Quasinormal ringing of Kerr black holes. III. Excitation coefficients for equatorial inspirals from the innermost stable circular orbit

The remnant of a black hole binary merger settles into a stationary configuration by “ringing down” through the emission of gravitational waves that consist of a superposition of damped exponentials with discrete complex frequencies - the remnant black hole’s quasinormal modes. While the frequencies themselves depend solely on the mass and spin of the remnant, the mode amplitudes depend on the merger dynamics. We investigate quasinormal mode excitation by a point particle plunging from the innermost stable circular orbit of a Kerr black hole. Our formalism is general, but we focus on computing the quasinormal mode excitation coefficients in the frequency domain for equatorial orbits, and we analyze their dependence on the remnant black hole spin. We find that higher overtones and subdominant multipoles of the radiation become increasingly significant for rapidly rotating black holes. This suggests that the prospects for detecting overtones and higher-order modes are considerably enhanced for highly spinning merger remnants.

💡 Research Summary

The paper presents a comprehensive frequency‑domain calculation of quasinormal‑mode (QNM) excitation coefficients for a point particle that plunges from the innermost stable circular orbit (ISCO) into a Kerr black hole. The authors focus on equatorial (circular) orbits, which are the most astrophysically relevant case for extreme‑mass‑ratio inspirals (EMRIs). Their work builds on the Teukolsky formalism for perturbations of Kerr spacetime, but instead of solving the long‑range Teukolsky radial equation directly, they employ the Sasaki‑Nakumura (SN) transformation to obtain a short‑range potential. This makes numerical integration far more stable and allows analytic continuation of the solutions into the complex‑frequency plane.

Key methodological steps are: (i) a derivation of the source term for a particle moving on the “critical plunge geodesic” – a trajectory that starts at the ISCO and follows the exact geodesic equations backward in time. The analytic expressions for the particle’s four‑velocity, energy, and angular momentum are inserted into the stress‑energy tensor, which is then projected onto the spin‑weighted spheroidal harmonics to produce the mode‑decomposed source T_{ℓmω}. (ii) Construction of the SN function X_{ℓmω} via a linear combination of the Teukolsky radial function R_{ℓmω} and its derivative, with carefully chosen α(r) and β(r) that guarantee a short‑range potential. (iii) Definition of a rescaled function ˜X_{ℓmω}=X_{ℓmω}√γ that eliminates the first‑derivative term, yielding a Schrödinger‑like equation d²˜X/d r_*²+˜F˜X=S_{ℓmω}√γ. (iv) Solution of the inhomogeneous equation by the Green’s‑function method, using homogeneous solutions that satisfy purely ingoing boundary conditions at the horizon (X_{r+}) and purely outgoing conditions at infinity (X_{∞}). The Wronskian of these solutions vanishes at the QNM frequencies ω_q, turning the integral into a sum over simple poles.

The excitation coefficient for each mode q≡(ℓ,m,n) is expressed as ˜C_{SN,q}=I_q B_q, where B_q is the source‑independent excitation factor (essentially the residue of the homogeneous solution at the pole) and I_q is a source‑dependent integral over the particle trajectory. The authors compute B_q analytically (Appendix B) and evaluate I_q numerically for a wide range of dimensionless spins a/M∈