ParSpec: Parametrized Spin Expansion Coefficients

• ParSpec is a framework that parametrizes deviations from Kerr quasinormal modes via a bivariate expansion in remnant spin and a beyond-GR coupling.
• It organizes observable shifts into systematic expansion coefficients, reducing source-dependent biases and enabling multi-event inference.
• Empirical analyses show that ParSpec’s perturbative approach yields tight GR-consistent constraints, paving the way for improved black hole spectroscopy with future data.

ParSpec, the Parametrized ringdown spin expansion coefficients formalism, is an observable-based, perturbative framework for black-hole spectroscopy that parametrizes departures from the Kerr/GR quasinormal-mode spectrum through a bivariate expansion in remnant spin and in a beyond-GR coupling. Rather than introducing ad hoc, source-dependent shifts in individual ringdown frequencies and damping times, it organizes deviations in coefficients tied to the spin-expansion structure of Kerr quasinormal modes and to a common coupling parameter, making the parametrization more source independent and better suited to combination across multiple events (Maselli et al., 2019).

1. Conceptual role in black-hole spectroscopy

ParSpec was introduced to address a specific limitation of beyond-GR ringdown tests: many earlier parametrizations were tied to specific theories, often neglected spin, and typically encoded deviations directly in observed ringdown quantities in a way that was source dependent. In GR, the Kerr black-hole quasinormal-mode spectrum is fully determined by the remnant mass $M$ and dimensionless spin $\chi$, so measuring more than one mode enables tests of the no-hair theorem. In generic modified gravity, however, spinning black holes are hard to model directly because the background geometry and perturbation equations are usually not known exactly. ParSpec addresses this by parametrizing the ringdown observables themselves rather than deriving a full rotating solution in each candidate theory (Maselli et al., 2019).

The framework is explicitly designed for multiple-event inference. If the remnant mass and spin are obtained from the full inspiral-merger-ringdown signal under GR, the number of observables is ${\cal O}=2Nq$ for $N$ sources and $q$ measured modes, because each mode contributes a frequency and a damping time. In the scale-free and single-dimensionful-coupling cases, one then requires $N>D+1$, where $D$ is the spin-expansion order. If one uses ringdown only, the number of independent observables is reduced to ${\cal O}=2N(q-1)$, and the required number of sources increases accordingly. This parameter-counting logic is why the original study emphasized that ParSpec requires $\mathcal{O}(10)$ ringdown detections and why the framework was presented as especially relevant for LISA and third-generation ground-based detectors (Maselli et al., 2019).

A central physical point is that ParSpec is perturbative in two senses. First, it assumes that beyond-GR effects are small, so GR is recovered as the coupling tends to zero. Second, it expands the quasinormal-mode spectrum in powers of spin. The original study showed that the spin expansion is systematic: for $D\ge 4$, the expansion is accurate to better than $\chi$0 for $\chi$1; for $\chi$2, this holds up to $\chi$3; and for $\chi$4, one would need $\chi$5 (Maselli et al., 2019).

2. Formal structure of the spin expansion

For a given quasinormal mode $\chi$6, ParSpec writes the ringdown frequency and damping time as

$\chi$7

Here $\chi$8 labels the quasinormal mode, $\chi$9 is the remnant black-hole mass, ${\cal O}=2Nq$0 is the dimensionless spin, ${\cal O}=2Nq$1 is the maximum spin-expansion order, ${\cal O}=2Nq$2 and ${\cal O}=2Nq$3 are the GR spin-expansion coefficients, and ${\cal O}=2Nq$4 and ${\cal O}=2Nq$5 are the deviation coefficients (Carullo, 2021).

This structure differs from a free mode-by-mode shift ansatz because the deviations are not inserted as arbitrary corrections to each observed quantity. Instead, they are embedded in a common expansion with theory-dependent coefficients multiplying a dimensionless coupling ${\cal O}=2Nq$6. The coefficients ${\cal O}=2Nq$7 and ${\cal O}=2Nq$8 are numerical fit coefficients encoding the Kerr spin dependence of the mode ${\cal O}=2Nq$9, while the deviation coefficients encode how a beyond-GR correction enters at each spin order. A key result emphasized in the high-spin analysis is that, at perturbative order, corrections to the black hole’s intrinsic parameters $N$0 caused by modified gravity can be absorbed into the deviation coefficients. One can therefore use the GR-inferred remnant mass and spin in the expansion, which reduces parameter correlations (Carullo, 2021).

The high-spin implementation extended the numerical fitting domain to $N$1, motivated by the fact that LIGO-Virgo posteriors often have support extending to much higher spins than the $N$2 range used in the earliest implementation. In that analysis, the same general expansion form was retained, with $N$3 for frequency fits and $N$4 for damping-time fits. The paper notes that damping times are harder to fit because their polynomial structure is more correlated and more numerically delicate (Carullo, 2021).

3. Couplings, scaling indices, and the EFT interpretation

The coupling parameter is the mechanism through which ParSpec relates observable ringdown deviations to a source-independent scale of new physics. In the dimensionful-coupling case,

$N$5

where $N$6 is the theory coupling, $N$7 is its mass dimension, $N$8 is the source-frame mass, $N$9 is the redshift, and $q$0 is the characteristic length scale of new physics. The original ParSpec study distinguished scale-free corrections with $q$1, single dimensionful couplings, and source-dependent charges or primary hair; the high-spin application then emphasized theory classes such as $q$2, $q$3, $q$4, and $q$5 (Maselli et al., 2019).

The extended formalism developed for GW250114 generalizes this setup by sampling both the characteristic length scale and the scaling index: $q$6 Its EFT-inspired action is written as

$q$7

and the general EFT motivation is

$q$8

Earlier ParSpec analyses typically fixed $q$9 to integer values motivated by specific theories or EFT operators. The extension replaces this theory-specific assumption with a theory-agnostic description inspired by effective-field theory to the Kerr ringdown spectrum. In this formulation, $N>D+1$0 is the common EFT length scale, $N>D+1$1 is a phenomenological scaling index, and non-integer $N>D+1$2 is allowed and interpreted as an effective fractional scaling behavior (Chen et al., 21 Jun 2026).

The dimensionless coupling then becomes

$N>D+1$3

The extended analysis explored the cuts $N>D+1$4, $N>D+1$5, and $N>D+1$6. The paper argues that $N>D+1$7 is the most natural prescription because it avoids an artificial distortion in parameter space. Rewriting the condition as

$N>D+1$8

the paper states that $N>D+1$9 corresponds to the clean boundary where the correction remains perturbative, the allowed $D$0 region is relatively uniform, and sampling in $D$1 does not strongly bias $D$2 (Chen et al., 21 Jun 2026).

4. Waveform modeling and inference implementation

The extended GW250114 analysis was performed with pyRing, a Bayesian time-domain ringdown inference package. The strain model is

$D$3

with mode contributions

$D$4

$D$5

Here $D$6 are mode amplitudes, $D$7 are phases, $D$8 are spin-weighted spheroidal harmonics, $D$9 and ${\cal O}=2N(q-1)$0 are orientation angles, and ${\cal O}=2N(q-1)$1 is the luminosity distance (Chen et al., 21 Jun 2026).

Two ringdown models were compared: a ${\cal O}=2N(q-1)$2-only model and a ${\cal O}=2N(q-1)$3 model. The ringdown start time was scanned over ${\cal O}=2N(q-1)$4 for the ${\cal O}=2N(q-1)$5-only model and ${\cal O}=2N(q-1)$6 for the ${\cal O}=2N(q-1)$7 model, with steps of ${\cal O}=2N(q-1)$8. The analysis used ${\cal O}=2N(q-1)$9, uniform priors $\mathcal{O}(10)$0 km, $\mathcal{O}(10)$1, and uniform priors $\mathcal{O}(10)$2 on $\mathcal{O}(10)$3 and $\mathcal{O}(10)$4. An appendix check with $\mathcal{O}(10)$5 was also reported (Chen et al., 21 Jun 2026).

The same study emphasizes the importance of informative priors on the remnant mass $\mathcal{O}(10)$6 and luminosity distance $\mathcal{O}(10)$7. In the pyRing waveform, the strain amplitude scales roughly like $\mathcal{O}(10)$8 times mode amplitudes, so $\mathcal{O}(10)$9 and $D\ge 4$0 are strongly degenerate with the amplitudes. The paper therefore restricts $D\ge 4$1 to the $D\ge 4$2 credible interval from NRSur7dq4 and adopts an informative prior on $D\ge 4$3 from inspiral-merger inference. This is especially important because $D\ge 4$4 depends directly on $D\ge 4$5, so uncertainty in $D\ge 4$6 propagates into $D\ge 4$7–$D\ge 4$8 degeneracies. The paper also shows that informative priors tighten overtone amplitudes, especially for the $D\ge 4$9 mode at early times (Chen et al., 21 Jun 2026).

5. Observational constraints and empirical behavior of the posteriors

Applied to GWTC-2 LIGO-Virgo observations, the high-spin ParSpec analysis found no statistically significant deviation from GR. It analyzed 17 events for frequency deviations and 14 for damping-time deviations, using pyRing and a ringdown-only time-domain likelihood. The fundamental mode $\chi$00 was better constrained than the first overtone $\chi$01, frequency deviations were more tightly constrained than damping-time deviations, and increasing the number of free deviation coefficients weakened single-parameter posteriors because of correlations. The headline result for dimensionless beyond-GR couplings was

$\chi$02

together with the upper bounds

$\chi$03

The paper reports negative log Bayes factors in favor of GR for most cases, with the GR hypothesis strongly favored in the $\chi$04 fundamental-frequency case (Carullo, 2021).

In the extended GW250114 study, the central empirical result is that the posterior of $\chi$05 remains largely prior dominated, indicating that current ringdown data cannot distinguish different scaling behaviors of the correction. The inferred constraint on $\chi$06 is instead mainly controlled by the geometry of the allowed parameter space induced by the $\chi$07 condition. Under the $\chi$08 prescription, the analysis finds stable but weak upper bounds of $\chi$09, with a plateau-like allowed region around $\chi$10–$\chi$11 km. The posterior on $\chi$12 consistently prefers the small-coupling regime, and there is no evidence for a statistically significant deviation from Kerr (Chen et al., 21 Jun 2026).

The paper further quantifies information gain with the Kullback–Leibler divergence,

$\chi$13

with $\chi$14. For $\chi$15, the KL divergence is small for both the raw prior and the effective prior. For $\chi$16 and $\chi$17, the KL divergence is $\chi$18 relative to the raw prior but drops to $\chi$19 once the effective prior induced by $\chi$20 is used. The paper therefore concludes that most of the apparent information gain comes from prior-volume reduction due to $\chi$21, not from the data themselves. At the current SNR level, the $\chi$22-only model provides a more informative constraint than the $\chi$23 model, and a joint analysis including GW231123 finds that the combined constraint is dominated by GW250114 (Chen et al., 21 Jun 2026).

6. Assumptions, interpretive cautions, and future development

ParSpec relies on several explicit assumptions. The original framework assumes perturbative deviations from Kerr/GR, a spin expansion about $\chi$24, and no new dominant extra polarizations in the observed ringdown. In the main proof-of-principle analysis, the authors considered fundamental modes only and neglected overtones because they are closely spaced, difficult to resolve, and not ideal for direct spectroscopy unless SNR is very high. The statistical treatment also assumed quasi-orthogonality of different $\chi$25 modes and GR-based estimates of source masses and spins (Maselli et al., 2019).

A recurring interpretive issue concerns what ParSpec constraints actually measure. The extended GW250114 analysis shows that the stable upper bound on $\chi$26 should not be read as a sharply data-driven measurement: it is largely driven by the geometry of the allowed prior space induced by the $\chi$27 cut. Likewise, the formal extension that promotes $\chi$28 to a free parameter does not imply that present data can infer a scaling law; in practice, current ringdown observations are not yet sensitive enough to distinguish theory-agnostic scaling laws in ParSpec (Chen et al., 21 Jun 2026).

The development path described across the three studies points in two directions. One is observational: more detectors and improved duty cycle in the LIGO-Virgo-KAGRA network, higher high-frequency sensitivity through proposed instruments such as NEMO, and the complementary curvature reach of space-based and ground-based detectors. The other is methodological: templates with more constrained amplitude models, alternative non-polynomial parametrizations, direct effective-metric descriptions, extra quasinormal-mode branches, multiple coupling constants, multiple secondary angular modes, and overtone extensions when sufficiently resolved. Within the extended formalism, a plausible implication is that future higher-SNR ringdowns and hierarchical multi-event analyses will be needed to break the $\chi$29–$\chi$30 degeneracy and turn prior-dominated upper bounds into genuinely data-driven tests of EFT-level corrections (Carullo, 2021).