Hyperboloidal foliations in extreme mass-ratio inspiral (EMRI) binary systems are an essential tool in the efficient computation of gravitational waveform signals at future null infinity. They provide a natural framework for enforcing radiation boundary conditions and extracting waveforms without contamination from artificial outer boundaries. While significant work has been done to apply these methods to spherically symmetric configurations, their application to axially symmetric Kerr spacetimes has so far been limited to quasi-circular, equatorial orbits.
In this work, we propose a framework that combines hyperboloidal compactifications with the frequency-domain Teukolsky equation to compute first‑order gravitational self‑force effects for eccentric equatorial orbits in Kerr spacetime. We employ a Fourier-domain worldtube puncture scheme in which the motion of the (non-rotating) secondary black hole is decomposed into radial and azimuthal modes that source the Teukolsky master scalar. Hyperboloidal compactifications are then used to regularize each mode and extract the waveform at null infinity.
We demonstrate the implementation of this scheme within the current numerical structures for equatorial, quasi-circular motion of the secondary and discuss its potential generalization to precessing and inclined orbits. By combining hyperboloidal foliations with dynamic worldtube methods, this work aims to provide a scalable pathway for accurate waveform modeling in EMRI systems, with direct relevance to future space‑based gravitational wave observatories.