Critical phenomena at the threshold of immediate merger in binary black hole systems: the extreme mass ratio case
In numerical simulations of black hole binaries, Pretorius and Khurana [Class. Quant. Grav. {\bf 24}, S83 (2007)] have observed critical behaviour at the threshold between scattering and immediate merger. The number of orbits scales as $n\simeq -γ\ln|p-p_*|$ along any one-parameter family of initial data such that the threshold is at $p=p_*$. Hence they conjecture that in ultrarelavistic collisions almost all the kinetic energy can be converted into gravitational waves if the impact parameter is fine-tuned to the threshold. As a toy model for the binary, they consider the geodesic motion of a test particle in a Kerr black hole spacetime, where the unstable circular geodesics play the role of critical solutions, and calculate the critical exponent $γ$. Here, we incorporate radiation reaction into this model using the self-force approximation. The critical solution now evolves adiabatically along a sequence of unstable circular geodesic orbits under the effect of the self-force. We confirm that almost all the initial energy and angular momentum are radiated on the critical solution. Our calculation suggests that, even for infinite initial energy, this happens over a finite number of orbits given by $n_\infty\simeq 0.41/η$, where $η$ is the (small) mass ratio. We derive expressions for the time spent on the critical solution, number of orbits and radiated energy as functions of the initial energy and impact parameter.