Challenges for testing the no-hair theorem with current and planned gravitational-wave detectors

General relativity’s no-hair theorem states that isolated astrophysical black holes are described by only two numbers: mass and spin. As a consequence, there are strict relationships between the frequency and damping time of the different modes of a perturbed Kerr black hole. Testing the no-hair theorem has been a long-standing goal of gravitational-wave astronomy. The recent detection of gravitational waves from black hole mergers would seem to make such tests imminent. We investigate how constraints on black hole ringdown parameters scale with the loudness of the ringdown signal—subject to the constraint that the postmerger remnant must be allowed to settle into a perturbative, Kerr-like state. In particular, we require that—for a given detector—the gravitational waveform predicted by numerical relativity is indistinguishable from an exponentially damped sine after time $t^{\mathrm{cut}}$. By requiring the postmerger remnant to settle into such a perturbative state, we find that confidence intervals for ringdown parameters do not necessarily shrink with louder signals. In at least some cases, more sensitive measurements probe later times without necessarily providing tighter constraints on ringdown frequencies and damping times. Preliminary investigations are unable to explain this result in terms of a numerical relativity artifact.

Figure 1

Scaling of ${\mathrm{GR}}_{+}$ confidence intervals with loudness (defined in Eq. (6)). The left-hand side are results for $\ell m=22$ while the right-hand side is for $\ell m=33$. The top (middle) panels plot ringdown frequency (damping time) as a function of loudness. The red shading shows the 95% confidence regions for ${\mathrm{GR}}_{+}$. The blue shading shows the 95% confidence region when $t_{\ell m}^{\mathrm{cut}}=6.5\mathrm{ms}$. The dashed black lines indicate the true parameter value. The red ${\mathrm{GR}}_{+}$ interval does not shrink monotonically suggesting a limit to our ability to measure ringdown parameters. The blue $t_{\ell m}^{\mathrm{cut}}=6.5\mathrm{ms}$ interval shrinks monotonically, but exhibits a bias so that the confidence interval eventually excludes the true value. The dashed red curves show the confidence intervals using the lower-resolution L5 waveform. The dashed green curves show the confidence intervals taking into account spheroidal-harmonic corrections. The bottom panels show $t_{\ell m}^{\mathrm{cut}}$ (Eq. (4)).

Figure 2

The absolute value of the strain time series for the $\ell m=22$ mode (top) and $\ell m=33$ mode (bottom). The solid blue curves show the numerical relativity waveforms $|h_{\ell m}^{\mathrm{NR}}(t)|$ from [16]. The dashed red curves show the residuals of the ${\mathrm{GR}}_{+}$ fit $|h_{\ell m}^{\mathrm{NR}}(t)-h_{\ell m}^{\mathrm{GR}+}(t)|$ with $t^{\mathrm{cut}}=10\mathrm{ms}$. The dotted green curves show the residuals of the spheroidal-spherical mismatch $|h_{\ell m}^{\mathrm{NR}}(t)-h_{\ell m}^{\mathrm{PT}}(t)|$. Deviation from linear perturbation theory is visible by eye in the envelope of the 33 mode (bottom blue).