Extreme mass ratio inspirals into black holes surrounded by matter: Resonance crossings

TL;DR

This work addresses the challenge of modeling EMRIs in a perturbed Schwarzschild background by comparing quadrupole, post-Newtonian, and Teukolsky-based fluxes, and by developing an efficient adiabatic-inspiral framework to study resonance crossings. It finds strong agreement between quadrupole and Teukolsky fluxes in near-integrable and moderately perturbed regimes, while PN breaks down outside integrable settings. The authors introduce interpolation and repetitiveness techniques to accelerate hundreds of adiabatic inspirals, enabling detailed mapping of resonance crossing behaviors (prolonged, transient, sustained) for the 2/3 and 4/5 resonances and revealing a sharp dependence on the mass ratio $q$. The results quantify how resonances contribute to waveform dephasing and show that resonance effects are most significant for EMRIs (small $q$) but rapidly diminish toward comparable-mass binaries, informing waveform modeling for LISA-era data analysis.

Abstract

The forthcoming space-based gravitational-wave observatory Laser Interferometer Space Antenna (LISA) should enable the detection of Extreme Mass Ratio Inspirals (EMRIs), in which a stellar-mass compact object gradually inspirals into a supermassive black hole while emitting gravitational waves. Modeling the waveforms of such systems is a challenging task, requiring precise computation of energy and angular momentum fluxes as well as proper treatment of orbital resonances, during which two fundamental orbital frequencies become commensurate. In this work, we perform a systematic comparison of fluxes derived from three approaches: the quadrupole formula, post-Newtonian approximations, and time-domain solutions of the Teukolsky equation. We show that quadrupole-based fluxes remain in good agreement with Teukolsky results across a broad range of orbital configurations, including perturbed orbits. Building on these insights, we explore the dynamical impact of resonance crossings within the adiabatic approximation. By introducing novel numerical methods, we reduce computational costs and uncover diverse resonance-crossing behaviors. These results contribute to the effort to understand theoretically and model adequately resonance crossings during an EMRI.

Figures (28)

• Figure 1: Illustration of motion on resonant (a) and quasiperiodic (b-d) tori in a two DoF system. Blue curves denote trajectories on the torus, while the Poincaré surfaces of section are in red. Each trajectory is evolved until it intersected the section 100 times. The resonant trajectory (a) corresponds to the resonant ratio $\omega^{1}/\omega^{2} = 4/5$. In contrast, the quasiperiodic trajectories (b-d) correspond to irrational ratios, yielding closed invariant curves that densely fill the torus over time.
• Figure 2: Top: Poincaré surface of section. Bottom: rotation curve along the $p^{r}/\mu=0$ line, with dominant resonances labeled by their frequency ratios. Parameters: ${L_z=4.0\mu M}$, ${E=0.98 \mu}$, ${\theta \left[ 0 \right]= \pi/2}$ and ${r \left[ 0 \right] \in \left( 6.298M; 50.098M \right)}$ with step size $0.2M$.
• Figure 3: Individual TK $m$ mode contributions to the energy (left two columns) and angular momentum (right two columns) fluxes for the motion in the perturbed Schwarzschild spacetime \ref{['eq: Metric']} as a function of the logarithm of the quadrupole perturbation parameter.
• Figure 4: Comparison of energy (left) and angular momentum (right) fluxes computed using the three different methods for the motion in the perturbed Schwarzschild spacetime \ref{['eq: Metric']}.
• Figure 5: Poincaré surfaces of section for the spacetimes used in the study of the 2/3 resonance (left) and in the subsequent resonance crossing through 2/3 resonance within adiabatic inspiral (right). The parameters are identical to those described in the text. Initial radii were sampled with step size $0.2M$ over the intervals $(6.266M,44.266M)$ (left) and $(6.298M,50.098M)$ (right).