Resonances in extreme mass-ratio inspirals: Asymptotic and hyperasymptotic analysis

Abstract

An expected source of gravitational waves for future detectors in space is the inspirals of small compact objects into much more massive black holes. These sources have the potential to provide a wealth of information about astronomy and fundamental physics. On short time scales the orbit of the small object is approximately geodesic. Generic geodesics for a Kerr black hole spacetime have a complete set of integrals and can be characterized by three frequencies of the motion. Over the course of an inspiral, a typical system will pass through resonances where two of these frequencies become commensurate. The effect of the resonance will be to alter significantly the rate of inspiral for the duration of the resonance. Understanding the impact of these resonances on gravitational wave phasing is important for the detection of these signals and for the exploitation of the observations for astrophysics and fundamental physics. Two differential equations that might describe the passage of an inspiral through such a resonance are investigated. These differences depending on whether it is the phase or the frequency components of a Fourier expansion of the motion that are taken to be continuous through the resonance. Asymptotic and hyperasymptotic analysis are used to find the late-time analytic behavior of the solution for a system that has passed through a resonance. Linearly growing (weak resonances) or linearly decaying (strong resonances) solutions are found depending on the strength of the resonance. In the weak-resonance case, frequency resonances leave an imprint (a resonant memory) on the gravitational wave frequency evolution. For frequency resonances, the transition between weak and strong resonances is characterized by a square-root-branch-cut singularity. On the strong resonance side of this singularity, solutions starting with different initial conditions bunch up into groups exponentially in the independent variable (time) and we show how this behavior can be understood by considering solutions near the singular point.