Extended parameterized spin expansion formalism for ringdown analysis with GW250114

Abstract: We extend the Parameterized Spin Expansion Coefficients (ParSpec) framework for ringdown tests of black holes by simultaneously sampling the characteristic length scale $\ell$ and the scaling index $p$, where the two parameters are replaced by $\tilde{p}$ and $\tilde{\ell}$ in subsequent analysis, respectively. Unlike previous analyses that fix $\tilde{p}$ to theory-motivated integer values, this extension provides a theory-agnostic description inspired by effective-field theory to the Kerr ringdown spectrum. Using \texttt{pyRing}, we apply the framework to GW250114 with informative priors on $M_{\rm f}$ and $D_{\rm L}$, comparing the $220$-only and $220+221$ mode models under different conditions on $γ$. The posterior distributions of $\tilde{\ell}, \tilde{p}$ and $γ$ are presented with different ringdown start times. We show that the posterior of $\tilde{p}$ remains largely prior dominated, indicating that current ringdown data cannot distinguish different scaling behaviors of the correction. The inferred constraint on $\tilde{\ell}$ is instead mainly controlled by the geometry of the allowed parameter space induced by the $γ$ condition. In particular, $γ<1$ provides the most natural prescription in the $(\tilde{\ell},\tilde{p})$ plane and yields stable but weak upper bounds of $\tilde{\ell}_{90}\simeq 83\, {\rm km}$. An analysis based on the Kullback-Leibler divergence further shows that, after accounting for the effective prior associated with $γ<1$, the data-driven information gain on $\tilde{\ell}$ and $\logγ$ is very small. At the current SNR level, the $220$-only model provides a more informative constraint than the $220+221$ model. We also perform a joint analysis including GW231123 and find that the combined constraint is dominated by GW250114.

• The paper introduces a generalized formalism for black hole ringdown analysis by sampling both the characteristic length scale and a fractional scaling index, widening the scope of GR tests.
• It employs Bayesian inference on high-SNR events like GW250114 with single- and multimode models to set stable, though weak, constraints on effective-field-theory corrections.
• The study highlights that current posterior results are largely influenced by prior geometry, indicating that higher-SNR detections are needed to resolve parameter degeneracies.

Extended Parameterized Spin Expansion Formalism for Black Hole Ringdown Spectroscopy

Framework Extension and Motivation

The paper introduces a substantive extension to the Parameterized Spin Expansion Coefficients (ParSpec) formalism for black hole ringdown analysis by allowing simultaneous Bayesian sampling over both the characteristic length scale $\ell$ and the scaling index $p$ of effective-field-theory (EFT) corrections. Previous applications typically fix $p$ to integer values corresponding to specific higher-curvature operators in modified gravity. In contrast, this generalized approach treats $p$ phenomenologically, admitting fractional (non-integer) values, thus providing a theory-agnostic parametrization capable of describing a broader class of potential deviations from General Relativity (GR).

This extension is motivated by the theoretical ambiguity inherent in EFT descriptions, where the action may contain multiple higher-order corrections with generic scaling indices. By decoupling $p$ from the constraints of specific model operators, the framework augments ringdown tests to probe a wider landscape of possible strong-field gravitational effects. The dimensionless coupling $\gamma$ is defined as $\gamma = (\ell c / GM_f)^p$, with $M_f$ denoting the remnant black hole mass, $z$ the cosmological redshift, and $c$ the speed of light. Flexible sampling over $p$0 and $p$1 permits direct inference on the characteristic length scale and scaling behavior of corrections, potentially informing both data-driven and theoretical constraints on gravity beyond GR.

Data Analysis and Bayesian Inference

The methodology leverages pyRing for time-domain Bayesian inference, applying the extended ParSpec formalism to GW250114, the highest-SNR ringdown event detected by LVK to date. Informative priors for $p$2 and luminosity distance $p$3 are adopted to suppress parameter degeneracy and enhance inference stability. Two ringdown models are considered: single-mode (220) and multimode (220+221), with systematic variation of ringdown start times. Posterior distributions of $p$4, $p$5, and $p$6 are analyzed under multiple coupling exclusion conditions ($p$7, $p$8, $p$9). Additionally, GW231123 is incorporated for joint-event analysis, emphasizing behavior in a multi-source setting.

The formalism applies perturbative corrections to the GR-predicted quasinormal mode (QNM) frequencies and damping times, parameterized as polynomials in the remnant spin. The action is generalized to admit fractional scaling indices, leading to effective higher-curvature operators of arbitrary order. Consistency constraints (e.g., $p$0) are explicitly checked during sampling to preserve the validity of perturbative EFT corrections.

Posterior Structure and Coupling Geometry

A core result concerns the geometry of the allowed parameter space under different $p$1 conditions. The upper bound $p$2 is shown to enforce the most natural prior structure, avoiding artificial correlations between $p$3 and $p$4 seen for $p$5 or $p$6. Sampling across a broad prior range demonstrates that the posterior for $p$7 remains nearly prior-dominated, indicating insensitivity of current ringdown data to the scaling behavior of EFT corrections. This is quantified via Kullback-Leibler (KL) divergences, with the effective prior accounting for parameter volume reduction caused by $p$8. The data-driven information gain on $p$9 and $p$0 is negligible relative to the reduction induced by the $p$1 prior.

Posterior distributions of $p$2 exhibit a plateau structure, with excluded regions outside $p$30–90 km at 3$p$4 significance. The inferred constraint on $p$5 is predominantly controlled by the geometry induced by the $p$6 condition rather than by direct data sensitivity. Mode content (single vs. multimode) and ringdown start time have only marginal impact on $p$7 across the sampled range, indicating robustness but also limited constraining power.

Numerical Results and Implications

The primary numerical result is a stable but weak upper bound on the characteristic length scale: $p$8–83 km under the $p$9 condition, regardless of ringdown start time or inclusion of overtone modes. The 220-only model yields marginally tighter constraints at early start times when SNR is highest, consistent with KL divergence analysis showing greater information gain for early single-mode data. Joint-event analyses combining GW250114 with GW231123 confirm that constraints are dominated by GW250114 due to a broader posterior from GW231123, and therefore multi-event combinations contribute limited improvement unless the additional events exhibit similar sensitivity.

No evidence for deviations from Kerr QNM spectra is found; the posterior consistently favors small coupling regimes. Sampling over a restricted prior range for $p$0 (e.g., $p$1 vs. $p$2) produces negligible shifts in $p$3, confirming that the prior-induced parameter space dominates current inference. The method's flexibility enables future hierarchical analyses as larger populations of high-SNR ringdowns are observed, increasing sensitivity to both $p$4 and $p$5.

Theoretical and Practical Implications

The extended ParSpec formalism presents a computationally efficient, model-independent strategy for ringdown tests of strong-field gravity. Practically, it can assimilate joint constraints across multiple events, facilitating hierarchical Bayesian inference on theory-agnostic EFT parameters. Theoretically, it exposes the degeneracy between scaling index and length scale in ringdown corrections, underscoring the need for higher-SNR detections or resolvable overtone/higher-mode content to break this degeneracy.

Current data are not sufficiently informative to constrain fractional scaling indices or differentiate among distinct higher-curvature scenarios. Future developments may focus on fixing one parameter (e.g., $p$6) while sampling the other, or leveraging events with statistically significant higher-mode contributions to enhance parameter inference. Such advances will be critical for robustly testing the strong-field regime of gravity and identifying subtle deviations from GR with gravitational waves.

Conclusion

The simultaneous sampling of scaling index and coupling length in the extended ParSpec formalism provides a highly general framework for ringdown analysis. Present observational data, even from the loudest available ringdowns, are dominated by prior geometry rather than by direct signal constraints. The resulting bounds on the characteristic length scale are weak but stable, and the framework is poised for improved sensitivity as the gravitational wave catalog grows. Enhanced measurement precision, mode decomposition, and hierarchical Bayesian strategies will progressively sharpen constraints, aiding the pursuit of empirical tests of strong-field gravity in the era of precision GW astronomy.