History dependence of self-force during an extreme mass-ratio inspiral

Perturbation models of gravitational waves produced by an extreme mass-ratio inspiral (EMRI) often involve simplifying assumptions that introduce accumulating errors. One common assumption, which we are correcting for, is that the small object in the EMRI system has historically followed a fixed geodesic. To improve this aspect of EMRI models, we apply a two-timescale expansion to the Teukolsky equation for quasi-circular non-spinning EMRIs. By including the next term in the two-timescale expansion, we will account for past properties of the orbit. We have already solved the Teukolsky equation for perfectly circular orbital motion, and compared these results with past work. We now turn our attention to the next term in the two-timescale expansion, which will account for history dependence of the gravitational perturbations and self-force.