Two-timescale analysis of extreme mass ratio inspirals in Kerr spacetime:: Orbital motion

Inspirals of stellar-mass compact objects into massive black holes are an important source for future gravitational wave detectors such as Advanced LIGO and LISA. The detection and analysis of these signals rely on accurate theoretical models of the binary dynamics. We cast the equations describing binary inspiral in the extreme mass ratio limit in terms of action-angle variables, and derive properties of general solutions using a two-timescale expansion. This provides a rigorous derivation of the prescription for computing the leading order orbital motion. As shown by Mino, this leading order or adiabatic motion requires only knowledge of the orbit-averaged, dissipative piece of the self-force. The two-timescale method also gives a framework for calculating the post-adiabatic corrections. For circular and for equatorial orbits, the leading order corrections are suppressed by one power of the mass ratio, and give rise to phase errors of order unity over a complete inspiral through the relativistic regime. These post-1-adiabatic corrections are generated by the fluctuating, dissipative piece of the first order self-force, by the conservative piece of the first order self-force, and by the orbit-averaged, dissipative piece of the second order self-force. We also sketch a two-timescale expansion of the Einstein equation, and deduce an analytic formula for the leading order, adiabatic gravitational waveforms generated by an inspiral.