Evolving extreme mass-ratio inspirals in a perturbed Schwarzschild spacetime

TL;DR

This work develops the Modified Teukolsky Formalism (MTF) to model GW emission from EMRIs around perturbed Schwarzschild BHs, enabling analysis beyond Ricci-flat, type-D spacetimes. By introducing a two-parameter expansion in the mass ratio $\epsilon$ and deformation strength $\zeta$, the authors derive decoupled Teukolsky-like equations for $\Psi_0^{(1,1)}$ and $\Psi_4^{(1,1)}$ with explicit source terms, constructed from perturbed NP quantities and the particle stress-energy. They provide a complete procedure to compute the sources from Regge-Wheeler and Zerilli-Moncrief perturbations and to obtain the modified GW flux at infinity (and horizon under certain assumptions), demonstrated in a proof-of-principle with a bumpy Schwarzschild background. The framework paves the way for EMRI-based tests of BH spacetimes in generic beyond-GR or environmental scenarios and sets the stage for extending to deformed Kerr spacetimes. This approach delivers a systematic, perturbative route to quantify how background geometry corrections imprint on EMRI waveforms, enabling parametrized tests of BH spacetimes with LISA-class detectors.

Abstract

In this work, we develop the modified Teukolsky formalism that describes the GW radiation from a point mass orbiting around a perturbed Schwarzschild BH. This perturbation of the background spacetime induces a secular change in the orbital phase of the point mass. In turn, this causes a modification in the GW flux, which can be used to probe the background spacetime. We explicitly apply this formalism to a bumpy Schwarzschild spacetime as a proof of principle. The results pave the way for the description of EMRIs in generic perturbed Kerr spacetime in future developments.

Figures (1)

• Figure 1: A schematic flowchart listing the relevant quantities and equations for evaluating the modified Teukolsky equation in Eqs. \ref{['eq:master_eqn_non_typeD_Psi0']} and \ref{['eq:master_eqn_non_typeD_Psi4']}. The final results of all the source terms are tabulated in Appendix \ref{['sec:SourceCoeffs']}. For quantities taking the same form at $\mathcal{O}(\zeta^1,\epsilon^0)$ and $\mathcal{O}(\zeta^0,\epsilon^1)$, one can replace $h_{\mu\nu}$ with $q_{\mu\nu}$ in the former's expression to get the latter, where we denote this procedure as $(h\to q)$. For $\mathcal{S}_{p}^{(1,1)}$ and $\mathcal{T}_{p}^{(1,1)}$, one also needs to set the rotation coefficients $a^{(0,1)}$ and $b^{(0,1)}$ to zero when making the replacement, as detailed in Sec. \ref{['sec:construct_source']}.