Learning about Black Hole Binaries from their Ringdown Spectra

The coalescence of two black holes generates gravitational waves that carry detailed information about the properties of those black holes and their binary configuration. The final coalescence cycles are in the form of a ringdown: a superposition of quasinormal modes of the merged remnant black hole. Each mode has an oscillation frequency and decay time that in general relativity is determined by the remnant’s mass and spin. Measuring the frequency and decay time of multiple modes makes it possible to measure the remnant’s mass and spin, and to test the waves against the predictions of gravity theories. In this Letter, we show that the relative amplitudes of these modes encode information about a binary’s geometry. Focusing on the large mass-ratio limit, which provides a simple-to-use tool for effectively exploring parameter space, we demonstrate how a binary’s geometry is encoded in the relative amplitudes of these modes, and how to parametrize the modes in this limit. Although more work is needed to assess how well this carries over to less extreme mass ratios, our results indicate that measuring multiple ringdown modes from coalescence may aid in measuring important source properties, such as the misalignment of its members’ spins and orbit.

Figure 1

Two inspirals and plunges into a Kerr black hole (with spin $a=0.5M$), the final waveform cycles that result, and some of the waveforms’ mode content. Panels on the left describe equatorial inspiral and plunge; those on the right are for an orbit inclined at $I=60°$ to the equatorial plane. Top panels show segments of the orbits’ trajectories. On the left, the small body is confined to $\theta =90°$, so not much detail is seen in the red worldline trace. On the right, the small body’s motion takes it through a range $30°\le \theta \le 150°$, plunging into the horizon at ${\theta }_{\text{fin}}=52°$. The middle panels show the final wave cycles ($+$ polarization in the equatorial plane) generated by these trajectories. Notice the highly modulated waves resulting from the inclined orbit’s more complicated motion. Bottom shows the amplitudes of the $\ell =2$ quasinormal modes present in these waveforms [see Eq. (2) for the definition of ${\mathcal{A}}_{\ell m0}$]. The equatorial case is dominated by $m=2$ and $m=1$ modes; the inclined case has contributions from all allowed $m$.

Figure 2

An example of how mode excitation varies as a function of ${\theta }_{\text{fin}}$, the polar angle at which the small body enters the event horizon. Both panels are for $\ell =m=2$ fundamental ($n=0$) mode excitation of a black hole with $a=0.5M$, and for orbit inclination $I=60°$. Top panel shows the mode’s amplitude ${\mathcal{A}}_{220}$ as a function of ${\theta }_{\text{fin}}$; bottom shows its phase ${\phi }_{220}$ in radians. In both plots, the red dots are cases for which the plunging body has ${\dot{\theta }}_{\text{fin}}>0$, blue dots have ${\dot{\theta }}_{\text{fin}}<0$.

Figure 3

Behavior of ${\mathcal{A}}_{220}$ for a black hole with $a=0.5M$ as a function of orbital inclination $I$ and final polar angle ${\theta }_{\text{fin}}$. As in Fig. 2, red (blue) data are for ${\dot{\theta }}_{\text{fin}}>0$ (${\dot{\theta }}_{\text{fin}}<0$). This surface is quite smooth and well behaved. Each orbit geometry cleanly picks out a particular mode amplitude.