Extreme mass ratio head-on collisions of black holes in Einstein-scalar-Gauss-Bonnet theory

Abstract

The evolution of the event horizon when two black holes merge can be determined by resorting to ray-tracing techniques on a single black hole spacetime, under the assumption that the binary's mass ratio is infinite and the underlying gravity theory respects the equivalence principle. We extend this analysis to the head-on collision of non-spinning hairy black holes in Einstein-scalar-Gauss-Bonnet gravity. In such theories the scalar field is coupled to a higher curvature operator, leading to possible modifications of the background geometry and consequently of photon propagation. We study three families of coupling functions: linear, quadratic, and a particular exponential form. The first choice enjoys a shift symmetry and forces the presence of scalar hair in the spectrum of black hole solutions. The latter two couplings break the shift symmetry and allow for spontaneously scalarized hairy black holes, which coexist with the Schwarzschild black hole. For all three classes of theories studied, we find a merger duration that is longer than the corresponding time in general relativity, when keeping the size of the small black hole fixed, and for viably small values of the coupling constant. However, the case of the exponential coupling yields a non-monotonic merger duration, which can become shorter than the general relativity value for a sufficiently large coupling constant. We observe that the merger duration and the area increment generically track the behavior of the small black hole's photon ring. Finally, we also compare our results with recent numerical simulations by other groups, despite the dissimilar mass ratios considered.

Figures (13)

• Figure 1: Single branch of BH solutions of EsGB with linear coupling, $f(\Phi)=\Phi/4$, plotted in the ($\Phi_h$, $\alpha/r_h^2$) parameter space. These develop in the ($\Phi_h$, $\alpha/r_h^2$)-plane from the point ($\Phi_h=0$, $\alpha/r_h^2=0$) until Eq. \ref{['eq:inequality']} is no longer satisfied, i.e. for $|\alpha|/r_h^2 \geq 1/\sqrt{6}$, represented by the red-dashed curves. The Schwarzschild spacetime, with $\Phi=0$, is a solution of the theory only for $\alpha/r_h^2=0$.
• Figure 2: (Top) Radial profile of the metric functions $e^{\Gamma(r)}$ and $e^{\Lambda(r)}$ for several EsGB BH solutions with a linear coupling. The Schwarzschild solution corresponds to the red dashed line. (Bottom) Radial profile of the scalar field, $\Phi(r)$, for the same BH solutions.
• Figure 3: First three branches of BH solutions of EsGB with quadratic coupling, $f(\Phi)=\Phi^2/4$, plotted in the ($\Phi_h$, $\alpha/r_h^2$) parameter space. The Schwarzschild solution, represented in blue along the $\Phi_h=0$ axis, is a solution for all $\alpha$, but it becomes unstable at specific values of $\alpha/r_h^2$, leading to the emergence of BHs with scalar hair. These hairy solutions branch out from the $\Phi_h=0$ axis until Eq. \ref{['eq:inequality']} is no longer satisfied. Hairy BHs are restricted to lie within the two red-dashed curves determined by $\Phi_h = \pm(24\alpha^2/r_h^4)^{-1/2}$.
• Figure 4: (Top) Radial profile of the metric functions $e^{\Gamma(r)}$ and $e^{\Lambda(r)}$ for several $n=0$ EsGB BH solutions with a quadratic coupling. The Schwarzschild solution corresponds to the red-dashed line. (Middle) Radial profile of the scalar field, $\Phi(r)$, for the same $n=0$ BH solutions. (Bottom) Radial profile of the scalar field, $\Phi(r)$, for $n=1$ and $n=2$ EsGB BH solutions with a quadratic coupling.
• Figure 5: First three branches of BH solutions of EsGB with exponential coupling, $f(\Phi)=(1-e^{-3 \Phi^2 /2})/6$, plotted in the ($\Phi_h$, $\alpha/r_h^2$) parameter space. The Schwarzschild solution, represented in blue in the $\Phi_h=0$ axis, is a solution for all coupling parameters $\alpha/r_h^2$. The Schwarzschild solution becomes unstable for $\alpha/r_h^2 > 0.362811$, leading to the emergence of BH solutions with scalar hair. For $n \geq 1$ solutions, these develop in the ($\Phi_h$, $\alpha/r_h^2$)-plane from the $\Phi_h=0$ axis until Eq. \ref{['eq:inequality']} is no longer satisfied. On the other hand, $n=0$ solutions exist for all values of $\Phi_h$. The red, dashed curves were obtained by solving numerically the equality in Eq. \ref{['eq:inequality']}.