Black-hole spectroscopy, the no-hair theorem, and GW150914: Kerr versus Occam
J. Calderón Bustillo, P. D. Lasky, Eric Thrane · 2021 · 91 citationsRead the paper
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 $(\ensuremath{\ell},m,n)$: the $(\ensuremath{\ell},m)$ 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 $\ensuremath{\sim}6.5$ 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.
What the full text shows
Shown in the lab- The Kerr nature of the GW150914 remnant is preferred over the hairy object hypothesis when using priors from binary black-hole merger models. — log B ~ 6.5 (hairy object disfavored < 1:600) (GW150914 data using IMRPhenomPv2 priors)
- Semi-agnostic tests (without priors) cannot provide strong evidence for the no-hair theorem due to the Occam penalty of adding overtones. — log B ~ 0.6 (Kerr only 1.8x more probable than exotic) (GW150914 data using semi-agnostic Kerr vs Hair models)
- The no-hair theorem is verified for GW150914 when incorporating the full set of overtones via binary black hole priors. — ~ 99.7% level (GW150914 data using IMRPhenomPv2)
- Parent binary masses can be recovered solely from the post-merger signal. — m1 = 41+10-5 M, m2 = 32+6-11 M (90% credible intervals) (GW150914 post-peak signal)
- The mass ratio of the parent binary is constrained using only the post-merger signal. — q < 2.29 (90% confidence) (GW150914 post-peak signal)
- Adding the first overtone corrects parameter estimation bias for low SNR signals. — log B ~ 0.5 (n=1 vs n=0) (NR simulation scaled to SNR 15)
- Setup
- Analysis performed on the GW150914 gravitational-wave event and a Numerical Relativity (NR) simulation scaled to SNRs of 15 and 100.
- How
- The authors performed Bayesian model selection comparing Kerr black hole models against 'Hair' models (damped sinusoids with free parameters). They compared semi-agnostic spectroscopy (no priors) against a strategy using IMRPhenomPv2 waveforms to impose priors on ringdown amplitudes and phases based on binary merger physics.
- Limitations
- Semi-agnostic tests are limited by the increasing number of free parameters needed to fit data. · Binary spin parameters (χeff) cannot be effectively recovered from the post-merger signal alone. · Some existing models for non-spinning BBHs show mismatches up to 0.02 with NR waveforms for mass ratios near unity.
Summarized by Gemma 4 from the open-access full text · 13 Oct 2026. Read it