This paper develops a practical way to compute gravitational waves when a small compact object spirals into a black hole in certain modified gravity theories. The authors build a “modified Teukolsky formalism” to describe the waves generated by the small object and use it to calculate how much energy carried by those waves goes down the black hole horizon and how much escapes to far away observers. As an example, they apply the method to a specific higher‑derivative theory called parity‑preserving cubic gravity and find that horizon absorption can be much larger than in General Relativity, while the flux to infinity is slightly reduced.

At a high level, the Teukolsky formalism is a standard tool in General Relativity for computing radiation from small perturbations of a black hole. Many modified gravity theories change both the background black hole and how waves are produced and travel. The authors extend the Teukolsky approach so it works perturbatively in those cases. Their modified equations become ordinary differential equations in the radial direction, driven by source terms that couple the beyond‑GR correction to the background with the usual GR perturbation produced by the small body. They solve those radial equations with Green’s‑function methods and take care to regularize high‑derivative terms that appear in the sources.

For this study the system is simplified to a non‑rotating black hole and a small companion on a circular, equatorial orbit. The calculation assumes two small parameters: the symmetric mass ratio η (small because the companion is much lighter than the central black hole) and a dimensionless coupling ζ that measures how far the theory departs from General Relativity. The work focuses on the leading correction that is first order in both η and ζ. The authors use ingoing Eddington–Finkelstein coordinates and a specific null tetrad (the Hawking–Hartle tetrad) to keep the equations regular, and they build on numerical metric perturbation data produced by existing codes for a point particle around a Schwarzschild black hole.