Black Hole Ringdown Amplitudescopy

Black hole ringdowns in extensions of General Relativity (GR) generically exhibit two distinct signatures: (1) theory-dependent shifts in the standard black-hole quasinormal modes, and (2) additional modes arising from extra fundamental fields – such as scalar, vector, or tensor degrees of freedom – that can also contribute to the gravitational-wave signal. As recently argued, in general both effects are present simultaneously, and accurately modeling them is essential for robust tests of GR in the ringdown regime. In this work, we investigate the impact of extra field-induced modes, which are often neglected in standard ringdown analyses, on the interpretation of gravitational-wave signals. To provide some concrete examples, we focus on dynamical Chern-Simons and Einstein-scalar-Gauss-Bonnet theories, well-motivated extensions of GR, characterized respectively by a parity-odd and a parity-even coupling between a dynamical scalar field and quadratic curvature invariants. We show that including extra field-induced modes improves the bounds on these theories compared to standard spectroscopy and also allows for equally constraining complementary tests not based on quasinormal mode shifts. Our analysis highlights the relevance of incorporating extra field-induced modes in ringdown templates and assesses their potential to either bias or enhance constraints on GR deviations.

💡 Research Summary

The paper investigates how extra degrees of freedom that appear in many extensions of General Relativity (GR) affect the gravitational‑wave (GW) ringdown signal of a newly formed black hole. In such theories two generic signatures coexist: (i) small shifts in the standard quasinormal‑mode (QNM) frequencies and damping times, and (ii) the excitation of additional modes associated with the extra fields (scalar, vector, or tensor). While most current ringdown analyses focus only on (i), the authors argue that (ii) can be equally important and must be incorporated for robust tests of GR.

To make the discussion concrete, the authors study two well‑known quadratic‑curvature theories: dynamical Chern‑Simons (CS) gravity, which couples a pseudoscalar to the parity‑odd Pontryagin density, and Einstein‑scalar‑Gauss‑Bonnet (GB) gravity, which couples a scalar (the dilaton) to the parity‑even Gauss‑Bonnet invariant. Both theories introduce a length‑scale coupling ℓ_CS or ℓ_GB (with α = ℓ^2) and modify the Kerr solution for rotating black holes, giving rise to a non‑trivial background scalar field.

The authors adopt the “ringdown amplitudescopy” framework introduced in earlier work. The GW strain is written as a sum of two families of damped sinusoids: (a) the usual gravitational QNMs with small fractional deviations δf_i and δτ_i from the Kerr values, and (b) extra‑field‑induced modes whose frequencies are taken to be the Kerr scalar QNMs (no shift at leading order) while their amplitudes ˆA_i scale linearly with the dimensionless coupling ζ = α^2/(M^4) and a theory‑dependent coefficient γ_i. The phases ˆϕ_i are left free. This separation isolates the new physics in the amplitudes alone, which the authors call “amplitudescopy” as opposed to the traditional “spectroscopy” that measures frequency shifts.

Using fitting formulas from the literature for the CS and GB QNM corrections (functions F_i(χ) and T_i(χ) that encode δf_i and δτ_i), the authors construct waveform models for a GW event similar to GW250114: a final black‑hole mass M≈60 M⊙, spin χ≈0, and a ringdown signal‑to‑noise ratio (SNR) of 20, appropriate for the upcoming O4 observing run. Two waveform families are considered:

• SpecPA – a standard GR‑based ringdown model that includes the dominant (ℓ=m=2) polar and axial modes, with the CS/GB frequency and damping corrections.
• AmplPA – the same as SpecPA but augmented by a single scalar extra mode (the ℓ=m=2 scalar QNM) with free amplitude γ and phase.

Bayesian parameter estimation is performed with the PyCBC‑Inference package on mock data generated with AmplPA. The coupling length is injected as ℓ≈35 km, corresponding to ζ≈0.024 for the 60 M⊙ black hole. The scalar amplitude coefficient γ is varied (0, 1, 5, 10) to explore its impact. Results show that when γ is small (0 or 1) the posterior on ℓ is essentially identical to that obtained with SpecPA, confirming that a weak extra mode does not improve constraints. However, for γ=5 or 10 the posterior tightens noticeably, yielding upper limits on ℓ that are ≈10 %–20 % stronger than current bounds (≈45 km for CS and ≈44 km for GB). This demonstrates that if the extra field is efficiently excited during merger, its inclusion can significantly enhance the sensitivity of ringdown tests.

The authors also discuss the bias that would arise if the extra mode were present in the data but omitted from the template: the inferred coupling would be systematically underestimated, potentially leading to a false confirmation of GR. They argue that future detectors with higher SNR (A+, Voyager, Einstein Telescope) will make the extra‑mode contribution non‑negligible for a wide range of theories, making amplitudescopy an essential component of the analysis pipeline.

In summary, the paper introduces a systematic way to incorporate extra‑field‑induced QNMs into ringdown analyses, validates the method on realistic mock data, and shows that doing so can both tighten constraints on quadratic‑curvature couplings and avoid biased inferences. The work paves the way for more comprehensive tests of gravity with forthcoming gravitational‑wave observations.