Black-hole spectroscopy, the no-hair theorem, and GW150914: Kerr versus Occam

The “no-hair” theorem states that astrophysical black holes are fully characterized by just two numbers: their mass and spin. The gravitational-wave emission from a perturbed black-hole consists of a superposition of damped sinusoids, known as quasinormal modes. Quasinormal modes are specified by three integers $(\ell ,m,n)$: the $(\ell ,m)$ integers describe the angular properties and ($n$) specifies the (over)tone. If the no-hair theorem holds, the frequencies and damping times of quasinormal modes are determined uniquely by the mass and spin of the black hole, while phases and amplitudes depend on the particular perturbation. Current tests of the no-hair theorem attempt to identify these modes in a semiagnostic way, without imposing priors on the source of the perturbation. This is usually known as black-hole spectroscopy. Applying this framework to GW150914, the measurement of the first overtone led to the confirmation of the theorem to 20% level. We show, however, that such semiagnostic tests cannot provide strong evidence in favor of the no-hair theorem, even for extremely loud signals, given the increasing number of overtones (and free parameters) needed to fit the data. This can be solved by imposing prior assumptions on the origin of the perturbed black hole that can further constrain the explored parameters: in particular, our knowledge that the ringdown is sourced by a binary black-hole merger. Applying this strategy to GW150914, we find a natural log Bayes factor of $\sim 6.5$ in favor of the Kerr nature of its remnant, indicating that the hairy object hypothesis is disfavored with $<1:600$ with respect to the Kerr black-hole one.

Figure 1

Left: Final mass and spin recovery for GW150914 (dashed) and a consistent simulation with optimal SNR of ${\rho }_{\mathrm{opt}}=15$ (solid) using 0,1 and 2 overtones (blue, orange, green). The contours obtained for both real and simulated data are in wide agreement. Right: final mass and spin estimates for the same injection as in the left panel, scaled to an SNR of 100. Two overtones are needed to obtain nonbiased estimates. For our injection analysis, we consider a single Advanced LIGO detector implementing the noise curve obtained by Bayes wave around the time of GW150914. The intersection of the black lines denotes the true values for our injection.

Figure 2

Overtone amplitudes for a GW150914 and a consistent simulation. Top: relative amplitudes $A_i/A_0$ of the overtones with respect to the fundamental tone. Bottom: amplitude of the fundamental mode. The optimal SNR of the injections is $\rho =15$ in the left panels and $\rho =100$ on the right ones.

Figure 3

Binary parameters of GW150914, inferred from the postpeak emission. Posterior distributions on individual masses, total mass, mass ratio, effective spin parameter, and luminosity distance obtained from the analysis of the postpeak signal of SXS:BBH:0305 (blue), the postpeak of GW150914 (orange), and the full GW150914 signal (green). All of the posteriors are consistent with one another. Notably, we can constrain the mass ratio of GW510914 to $q<2.29$ at 90% confidence analyzing solely the postpeak portion of the signal, while the effective-spin parameter ${\chi }_{\mathrm{eff}}$ cannot be measured.

Figure 4

Frequencies and damping times for GW150914 and a numerical injection consistent with it, with optimal SNR $\rho =15$. We report the frequency and damping time of the fundamental mode (left) and the next mode (right), when the signal is compared to one (blue) and two damped sinusoids (orange). When using two damped sinusoids, $f_0$ matches the ringdown frequency of the fundamental (2,2,0) mode, while $f_1$ is consistent with the merger frequency. For GW150914, we find $f_0=254_{-7}^{+6}\mathrm{Hz}$, consistent with [40]. For both the injection and the GW150914 case, ${\tau }_0$ shows significant support for values larger than that of the (2,2,0) mode. The value of $f_1$ obtained in both cases is consistent with the merger frequency. For GW150914, we obtain $f_1=150_{-12}^{+16}\mathrm{Hz}$.