Hyperbolic black hole encounters probe strong-field gravity and offer an additional testing ground for General Relativity. This work presents the first fully nonlinear numerical simulations of black hole scattering in Einstein-scalar-Gauss-Bonnet theory—an effective gravity model with an extra scalar field. The scattering angle from simulations was compared to an analytic prediction within the effective one-body formalism. Excellent agreement was found, demonstrating accurate modeling of strong-field scalar-gravitational dynamics. This paves the way for semi-analytical templates of gravitational-wave signals from compact binaries in modified theories of gravity.
Black holes are ideal laboratories for testing gravity in the strong-field regime. Since the first detection of gravitational waves from a pair of black holes by the LIGO detectors in 2015, we have obtained abundant evidence of the accuracy of the theory laid down by Albert Einstein. However, inconsistencies—from the dark matter problem to the quantization of gravity—motivate the search for deviations. Einstein-scalar-Gauss-Bonnet theory adds a scalar field coupled to the spacetime curvature, endowing black holes with 'scalar hair.' Hyperbolic encounters—fly-bys of black holes past each other—provide an additional test channel, as the scattering angle is a gauge-invariant observable. Previously, there were no nonlinear simulations of such encounters in this theory, and analysts relied only on approximations.
The calculations were conducted along two fronts. On the analytical side, using the effective one-body formalism, corrections to the potential up to the third post-Minkowskian order were derived, drawing on results from effective field theory. On the numerical side, the code GRFolres, which extends GRChombo for theories with higher derivatives, was employed. The equations were solved in the mCCZ4 formalism with a modified harmonic gauge, ensuring the well-posedness of the problem. For initial data in general relativity, TwoPunctures was used—the scalar field was set to zero, which produced a short-lived transient. An adaptive mesh was refined near singularities and along wavefronts. Five simulations were conducted, varying the coupling constant λ and the impact parameter, and scattering angles were extracted via polynomial fitting of trajectories.
For an impact parameter b=9.7M, where the scattering angle exceeded 200°, the difference between numerical and analytical angles was less than 1 degree—within simulation errors. For weak coupling (λ/M²=0.0175), the 2PM analytical prediction already gave a deviation δχ≈-0.2, and for strong coupling (λ/M²=0.0325), going from 2PM to 3PM improved accuracy fourfold: δχ changed from -3.07 to -0.72. For a larger impact parameter b=11.0M, the effects of modified gravity weakened, and results approached GR predictions. Remarkably, unlike bound orbits, where a merger delay was observed, on hyperbolic trajectories the imperfection of initial data is negligible. This makes the scattering regime ideal for testing theories. Moreover, the evolution of irreducible masses, computed via Wald's entropy, which generalizes ideas of Stephen Hawking, confirmed the rapid scalarization of the holes.
The agreement between numerical and analytical results demonstrates that the effective one-body formalism, developed for GR, can be successfully applied to modified theories—at least for non-spinning equal masses. This paves the way for fast semi-analytical models of gravitational-wave signals without costly full simulations. Such templates are critically important for detection and parameter estimation of events in future observing runs of LIGO, in the development of which Kip Thorne participated.
In the future, scientists plan to include radiation corrections from scalar dipole emission, which are relevant for systems with unequal masses or spin. It is also necessary to construct fully self-consistent initial data in Einstein-scalar-Gauss-Bonnet theory to eliminate transient artifacts. Extension to spinning black holes and realistic astrophysical scenarios will allow the creation of accurate templates for searching for GR violations in gravitational-wave data.
The results will impact gravitational-wave astrophysics (improving GR tests), fundamental gravity theory (understanding strong-field effects in theories with higher curvatures), and computational physics (advancing numerical relativity methods for modified theories).
Increase simulation accuracy by improving initial conditions; include scalar radiation in the analytics; extend the study to spinning configurations and systems with unequal masses.
The search for deviations from GR is directly motivated by the unsolved mysteries of dark matter and dark energy. Theories with a dynamical scalar field, such as Einstein-scalar-Gauss-Bonnet, can leave characteristic signatures in gravitational waves, allowing these hypotheses to be tested on astrophysical scales.
🎯 In Einstein-scalar-Gauss-Bonnet theory, black holes acquire 'scalar hair,' which violates the famous no-hair theorem of GR—their properties begin to depend on the additional field.
🎬 In the movie 'Interstellar,' the characters use the black hole Gargantua for time travel. If a pervasive scalar field existed, the properties of such black holes could become even more exotic, exactly as in this modified theory.