Gravitational waves from the late inspiral, transition, and plunge of small-mass-ratio eccentric binaries

TL;DR

This work probes the ringdown of small-mass-ratio eccentric binaries in Kerr spacetime by constructing eccentric inspiral–transition–plunge worldlines and solving the time-domain Teukolsky equation to generate gravitational-wave waveforms. The analysis isolates the ringdown into Kerr quasinormal modes and power-law tails, and develops a QNM extraction pipeline that accounts for spheroidal–spherical mode mixing, with tails modeled in the Weyl scalar $oldsymbol{ abla^2\, ext{psi}_4}$ domain. The key finding is that eccentricity and, crucially, the radial anomaly angle controlling the final plunge strongly modulate QNM excitations and tail amplitudes, sometimes yielding ringdowns nearly indistinguishable from circular mergers, and other times selecting subdominant modes such as $(2,1)$, depending on the final kinematics. The results underscore the sensitivity of late-time gravitational-wave structure to the detailed plunge dynamics and motivate extensions to generic inclinations, higher mass ratios, and cross-validation with numerical relativity and other perturbative approaches for robust gravitational-wave modeling relevant to LISA.

Abstract

Black hole binaries with small mass ratios will be important sources for the forthcoming Laser Interferometer Space Antenna (LISA) mission. Models of such binaries also serve as useful tools for understanding the dynamics of compact binary systems and the gravitational waves they emit. Using an eccentric Ori-Thorne procedure developed in previous work, we build worldlines that describe the full inspiral and plunge of a small body on an initially eccentric orbit of a Kerr black hole. We now calculate the gravitational waves associated with these trajectories using a code that solves the Teukolsky equation in the time domain. The final cycles of these waveforms, the ringdown, contains a superposition of Kerr quasinormal modes followed by a power-law tail. In this paper, we study how a binary's eccentricity and orbital anomaly angle affect the excitation of both quasinormal modes and late-time tails. We find that the relative excitation of quasinormal modes varies in an important and interesting way with these parameters. For some anomaly angles, the relative excitations of quasinormal modes are essentially indistinguishable from those excited in quasi-circular coalescences. Consistent with other recent studies, we find that eccentricity tends to amplify the late-time power-law tail, though the amount of this amplification varies significantly with orbital anomaly. We thus find that eccentricity has an important impact on the late-time coalescence waveform, but the interplay of eccentricity and orbit anomaly complicates this impact.

Figures (13)

• Figure 1: Example of a prograde inspiral-transition-plunge worldline built using BH25 and its corresponding waveform. The binary has a mass ratio of $\eta =10^{-4}$ and spin $a = 0.7M$. Top panel shows orbital radius versus Boyer-Lindquist coordinate time. The interval shown includes the final cycles of adiabatic inspiral, ending with an eccentricity of $e_{\rm IT} = 0.304$. The secondary then completes a radial cycle in the transition, eventually crossing the LSO (the black bar on the plot). Past the LSO, we switch to a plunging geodesic, and the small body ultimately freezes to the horizon coordinate $r_{\rm H} =1.7414M$. Before freezing to the horizon, the secondary also crosses the prograde equatorial light ring at $t_{\rm PLR}$. Middle panel includes the plus-polarization waveform according to an "edge-on" observer in the primary's equatorial plane. Bottom panels focus on the ringdown of the $\ell = m = 2$ component of the waveform: the bottom left panel is the plus polarization, and the bottom right panel is the cross polarization.
• Figure 2: Prograde (left panels) and retrograde (right panels) spheroidal mode excitations for plunges at $\eta = 10^{-4}$ into a black hole with $a = 0.3M$. All data are for $k=2$ and $n = 0$ modes, with $m\in \{-2, -1, 0, 1, 2\}$. Top row corresponds to systems with $e_{\rm IT} = 0.09$; bottom is for $e_{\rm IT} = 0.50$. The data points denote the average mode excitation across a sample of 36 trajectories that differ only by $\chi_{r\rm IT}$, and the vertical bars encode the spread in the amplitude from variations in $\chi_{r\rm IT}$ (the top of the bar gives the maximum amplitude and the bottom of the bar gives the minimum).
• Figure 3: Mode excitation magnitude $\mathcal{A}_{km0}$ for spheroidal QNMs with $k = 2, \;m\in \{-2, -1, 0, 1, 2\}$ and $n=0$ as a function of the radial anomaly angle at the end of adiabatic inspiral, $\chi_{r\rm IT}$. Eccentricities in the strong field vary on the range $e_{\rm IT} \in [0.02, 0.56]$, with red dots denoting the lowest eccentricities, blue squares denoting the highest, and yellow diamonds and green asterisks denoting intermediate values. Top panels show results for black hole spin $a = 0$, middle for $a = 0.3M$ and bottom for $a = 0.7M$. From left to right, $m$ varies from $-2$ to $2$. Note that the scale of the vertical axes differ between panels. In all cases, the spread in the mode amplitudes increases with eccentricity due to the dependence on $\chi_{r\rm IT}$.
• Figure 4: Worldlines and their resulting mode excitation magnitudes for spheroidal QNMs with $k = 2$, and $m = 1$ or $2$. Top panel includes trajectories for inspirals and plunges into a Schwarzschild black hole with $e_{\rm IT} = 0.56$ and $\eta = 10^{-4}$. Each trajectory has a slightly different anomaly angle in the strong field delineated by the plot legend ($\Delta \chi_{r\rm IT} \approx (10^{-2})^{\circ}$). Bottom panels show the amplitudes of the $(2,1)$ (left) and $(2,2)$ mode (right).
• Figure 5: Mode excitation magnitude $\mathcal{A}_{km0}$ for spheroidal QNMs with $k = 2, \;m\in \{-2, -1, 0, 1, 2\}$, $n=0$ as a function of the radial velocity at the prograde equatorial light ring, $|dr/d\tau|_{r = r_{\rm PLR}}$. Eccentricities in the strong field vary on the range $e_{\rm IT} \in [0.02, 0.56]$, with red dots denoting the lowest eccentricities, blue squares the highest, and yellow diamonds and green asterisks denoting intermediate values. Top panels show results for black hole spin $a = 0$, middle panels $a = 0.3M$ and bottom panels $a = 0.7M$. From left to right, $m$ varies from $-2$ to $2$. Note that the scale of the vertical axes differs between panels.