Scalar self-force for highly eccentric equatorial orbits in Kerr spacetime

If a small "particle" of mass $\mu M$ (with $\mu \ll 1$) orbits a black hole of mass $M$, the leading-order radiation-reaction effect is an $\mathcal{O}(\mu^2)$ "self-force" acting on the particle, with a corresponding $\mathcal{O}(\mu)$ "self-acceleration" of the particle away from a geodesic. Such "extreme--mass-ratio inspiral" systems are likely to be important gravitational-wave sources for future space-based gravitational-wave detectors. Here we consider the "toy model" problem of computing the self-force for a scalar-field particle on a bound eccentric orbit in Kerr spacetime. We use the Barack-Golbourn-Vega-Detweiler effective-source regularization with a 4th order puncture field, followed by an $e^{im\phi}$ ("m-mode") Fourier decomposition and a separate time-domain numerical evolution in $2+1$ dimensions for each $m$. We introduce a finite worldtube that surrounds the particle worldline and define our evolution equations in a piecewise manner so that the effective source is only used within the worldtube. Viewed as a spatial region, the worldtube moves to follow the particle's orbital motion. We use slices of constant Boyer-Lindquist time in the region of the particle's motion, deformed to be asymptotically hyperboloidal and compactified near the horizon and $\mathcal{J}^+$. We present numerical results for a number of test cases with orbital eccentricities as high as $0.98$. In some cases we find large oscillations ("wiggles") in the self-force shortly after periastron passage.