A multi-parameter expansion for the evolution of asymmetric binaries in astrophysical environments

Compact binaries with large mass asymmetries - such as Extreme and Intermediate Mass Ratio Inspirals - are unique probes of the astrophysical environments in which they evolve. Their long-lived and intricate dynamics allow for precise inference of source properties, provided waveform models are accurate enough to capture the full complexity of their orbital evolution. In this work, we develop a multi-parameter formalism, inspired by vacuum perturbation theory, to model asymmetric binaries embedded in general matter distributions with both radial and tangential pressures. In the regime of small deviations from the Schwarzschild metric, relevant to most astrophysical scenarios, the system admits a simplified description, where both metric and fluid perturbations can be cast into wave equations closely related to those of the vacuum case. This framework offers a practical approach to modeling the dynamics and the gravitational wave emission from binaries in realistic matter distributions, and can be modularly integrated with existing results for vacuum sources.

💡 Research Summary

The paper presents a novel perturbative framework for modeling extreme‑mass‑ratio (EMRI) and intermediate‑mass‑ratio (IMRI) binaries that evolve within realistic astrophysical environments. Traditional self‑force (SF) approaches expand the Einstein equations solely in the small mass‑ratio q, treating the secondary as a point particle moving in a vacuum Schwarzschild (or Kerr) background. However, many astrophysical settings—dark‑matter halos, accretion disks, scalar clouds—introduce additional, typically weak, matter fields that can subtly affect the binary’s orbital dynamics and the emitted gravitational‑wave (GW) signal.

To capture these effects, the authors introduce a second bookkeeping parameter ε that quantifies the strength of the environmental stress‑energy relative to the central black‑hole’s mass and size. Assuming ε ≪ 1 (the “low‑density” regime) and q ≪ 1, they expand the metric and stress‑energy tensors simultaneously in powers of (q, ε):

g = g^{(0,0)} + q g^{(1,0)} + ε g^{(0,1)} + qε g^{(1,1)} + …,

T^{e} = ε T^{(0,1)} + qε T^{(1,1)}, T^{p} = q T^{(1,0)} + qε T^{(1,1)}.

Here g^{(0,0)} is the Schwarzschild background, g^{(1,0)} encodes the standard vacuum SF perturbations, g^{(0,1)} describes the pure environmental deformation of the background, and g^{(1,1)} captures the mixed interaction between the particle and the fluid.

The authors derive the field equations order‑by‑order. The (0,1) sector satisfies G^{(1)}