Bayesian model selection for testing the no-hair theorem with black hole ringdowns

In this paper we examine the extent to which black hole quasinormal modes (QNMs) could be used to test the no-hair theorem with future ground- and space-based gravitational-wave detectors. We model departures from general relativity (GR) by introducing extra parameters which change the mode frequencies or decay times from their values in GR. With the aid of Bayesian model selection, we assess the extent to which the presence of such a parameter could be inferred, and its value estimated. We find that it is harder to measure the departure of the mode decay times from their GR values than it is with the mode frequencies. The Einstein Telescope (ET, a third generation ground-based detector) could detect departures of as little as 8% in the frequency of the dominant QNM mode of a $500M_{\odot }$ black hole, out to a maximum range of $\simeq 6\mathrm{Gpc}$ ($z\simeq 0.91$). The New Gravitational Observatory (NGO, an ESA space mission to detect gravitational waves) can detect departures of $\sim 0.6\%$ in a ${10}^8{M}_{\odot }$ black hole to a luminosity distance of 50 Gpc ($z\simeq 5.1$), and departures of $\sim 10\%$ in a ${10}^6{M}_{\odot }$ black hole to a luminosity distance of $\simeq 6\mathrm{Gpc}$. In this exploratory work we have made a specific choice of source position (overhead), orientation (inclination angle of $\pi /3$) and mass ratio of progenitor binary ($m_1/m_2=2$). A more exhaustive Monte Carlo simulation that incorporates progenitor black hole spins and a hierarchical model for the growth of massive black holes is needed to evaluate a more realistic picture of the possibility of ET and NGO to carry out such tests.

Figure 1

Fits to the amplitudes of modes excited in the process of a binary black hole merger as a function of the symmetric mass ratio of the progenitor binary.

Figure 2

Left: Strain amplitude of a quasinormal mode signal from a black hole that forms from the merger of a binary of (observed) total mass $5\times {10}^6{M}_{\odot }$ and mass ratio $q=2$ at 1 Gpc. We have plotted the first four dominant modes, 21, 22, 33 and 44, together with their superposition. Right: The signal-to-noise ratio integrand of the same signal $d{\rho }^2/df=|H(f){|}^2/S_h(f)$, where $S_h(f)$ is taken to be that of NGO. The presence of the 33 and 44 mode can be clearly seen in the overall spectrum; 21, however, is buried under 22.

Figure 3

Distribution of the signal-to-noise ratios in different quasinormal modes (21, 22, 33 and 44) for sources located at random positions on the sky, with random inclination and polarization angles. The left plot is for QNM resulting from the merger of a 500 solar mass binary observed in ET, and the right plot is for a 5 million solar mass binary observed in NGO. In both cases the mass ratio of the binary is assumed to be $q=2$ and the source is assumed to be at 1 Gpc.

Figure 4

Projections in the $(M,j)$-plane of the 90% confidence limits on ${\omega }_{22}$, ${\tau }_{22}$ and ${\omega }_{33}$ (black, red dashed and green dot-dashed lines respectively) for injections of signals consistent with GR for $M=500M_{\odot }$ (top-left at 125 Mpc; $\mathrm{SNR}=2888$), $M=1000M_{\odot }$, (top-right at 225 Mpc; $\mathrm{SNR}=2423$); $M={10}^5{M}_{\odot }$ (middle-left at 125 Mpc; $\mathrm{SNR}=63$), $M={10}^6{M}_{\odot }$ (middle-right at 125 Mpc; $\mathrm{SNR}=1756$), $M=5\times {10}^6{M}_{\odot }$ (bottom-left at 1 Gpc; $\mathrm{SNR}=6377$) and $M={10}^8{M}_{\odot }$ (bottom-right at 1 Gpc; $\mathrm{SNR}=115154$). The injected value is denoted in each case by a diamond.

Figure 5

Projections in the $(M,j)$-plane of the 90% confidence limits on ${\omega }_{22}$, ${\tau }_{22}$ and ${\omega }_{33}$ [black, red dashed and green dot-dashed lines, respectively for non-GR injections of $M=500M_{\odot }$ (top at 125 Mpc; with $\Delta {\widehat{\omega }}_{22}=-0.01$, $\mathrm{SNR}=2867$ (left) and $\Delta {\widehat{\omega }}_{22}=-0.05$, $\mathrm{SNR}=2779$ (right)], $M={10}^6{M}_{\odot }$ [middle at 125 Mpc; with $\Delta {\widehat{\omega }}_{22}=-0.01$, $\mathrm{SNR}=1753$ (left) and $\Delta {\widehat{\omega }}_{22}=-0.05$, $\mathrm{SNR}=1735$ (right)] and $M={10}^8{M}_{\odot }$ [bottom at 1 Gpc; with $\Delta {\widehat{\omega }}_{22}=-0.001$, $\mathrm{SNR}=115130$ (left) and $\Delta {\widehat{\omega }}_{22}=-0.005$, $\mathrm{SNR}=115031$ (right)]. The injected value is denoted in each case by a diamond.

Figure 6

Width of the 90% confidence intervals for $\Delta {\widehat{\omega }}_{22}$,$\Delta {\widehat{\omega }}_{33}$ and $\Delta {\widehat{\tau }}_{22}$ (black, green dot-dashed and red dashed lines respectively) against luminosity distance for injections of 500 (top-left), 1000 (top-right), ${10}^5$ (middle-left), ${10}^6$ (middle-right), $5\times {10}^6$ (bottom-left) and ${10}^8{M}_{\odot }$ (bottom-right) for ET (top) and NGO (middle and bottom) simulations. The limiting values in the ${10}^5{M}_{\odot }$ plot are due to the posteriors expanding to fill the prior distributions when they are no longer resolvable.

Figure 7

The differences in evidence between the GR and non-GR models against $\Delta {\widehat{\omega }}_{22}$ and $\Delta {\widehat{\tau }}_{22}$ (black and red dashed lines respectively). Left: For injections of $M=500$ at 125 Mpc in ET, with SNR between 2664 and 2888. Right: For injections of ${10}^8{M}_{\odot }$ in NGO at 1 Gpc, with SNR between 114909 and 115154. Center: For injections of ${10}^6{M}_{\odot }$ in NGO at 125 Mpc, with SNR between 1706 and 1757.

Figure 8

The differences in evidence for the GR and non-GR models is plotted against the luminosity distance to the source, for observation of ringdown signals from a $500M_{\odot }$ black hole in ET (top panels), ${10}^8$ and ${10}^6{M}_{\odot }$ black hole in NGO (middle and bottom panels, respectively). The non-GR model is simulated by varying either $\Delta {\widehat{\omega }}_{22}$ (left panels) or $\Delta {\widehat{\tau }}_{22}$ (right panels) as compared to their GR values of 0. $\Delta {\widehat{\omega }}_{22}$ and $\Delta {\widehat{\tau }}_{22}$ are varied over the range 0 to $-0.1$ in steps of $-0.01$ in the case of ET and the ${10}^6{M}_{\odot }$ NGO system, and over the range 0 to $-0.01$ in steps of $-0.001$ in the case of the ${10}^8{M}_{\odot }$ NGO system, while keeping all other parameters as for GR. The different curves in each panel correspond to different values of the non-GR parameter, starting with 0 for the top most curve and changing by either $-0.01$ (top and bottom panels) or $-0.001$ (middle panels). A difference in evidence of $-10$ or smaller is considered good enough to discriminate a non-GR model from GR model.