Observational black hole spectroscopy: A time-domain multimode analysis of GW150914

The detection of the least damped quasinormal mode from the remnant of the gravitational wave event GW150914 realized the long-sought possibility of observationally studying the properties of quasistationary black hole spacetimes through gravitational waves. Past literature has extensively explored this possibility, and the emerging field has been named “black hole spectroscopy.” In this study, we present results regarding the ringdown spectrum of GW150914, obtained by application of Bayesian inference to identify and characterize the ringdown modes. We employ a pure time-domain analysis method, which infers from the data the time of transition between the nonlinear and quasilinear regimes of the postmerger emission in concert with all other parameters characterizing the source. We find that the data provide no evidence for the presence of more than one quasinormal mode. However, from the central frequency and damping time posteriors alone, no unambiguous identification of a single mode is possible. More in-depth analysis adopting a ringdown model based on results in perturbation theory over the Kerr metric confirms that the data do not provide enough evidence to discriminate among an $l=2$ and the $l=3$ subset of modes. Our work provides the first comprehensive agnostic framework to observationally investigate astrophysical black holes’ ringdown spectra.

Figure 1

BH spectroscopy from the two-dimensional posterior for central frequency and damping time obtained with a single damped sinusoid ringdown model. The colored contours are the 90% credible intervals for particular $(\ell ,m,n)$ Kerr modes, derived from the LVC reported remnant mass $M_f$ and spin $a_f$ (derived from inspiral). We show the $\ell =2$ and $\ell =3$ modes as other $\ell$’s do not overlap with the posterior distribution.

Figure 2

Posterior distribution over $t_{\text{start}}$ obtained assuming a uniform prior distribution in $[10,20]M_f$.

Figure 3

Orthographic projection of the 90% two-dimensional contour for the sky position of GW150914 obtained by our analysis (in black). For a comparison, we also show the publicly available posterior samples from a full IMR analysis by the LVC [37] (in purple, completely overlapping with our result). Regardless of the particular model chosen for the analysis, we always observe the same posterior.

Figure 4

Reconstructed whitened waveform superimposed on LIGO-Hanford (top panel) and LIGO-Livingston (bottom panel) data. The solid line shows the median recovered waveform, while shaded regions represent 90% credible intervals.

Figure 5

Posterior distribution for $M_f$ and $a_f$ assuming the presence of individual modes, $(\ell ,m,n)=(2,2,0)$ (dotted orange line) or $(\ell ,m,n)=(3,-3,0)$ (blue dotted line), and from a two-modes analysis $\{(2,2,0),(3,-3,0)\}$ of which the posterior density is represented in shades of gray. In green, the posterior for $M_f$ and $a_f$ from the LVC IMR analysis. The inferred mass and spin are consistent among the three cases considered, as well as with the LVC posteriors, in line with the Bayes factors, Table 1, which indicate no preference towards any of them. See the text for a more in-depth explanation.

Figure 6

Mode identification probability as a function of the lower prior bound on the transition time. Each histogram represents the probability that the identified frequency and damping time correspond to the $k$th mode as predicted assuming the LVC final mass and spin measurements. The color scheme is the same as in Fig. (3) to ease comparison. As expected, starting the analysis at early times returns frequencies that are lower than that expected for the (2,2,0) mode and lead to other modes being preferred. In particular, the (2,1,0) mode is the most probable until the start time is smaller than $10M_f$ (green histogram). For times between $10M_f$ and $25M_f$, the most probable modes are the (2,2,0), the $(3,-3,0)$, and the $(3,-2,0)$, aquamarine, khaki, and yellow histograms, respectively. Finally, for times greater than $25M_f$, the GW150914 signal is too quiet to reliably identify any specific mode. The leftmost bars summarize the mode probabilities.

Figure 7

Effect of the start time prior on the posterior on the reconstructed frequency and damping time, when assuming a single damped sinusoid. The solid black band marks the prediction obtained by using the LVC posteriors and assuming the excited mode was $(\ell ,m,n)=(2,2,0)$. Using a lower bound on the start time of $30M_f$, the obtained reconstruction is flat; thus, it is not shown in the plot.

Figure 8

Effect of the start time prior on the posterior on the reconstructed final mass and spin, when assuming a $(\ell ,m,n)=(2,2,0)$ Kerr mode. The solid black band marks the prediction obtained by using the LVC posteriors. Using a lower bound on the start time of $30M_f$, the obtained reconstruction is flat; thus, it is not shown in the plot.