Adiabatic and post-adiabatic approaches to extreme mass ratio inspiral

Extreme mass ratio inspirals (EMRIs) show a strong separation of timescales, with the time characterizing inspiral, $T_{\rm i}$, much longer than any time $T_{\rm o}$ characterizing orbital motions. The ratio of these timescales (which is essentially an EMRI's mass ratio) can be regarded as a parameter that controls a perturbative expansion. Here we describe the value and limitations of an "adiabatic" description of these binaries, which uses only the leading terms arising from such a two-timescale expansion. An adiabatic approach breaks down when orbits evolve through resonances, with important dynamical and observational consequences. We describe the shortfalls of an approach that only includes the adiabatic contributions to EMRI evolution, and outline what must be done to evolve these systems through resonance and to improve our ability to model EMRI systems more generally.