Evolution of circular, non-equatorial orbits of Kerr black holes due to gravitational-wave emission: II. Inspiral trajectories and gravitational waveforms

TL;DR

The paper addresses how circular, inclined EMRIs evolve in the strong-field Kerr spacetime and how their gravitational waves encode black-hole properties, using a Teukolsky-based, adiabatic radiation-reaction framework. It builds a radiation-reaction grid in the strong-field phase space and integrates trajectories with spline interpolation to produce time-domain waveforms decomposed into multiple harmonic voices. The key contributions include detailed EMRI inspiral trajectories for several spins, demonstration of horizon-induced tidal coupling that can prolong or shorten inspirals, and the emergence of a multi-voice chirp structure in the GW signal, with implications for voice-by-voice searches in LISA data. The work provides foundational waveforms and insights for testing Kerr geometry and guiding future data-analysis strategies for extreme-mass-ratio sources.

Abstract

The inspiral of a ``small'' ($μ\sim 1-100 M_\odot$) compact body into a ``large'' ($M \sim 10^{5-7} M_\odot$) black hole is a key source of gravitational radiation for the space-based gravitational-wave observatory LISA. The waves from such inspirals will probe the extreme strong-field nature of the Kerr metric. In this paper, I investigate the properties of a restricted family of such inspirals (the inspiral of circular, inclined orbits) with an eye toward understanding observable properties of the gravitational waves that they generate. Using results previously presented to calculate the effects of radiation reaction, I assemble the inspiral trajectories (assuming that radiation reacts adiabatically, so that over short timescales the trajectory is approximately geodesic) and calculate the wave generated as the compact body spirals in. I do this analysis for several black hole spins, sampling a range that should be indicative of what spins we will encounter in nature. The spin has a very strong impact on the waveform. In particular, when the hole rotates very rapidly, tidal coupling between the inspiraling body and the event horizon has a very strong influence on the inspiral time scale, which in turn has a big impact on the gravitational wave phasing. The gravitational waves themselves are very usefully described as ``multi-voice chirps'': the wave is a sum of ``voices'', each corresponding to a different harmonic of the fundamental orbital frequencies. Each voice has a rather simple phase evolution. Searching for extreme mass ratio inspirals voice-by-voice may be more effective than searching for the summed waveform all at once.

Figures (11)

• Figure 1: Circular orbit radiation reaction data near the last stable orbits (LSO) of a Kerr black hole with $a = 0.998 M$. Any point $(r,\iota)$ in this plot is a circular geodesic orbit. The arrows are proportional to the vector $[(M/\mu)\dot r, (M^2/\mu)\dot\iota]$: the orientation gives the direction in which gravitational-wave emission drives the orbit, and the magnitude is proportional to the rate at which it is so driven. The dotted line is the LSO --- orbits above and to the left of this line are dynamically unstable and rapidly plunge into the black hole. The arrows get longer as this line is approached and their stability decreases. (Additional data were produced representing radiation reaction for orbits on the LSO; these orbits are so unstable that their radiation reaction vectors do not fit on this plot. These data are used in all computations, however.)
• Figure 2: Inspiral trajectories in the strong field of a Kerr black hole with $a = 0.998M$. To make this plot, the data shown in Fig. \ref{['fig:rrfield_0.998']} were integrated using the procedures discussed in Sec. \ref{['subsec:integrate']}, assuming that the black hole has mass $M = 10^6\,M_\odot$ and that the inspiraling body has mass $\mu = 1\,M_\odot$. The trajectory shapes are independent of the two masses, so the inspiral times and accumulated number of cycles $N_{\phi,\theta}$ can be rescaled to other masses quite easily: $T_{\rm inspiral} \propto M^2/\mu$, $N_{\phi,\theta} \propto M/\mu$. Notice that the trajectories are nearly flat --- $\iota$ decreases, but not very much, over these inspirals.
• Figure 3: Inspiral trajectories in the strong field of a Kerr black hole with $a = 0.998M$, ignoring the flux down the horizon --- cf. Eq. (\ref{['eq:turnHflux_onoff']}), with $\eta = 0$. The figure is otherwise identical to Fig. \ref{['fig:traj_H_0.998']}. The trajectory shapes change very slightly ($\iota$ does not decrease quite as much over the inspiral), though that effect is very small. Much more interestingly, the inspiral is markedly faster: especially at shallow inclination angle, the small body takes several weeks less spiraling to the LSO, and executes many thousands fewer orbits. As discussed in Sec. \ref{['subsec:traj_0.998']}, this illustrates how tidal coupling between the small body and the event horizon strongly impacts the inspiral: a tidal bulged is raised on the hole, which, due to the hole's rapid rotation in this case, transfers rotational kinetic energy to the small body's orbit.
• Figure 4: The evolution of the adiabaticity parameters ${\cal N}_{\phi,\theta} = \Omega_{\phi,\theta}^2/2\pi{\dot\Omega}_{\phi,\theta}$ for the inspiral track beginning at $\iota = 20^\circ$ shown in Fig. \ref{['fig:traj_H_0.998']}. The solid line is ${\cal N}_\phi$, the dotted line ${\cal N}_\theta$. Except at the extreme end, ${\cal N}_{\phi,\theta}\gg 1$, indicating that the inspiral is indeed adiabatic, as required. Note that ${\cal N}_{\phi,\theta}\propto M/\mu$, indicating that the adiabatic requirements are unlikely to be met if $M/\mu < 1000$. The divergence in ${\cal N}_\theta$ is due to ${\dot\Omega}_\theta$ passing through zero and changing sign during the inspiral --- it ceases chirping up, and begins chirping down.
• Figure 5: Inspiral trajectories in the strong field of a Kerr black hole with $a = 0.3594M$. The top panel includes the effects of the down-horizon flux; the bottom panel does not. The span of data is chosen so that the total inspiral duration is similar to that shown in Fig. \ref{['fig:traj_H_0.998']}. For this spin, inspiral is nearly identical with the horizon flux included or disincluded: inspiral is slightly faster without horizon flux, but not nearly so much faster as when $a = 0.998M$. This is largely because the spin is not fast enough to drag the tidal bulge on the horizon so far forward. In fact, at the innermost orbits the bulge should be essentially in perfect face-on lock with the orbiting body, since $a = 0.3594M$ is the spin value at which the horizon spin frequency matches the innermost orbital frequency.