Resonances in b-EMRIs: playing the black hole piano

TL;DR

This work demonstrates that binary extreme-mass-ratio inspirals can resonantly excite the quasinormal modes of a nearby supermassive black hole, with the strongest response occurring when the stellar binary is near the light ring and oriented to feed the corresponding photon-ring region. Using a frequency-domain Teukolsky framework and Dixon-based quadrupole modeling, the authors show that resonant energy fluxes peak close to, but not exactly at, the real parts of the SMBH QNM frequencies, and that the offset grows as the binary moves away from the black hole. In Kerr spacetimes the spectrum becomes denser and the resonances more intricate, though individual mode identification remains challenging. The results provide a concrete mechanism for BH spectroscopy via gravitational tuning forks and offer guidance on observational signatures and extensions, including backreaction and overtone resonances, with potential implications for LISA detections of SMBH environments.

Abstract

Stellar-mass binaries evolving in the vicinity of supermassive black holes (SMBHs) may be common in the universe, either in active galactic nuclei or in other astrophysical environments. Here, we study in detail the resonant excitation of SMBH modes driven by a nearby stellar-mass binary. The resulting resonant energy fluxes vary with the orbital location and frequency of the binary, exhibiting a rich and complex structure. In particular, we find that the total energy flux radiated to infinity is maximized at a gravitational-wave frequency that is close to, but not exactly equal to, the real part of the corresponding quasinormal-mode frequency. Moreover, as the binary is moved farther away from the SMBH, this offset from the mode frequency becomes increasingly pronounced. In addition, for suitable orientations, the binary can effectively ``feed'' the light ring of the SMBH, selectively exciting particular oscillation modes. For rotating (Kerr) black holes, the mode spectrum is significantly more intricate; however, individual modes are also less strongly damped, leading to an enhanced - but more difficult to interpret - resonant response.

Figures (14)

• Figure 1: Depiction of the binary extreme-mass-ratio inspiral (b-EMRI) considered herein. Adapted from Ref. Cardoso:2021vjq. A Kerr SMBH sits at the center and acts as the primary of a b-EMRI system. The ISCO, light ring and horizon of the SMBH all play a role in GW emission from this system. The secondary binary (SB, not to scale) lies in the vicinity of the SMBH. The tuning fork is the stellar binary, whose intrinsic frequency varies as the binary evolves, possibly resonating with the modes of the central BH. Such systems may be ubiquitous in the cosmos.
• Figure 2: The geometrical setup that we consider in Sec. \ref{['sec:LR']}. The primary is a supermassive, non-spinning BH. A reference SB is placed at the photon sphere, with intrinsic spin parallel to the $\hat{z}$-axis of a reference frame centered at the supermassive BH (SB I), and we also study a configuration with the intrinsic spin aligned with the equatorial light ring (SB II). As we discuss below, most of the radiation (indicated by spiraling blue lines) is emitted in the direction of the SB intrinsic spin, and so the relative orientation between this spin and the $\hat{z}$-axis determines the excitation of different angular modes of the SMBH. The polar and equatorial light rings are depicted by red circles. Thus, SB I is emitting most of its radiation onto the polar light ring, whereas SB II is sending most of the radiation to the equatorial light ring.
• Figure 3: Energy flux in different $\ell$ modes, normalized to total flux $\dot E _Q$, for a Newtonian binary system with orbital frequency $\Omega_{\rm SB}$ in flat space, with the CoM at $x=r_0=10$ (in arbitrary units), $y=z=0$. The spin of the binary points in the $\hat{z}$ direction; we use the quadrupole approximation, so that the GW frequency is simply $2\Omega_{SB}$. Still, the signal is not uniquely described by $\ell=m=2$ harmonics, as would be the case if the CoM were at the origin. Each curve has a global maximum for $2 r_0 \Omega_{\rm SB} \approx 1.1 \sqrt{\lambda_\ell}$, where $\lambda_\ell= \ell(\ell+1)-2$ is the eigenvalue of the spin $-2$ harmonics with angular number $\ell$.
• Figure 4: Total energy flux from SB I (cf. Fig. \ref{['fig:BinaryGeometry']}) around a nonspinning BH, at infinity ( top) and at the horizon (bottom), normalized to the geometric-optics expression $\dot{E}^Q$ in Eq. \ref{['eq:EQ']}. The SB is static and placed at $r_0=3M$. Vertical dashed lines indicate the fundamental QNM frequencies $\omega_{\ell m 0}^{\rm QNM}$ for $\ell=m=2,\ldots,9$. The gravitational radiation is monochromatic with frequency $\omega=2\Omega_{\rm SB}$. For $M\omega \gg 1$, fluxes at infinity and the horizon asymptote to roughly $\dot{E}^Q/2$, as expected, since at the light ring half the gravitons escape to infinity whereas the other half fall into the BH. For small $M \omega$, $\dot{E}^Q$ is outside its regime of validity, but we checked that both $\dot E^{\infty,H}\to 0$, while $\dot E^{H}/\dot E ^{\infty}\to \infty$ when $\omega\to 0$; the latter is an instance of the horizon dominance effect first reported in Ref. Santos:2024tlt. The resonances with the QNMs lead to an excess in the energy flux, easily identified above.
• Figure 5: Energy flux from the binary in Fig. \ref{['fig:Sch_LR_total']} for different $\ell$ modes, normalized to the total energy flux at infinity ( top) and on the horizon ( bottom), respectively. Different colors indicate different $\ell$ modes. Vertical dashed lines mark the real part of the QNM frequencies, while the dotted lines are the individual $\dot E_{\ell m}$ (rescaled by $2 \ell +1$) contributing to $\dot E_\ell$. Each $\ell$ mode peaks when it crosses the corresponding QNM frequency. The wiggles for large $\Omega_{\rm SB}$ are "anti-resonances" caused by peaks in the denominator; see Fig. \ref{['fig:Sch_LR_total']}.