorted as it approaches the BH and how it eventually is almost totally swallowed up by the BH. Animations are available online [65], and also in the Supplemental Material [66].

Figure 3

Snapshots of evolution, depicting the scalar field absolute value $|\mathrm{\Phi }|$ for the initial data IVB in Table 1. From left to right, the snapshots are taken at instants $t=0.0$, 349.6, 389.6, 408.8 in the top row, and at $t=415.2$, 436.8, 450.4, 480.8 in the bottom row. Pink lines depict contours of constant lapse function $\alpha =0.2$, which, as in the previous figure, indicate the horizon location; this region is much smaller (and difficult to see) than the corresponding one of Fig. 2 since the BH is much smaller. Notice how the BS pulls back the BH during the collision process (panels 5 and 6) and eventually is swallowed up by the BH. Animations are available online [65], and also in the Supplemental Material [66].

Figure 4

Accretion of a scalar onto the BH. Top panel: normalized BH irreducible mass $M_{\mathrm{irr}}/M_{\mathrm{BH}}$ for simulations IB and IVB. The gray lines are the total mass $M_{\mathrm{tot}}$ normalized by the initial mass $M_{\mathrm{BH}}$ given in Table 1. At late times the BH mass approaches $M_{\mathrm{tot}}$; thus, the BH ends up accreting the entire BS. Bottom panel: accretion rate for the two different initial data. Notice that there are two accretion stages for simulation IVB, which we argue to be due to tidal effects. The accretion rate is larger than expected for a stationary regime (see main text).

Figure 5

Location $z$ and the speed $v$ of the BH for IB and IVB, as read from the puncture location. The interaction between the BH and the BS is clear from these data, and it translates into a deceleration starting at $t\sim 400$ and lasting for 20, roughly the time taken to cross the bulk of the BS. Notice also that the BH velocity is negative for a small amount of time in simulation IVB: the BH is tidally captured by the boson star.

Figure 6

Energy flux $F=\frac{d{E}^{\mathrm{rad}}(t)}{dt}$ of the GW, which is the integration of ${\mathrm{\Psi }}_4$ on the sphere $r=500$. The blue dashed line is multiplied by 10.

Figure 7

Real part of the $l=m=0$ multipole of the scalar field on sphere $r=r_{\mathrm{BH}}+1$ around the BH for IB and IVB, where the origin is the position of the BH and $r_{\mathrm{BH}}$ is the BH horizon radius. The dashed gray line indicates a “merger instant” (somewhat arbitrarily) defined as the instant where $|{\varphi }_{00}|>{10}^{-2}$ on the sphere $r=r_{\mathrm{BH}}+1$. The monopolar component grows once the BH plunges into the BS, after which we see an exponentially decaying stage of a rate (${\omega }_I\sim -0.15$ for IB and ${\omega }_I\sim -0.075$ for IVB, in rough agreement with expectations from the quasi-bound state calculation within a linearized approach). Our calculations indicate that this is the first overtone and that a “gravitational atom” was created. At late times, we see a power-law decay, ${\varphi }_{00}\sim t^{-1.5}$ for simulations IB, as expected for massive fields [74, 75]. For simulation IVB we do not have clear control on the very-late-time behavior: this would require a longer simulation, which for these extreme values is very challenging to achieve, while keeping precision under control.

Figure 8

Top: convergence of the Hamiltonian constraint violation at $t=0$ for IB. The green line is multiplied by $Q_4=1.82$, the expected factor for fourth-order convergence. Bottom: Richardson extrapolation used to obtain the value of the Hamiltonian constraint as $\mathrm{\Delta }\to 0$.

Figure 9

Violation of the Hamiltonian and momentum constraints as functions of time for run IB.

Figure 10

Convergence analysis of the $l=0$, $m=2$ multipole of ${\mathrm{\Psi }}_4$, extracted at $r=500M$. The blue line shows the expected result for third-order convergence ($Q_3=1.49$), while the green line shows the expected result for fourth-order convergence ($Q_4=1.82$).

Figure 11

Puncture velocity of a single boosted BH with $M=1$ and $v=0.5$. The dashed gray line is $v=0.5$.