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

TL;DR

This work investigates how quasi-normal modes from black-hole ringdowns can test General Relativity's no-hair theorem by introducing independent non-GR deviations in mode frequencies and damping times and applying two testing strategies: (i) mode-parameter consistency checks and (ii) Bayesian model selection between GR and generalized, hairy models. Using projected ET and NGO sensitivities, the study shows that Bayesian model selection can detect deviations at the one-to-two percent level for representative high-mass systems across cosmological distances, with NGO offering substantially greater reach than ET. The dominant $l=2$, $m=2$ mode enables mass–spin inferences, while additional modes (e.g., $l=3$, $m=3$) enable consistency tests or, under the generalized model, evidence-based discrimination between GR and non-GR scenarios. The findings indicate NGO's strong potential for robust no-hair tests over a wide redshift range, whereas ET would test only relatively nearby or rare events; limitations include the simplified non-spinning-progenitor assumption and fixed source localization, suggesting future work should address spins and broader parameter spaces.

Abstract

General relativity predicts that a black hole that results from the merger of two compact stars (either black holes or neutron stars) is initially highly deformed but soon settles down to a quiescent state by emitting a superposition of quasi-normal modes (QNMs). The QNMs are damped sinusoids with characteristic frequencies and decay times that depend only on the mass and spin of the black hole and no other parameter - a statement of the no-hair theorem. In this paper we have examined the extent to which 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 general relativistic values. With the aid of numerical simulations and 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 decipher the departure of decay times from their GR value than it is with the mode frequencies. Einstein Telescope (ET, a third generation ground-based detector) could detect departures of <1% in the frequency of the dominant QNM mode of a 500 Msun black hole, out to a maximum range of 4 Gpc. In contrast, the New Gravitational Observatory (NGO, an ESA space mission to detect gravitational waves) can detect departures of ~ 0.1% in a 10^8 Msun black hole to a luminosity distance of 30 Gpc (z = 3.5).

Figures (8)

• 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 quasi-normal 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 quasi-normal modes (21, 22, 33 and 44) for sources located at random positions on the sky, with random inclination and polarisation angles. The left plot is for QNM resutling 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 $\omega_{33}$ (blue, blue dotted and red lines respectively) for injections of signals consistent with GR for $M = 500\,M_{\odot}$ (top-left at 125 Mpc; SNR = $2\,888$), $M=1000\,M_{\odot}$, (top-right at 225 Mpc; SNR = $2\,423$); $M=10^{5}\,M_{\odot}$ (middle-left at 125 Mpc; SNR = $63$), $M=10^{6}\,M_{\odot}$ (middle-right at 125 Mpc; SNR = $1\,756$), $M=5 \times 10^{6}\,M_{\odot}$ (bottom-left at 1 Gpc; SNR = $6\,377$) and $M=10^{8}\,M_{\odot}$ (bottom-right at 1 Gpc; SNR = $115\,154$). 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 $\omega_{33}$ (blue, blue dotted and red lines, respectively) for non-GR injections of $M = 500\,M_{\odot}$ (top at 125 Mpc; with $\Delta\hat{\omega}_{22} = -0.01$, SNR = $2\,867$ (left) and $\Delta\hat{\omega}_{22} = -0.05$, SNR = $2\,779$ (right)), $M = 10^{6}\,M_{\odot}$ (middle at 125 Mpc; with $\Delta\hat{\omega}_{22} = -0.01$, SNR = $1\,753$ (left) and $\Delta\hat{\omega}_{22} = -0.05$, SNR = $1\,735$ (right)) and $M = 10^{8}\,M_{\odot}$ (bottom at 1 Gpc; with $\Delta\hat{\omega}_{22} = -0.001$, SNR = $115\,130$ (left) and $\Delta\hat{\omega}_{22} = -0.005$, SNR = $115\,031$ (right)). The injected value is denoted in each case by a diamond.