Spin-induced dynamical scalarization, descalarization, and stealthness in scalar-Gauss-Bonnet gravity during a black hole coalescence

Particular couplings between a scalar field and the Gauss-Bonnet invariant lead to spontaneous scalarization of black holes. Here, we continue our work on simulating this phenomenon in the context of binary black hole systems. We consider a negative coupling for which the black-hole spin plays a major role in the scalarization process. We find two main phenomena: (i) dynamical descalarization, in which initially scalarized black holes form an unscalarized remnant, and (ii) dynamical scalarization, whereby the late merger of initially unscalarized black holes can cause scalar hair to grow. An important consequence of the latter case is that modifications to the gravitational waveform due to the scalar field may only occur postmerger, as its presence is hidden during the entirety of the inspiral. However, with a sufficiently strong coupling, we find that scalarization can occur before the remnant has even formed. We close with a discussion of observational implications for gravitational-wave tests of general relativity.

Figure 1

Absolute value of the critical coupling, ${\beta }_{\mathrm{c}}$, for spin-induced scalarization of a single BH as a function of the dimensionless spin $\chi$. We show the numerical data of Ref. [74] and the fitting formula (15). The inset shows the relative error between the fit and the data. We see that the error is less than 15% in the range $0.5\le \chi <1$ and less than 5% for $\chi \lesssim 0.74$.

Figure 2

Binary BH simulations, where $s$ ($\overline{s}$) stands for initial or final BH states that are scalarized (unscalarized) and with spin along the positive ($\uparrow$) or negative ($\downarrow$) $z$-direction (i.e., aligned or antialigned with the orbital angular momentum, assuming the latter is $\uparrow$). BH states without an arrow are nonspinning. Panel 2 illustrates a process of spin-induced dynamical scalarization: two initially unscalarized BHs produce a spinning, scalarized remnant. Panel 2 illustrates a process of dynamical descalarization: two initially rotating, scalarized BHs whose spin is antialigned with the orbital angular momentum merge into a rotating BH with a smaller spin magnitude. Consequently, the remnant descalarizes.

Figure 3

Evolution of the scalar field monopole (top panel), scalar field $\ell =2$ multipoles (middle panel) and the gravitational waveform of the background spacetime (bottom panel) for Setup A in Table 1. We rescale the multipoles by the extraction radius $r_{\mathrm{ex}}=100M$, and shift them in time such that $(t-r_{\mathrm{ex}}-t_{\mathrm{M}})/M=0$ indicates the time of merger, determined by the peak of the gravitational waveform.

Figure 4

Profiles of the GB invariant (top panel) and of the scalar field (bottom panel), corresponding to Setup A in Table 1, along the $z$-axis in a close-up region near the CAH. The curves correspond to different times throughout the evolution. The shaded region indicates the CAH, shown $t=100M$ after its formation when the final BH has relaxed to its stationary state. The GB invariant becomes negative during the BHs’ last orbit before merger, and settles to its profile around the final rotating BH with dimensionless spin ${\chi }_f=0.68$. In response, the scalar field becomes unstable.

Figure 5

Snapshots of the scalar field, $\mathrm{\Phi }$, and the GB invariant in the $xz$-plane corresponding to Setup A in Table 1. The color map indicates the amplitude of the scalar field. The isocurvature contours of the GB invariant correspond to $|𝒢M^4|=1$ (solid line), $|𝒢M^4|={10}^{-1}$ (dashed line), $|𝒢M^4|={10}^{-2}$ (dot-dashed line), $|𝒢M^4|={10}^{-3}$ (dotted line), black (red) lines correspond to positive (negative) values of $𝒢$. We show the inspiral (top left), half an orbit before merger (top right), formation of the first CAH (bottom left) and about $200M$ after the merger.

Figure 6

Evolution of the $\ell =m=0$ (solid line), $\ell =2$, $m=0$ (dashed line) and $\ell =m=2$ (dot-dashed line) scalar field multipoles for the coupling $\beta =-{10}^3$; cf. Setup A1 in Table 2. We rescale the multipoles by the extraction radius $r_{\mathrm{ex}}=50M$ and shift them such that $(t-r_{\mathrm{ex}}-t_{\mathrm{M}})/M=0$ indicates the time of merger determined by the peak in the gravitational waveform. For comparison, we also show the formation of the CAH (dotted line). We observe that the scalar field grows exponentially about $20M$ prior to the merger.

Figure 7

Same as Fig. 4, but for Setup A1 in Table 2. We see that the GB invariant (top panel) becomes negative and triggers the excitation of the scalar field (bottom panel) before the formation of the CAH, indicated by the gray region.

Figure 8

Snapshots of the scalar field, $\mathrm{\Phi }$, and the GB invariant, $𝒢$, in the $xz$-plane, corresponding to Setup B in Table 2. The color map represents the amplitude of the scalar field. The isocurvature contours indicate the magnitude of the GB invariant with $|𝒢M^4|=1$ (solid line), $|𝒢M^4|={10}^{-1}$ (dashed line), $|𝒢M^4|={10}^{-2}$ (dot-dashed line), $|𝒢M^4|={10}^{-3}$ (dotted line), with positive (negative) values shown in black (red). We show the inspiral (top left), half an orbit before merger (top right), $10M$ after the CAH formation (bottom left) and about $100M$ after the merger (bottom right).

Figure 9

Profiles of the GB invariant (top panel) and of the scalar field (bottom panel) for Setup B in Table 2 along the $z$-axis. The lines correspond to different times during the evolution. The shaded region indicates the CAH, shown $100M$ after its formation. The GB invariant becomes positive outside the horizon when the CAH is first formed. Consequently, the scalar field magnitude decreases and the remnant BH descalarizes.

Figure 10

Evolution of the scalar field monopole (top panel) and quadrupole (middle panel) and gravitational quadrupole (bottom panel) for Setup B in Table 1. The waveforms are rescaled by the extractions radius $r_{\mathrm{ex}}=100M$ and shifted in time such that $(t-r_{\mathrm{ex}}-t_{\mathrm{M}})/M=0$ at the merger. In the insets we show the absolute values of the multipoles, in logarithmic scale, during the merger and ringdown.

Figure 11

Convergence plots for the $\ell =m=2$ mode of the gravitational waveform (left panel) and the $\ell =m=0$ mode of the scalar field (right panel). In both panels, we show the difference between the low and medium resolution run (solid line) and the medium and high resolution run (dashed line). The latter is rescaled by $Q_4=1.94$, indicating fourth order convergence. The lines are shifted in time such that $(t-r_{\mathrm{ex}}-t_{\mathrm{M}})/M=0$ indicates the time of merger and they are rescaled by the extraction radius $r_{\mathrm{ex}}=100M$.

Figure 12

Hamiltonian constraint along the z-axis during the late-inspiral (solid black), half an orbit before merger (dashed red), at the time of merger from the peak of the gravitational waveform (dash-dot blue) and $100M$ after merger (dotted green). The shaded region indicates the CAH, shown $100M$ after merger.