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Beyond black hole spectroscopy: Quasinormal mode contamination by massless scalars

T0 review · 0 major / 5 minor · reviewed 2026-07-10 · glm-5.2

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Pith's one-line read Scalar fields can dominate black hole ringdowns before frequency shifts appear

desk verdict Clean perturbative calculation showing scalar QNM contamination can dominate over frequency shifts in Horndeski ringdowns — deserves a serious referee. read the letter →

arxiv 2607.08619 v1 pith:V2SFLOXZ submitted 2026-07-09 gr-qc astro-ph.HEhep-th

Miguel Yulo Asuncion , Giovanni D'Addario , Thomas P. Sotiriou This is my paper · ORCID
classification gr-qcastro-ph.HEhep-th
keywords scalarblackcontaminationholeordershiftsamplitudeappear
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

When a black hole rings down after a merger, its signal is a sum of characteristic frequencies called quasinormal modes. The standard approach to testing General Relativity (GR) with these signals looks for small shifts in the expected frequencies. This paper argues that if a new massless scalar field exists and couples to gravity, the ringdown signal will also be contaminated by the scalar field's own frequencies, which are generically different from the gravitational ones. Working within shift-symmetric Horndeski gravity, the most general theory coupling a massless scalar to gravity at second order, the authors show that under the standard assumption that the scalar amplitude is suppressed by the charge parameter q, both the frequency shift and the contamination appear at the same order in the perturbative expansion, order q squared. The only coupling that drives both effects at that order is the linear coupling between the scalar and the Gauss-Bonnet invariant, a topological quantity built from curvature. If the suppression assumption is relaxed, contamination enters at order q, one step earlier than the frequency shift, and can dominate the beyond-GR signal. In that case a second coupling, from the quartic Horndeski sector, also contributes subleading corrections to the contamination.

What carries the argument

The central mechanism is a double perturbative expansion: a linear expansion in dynamical perturbations (parameter epsilon) and an expansion in the scalar charge per unit black hole mass (parameter q). The scalar charge q is not free but is set by the black hole mass and the theory's coupling constants, with the linear scalar-Gauss-Bonnet coupling alpha being the dominant contributor. The perturbation hierarchy separates cleanly because the operator acting on each perturbation order is always a background quantity: the d'Alembertian for the scalar and the linearized Einstein tensor for the metric. At each order, one can trace which coupling constants source which parts of the ringdown ansatz

What would settle it

A ringdown signal from a black hole with known mass and spin that shows no evidence of extra frequencies beyond the Kerr quasinormal mode spectrum, at a sensitivity level where the predicted contamination amplitude should be detectable, would constrain the scalar-Gauss-Bonnet coupling alpha and the charge q to be below the threshold where contamination is observable.

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Extended reading notes

Core claim

In shift-symmetric Horndeski gravity, the ringdown signal of a hairy black hole receives two distinct types of beyond-GR correction: a shift of the gravitational quasinormal mode frequencies and contamination from the scalar field's own quasinormal mode frequencies. Under the standard amplitude-suppression assumption, both effects appear at order q squared and are controlled solely by the scalar-Gauss-Bonnet coupling. Without that assumption, contamination appears at order q and dominates over the frequency shift, with subleading corrections from the quartic Horndeski coupling tau_4.

Load-bearing premise

The analysis assumes a single new energy scale in the theory, which forces the Horndeski coupling constants to scale with the charge parameter q in a specific way. Theories with two widely separated scales are known to exist and would break this hierarchy, potentially changing which couplings dominate and at what perturbative order each effect appears.

Editorial extensions

If this is right

  • Ringdown searches for beyond-GR physics that only model frequency shifts may carry a systematic bias if scalar contamination is present but unmodelled
  • If scalar amplitudes are not suppressed by q, contamination rather than frequency shifts would be the leading observable beyond-GR effect in ringdown signals
  • Only two coupling constants, alpha and tau_4, need to be retained to model massless scalar effects on ringdowns up to order q squared, simplifying theory-specific searches
  • The scaling q proportional to M inverse squared means supermassive black hole ringdowns probed by space-based detectors are poor probes of these effects, while solar-mass ringdowns probed by next-generation ground detectors are more promising
  • Nonlinear merger dynamics or spontaneous scalarization scenarios could amplify scalar amplitudes, making the contamination-dominated regime physically realizable

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If future ringdown observations detect extra frequencies that do not match any Kerr quasinormal mode, the contamination framework provides a direct way to distinguish scalar-field signatures from other beyond-GR effects such as modified dispersion relations
  • The dominance of the scalar-Gauss-Bonnet coupling at leading order suggests that null results from current ringdown tests can be recast as direct bounds on alpha rather than on a generic deviation parameter, tightening the link between observation and theory
  • The hierarchy where contamination precedes frequency shifts when amplitudes are unsuppressed implies that early-time ringdown data, where higher overtones are more visible, might be especially sensitive to contamination since overtone amplitudes could be less suppressed than the fundamental mode
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

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Summary. This paper studies black hole ringdown perturbations in shift-symmetric Horndeski gravity, focusing on how massless scalar fields affect quasinormal mode (QNM) frequencies. The authors employ a double perturbative expansion in dynamical perturbations ($ϵ$) and scalar charge per unit mass ($q$), extending prior work to $O(ϵq^2)$. The central result is that, under the assumption that the scalar amplitude is suppressed by $q$ (i.e., $ϕ^{(1,0)}=0$), the linear coupling $αϕG$ between the scalar and the Gauss-Bonnet invariant is the only interaction contributing to both frequency shifts and contamination at $O(ϵq^2)$, with both effects appearing at the same perturbative order. If this suppression assumption is relaxed, contamination appears at $O(ϵq)$ and can dominate over frequency shifts, with subleading corrections from the quartic coupling $τ_4$. The perturbative framework is systematic: the hierarchical field equations (Eqs. 18–21) are derived step by step, and the ringdown ansatz (Eq. 31) is constructed by tracing which source terms survive at each order. The key structural simplification—that the background d'Alembertian (scalar sector) and linearized Einstein tensor (metric sector) serve as operators at each perturbative order—is consistent, and the claim that only $α$ and $τ_4$ contribute at $O(ϵq^2)$ follows directly from the source term structure in Eq. (21).

Significance. The paper addresses a timely and important question in gravitational-wave physics: whether the standard black hole spectroscopy program, which models beyond-GR effects purely as frequency shifts, is systematically biased by missing contamination from additional field modes. The result that contamination and frequency shifts generically appear at the same perturbative order (under standard assumptions) provides concrete theoretical backing for the theory-agnostic claims of Ref. [37] and motivates updating ringdown search templates. The identification of exactly which Horndeski couplings ($α$ and $τ_4$) are relevant at $O(ϵq^2)$ is a useful simplification for EFT-based modeling. The falsifiable prediction that contamination can dominate over shifts when $ϕ^{(1,0)}≠0$ is a concrete, testable claim that could be checked with numerical relativity simulations of mergers. The framework is parameter-efficient: only two coupling constants need to be constrained at this order.

minor comments (5)
  1. §IV, Eq. (31): The relabeling of the shift-term coefficient from $G_n$ in Eq. (30) to $ẽA_n$ in Eq. (31) is explained, but the notation $ẽA_n$ (with a tilde over 'eA') is unusual and could be confused with a derivative or operator. Consider using a more standard notation such as $A_n^{(2)}$ or $Ã_n$ to improve readability.
  2. §II.A, paragraph after Eq. (9): The statement that $O(X^2)$ terms do not contribute at the perturbative orders considered is confirmed later (end of §III), but it would help the reader to briefly note already at this point that this will be verified a posteriori, to avoid apparent inconsistency with the claim that all second-order-equation interactions are included.
  3. §IV, Eq. (29): The statement that the spatial mode function shift can be absorbed into amplitudes when evaluating at null infinity is reasonable, but a brief justification or reference for why this absorption is valid specifically for QNM mode functions (which are not normalizable) would strengthen this point.
  4. §V: The discussion of when $ϕ^{(1,0)}≠0$ might arise (nonlinearities during merger, spontaneous scalarization) is interesting but speculative. A brief mention of whether any existing numerical relativity results in shift-symmetric Horndeski already provide evidence one way or the other would contextualize this, even if the answer is currently unknown.
  5. The abstract in the manuscript appears to be cut off mid-sentence ('...from an additional coupling constant'). This is likely a formatting artifact but should be checked.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the derivation is a self-contained perturbative calculation

full rationale

The paper's central claims follow directly from an explicit perturbative expansion of the shift-symmetric Horndeski field equations. The derivation chain is: (1) write the action (Eq. 8), (2) expand coupling functions in X (Eq. 9), (3) perturb metric and scalar in (ε, q) (Eqs. 12–13), (4) derive perturbation equations order by order (Eqs. 14–21), (5) trace which source terms survive at each order to build the ringdown ansatz (Eqs. 23–31). The conclusion that only α and τ₄ contribute at O(εq²) follows from inspecting which terms in Eq. 21 are non-zero: S^(1,2)[h^(1,0)] and S^(1,2)[ϕ^(1,1)] involve only α (since ϕ^(1,1) is sourced only by α via Eq. 17), while S^(1,2)[ϕ^(1,0)] introduces τ₄ but vanishes when ϕ^(1,0)=0. This is a direct consequence of the equations, not a definition or fit. The self-cited results (Ref [49] for q-scaling via the horizon integral Eq. 5; Ref [56] for the perturbative formalism; Ref [37] for the theory-agnostic ansatz) are themselves derived mathematical results, not fitted parameters or ansätze that would make the present conclusions tautological. The paper confirms consistency with Ref [37] rather than importing its conclusion. The single-scale assumption forcing τ_i ~ q is explicitly stated as an EFT assumption with stated caveats (theories with two scales are excluded), and all claims are conditional on it. No step in the derivation reduces to its own inputs by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new entities are postulated. The scalar field ϕ and the Horndeski couplings are standard in the literature. The Gauss-Bonnet invariant is a standard geometric quantity. The perturbative orders and bookkeeping parameters (ϵ, q) are computational tools, not physical entities.

free parameters (3)
  • α (Gauss-Bonnet coupling)
    Dimensionful coupling constant governing the ϕG interaction; not fitted but treated as a theory parameter to be constrained.
  • τ₄ (quartic Horndeski coupling)
    Dimensionless coefficient of the leading G₄(X)R term; contributes to contamination at subleading order.
  • q (scalar charge per unit mass)
    Not a free parameter but determined by α, M, and the horizon integral (Eq. 5); serves as the expansion parameter.
assumptions (5)
  • domain assumption Single new scale in the theory, forcing τ₃,τ₄ ~ q and τ₅ ~ q²
    Stated in §II.A after Eq. (9); uses EFT dimensional analysis. Theories with two disparate scales are explicitly excluded.
  • domain assumption Local Lorentz symmetry is respected (constant ϕ for flat spacetime)
    Stated in §II.A; restricts the class of theories considered.
  • domain assumption Scalar amplitude suppression by q (ϕ^(1,0) = 0) in the primary scenario
    Assumed in the main analysis and in Ref. [37]; relaxed in the alternative scenario discussed in §V.
  • standard math Linear perturbation theory in ϵ is valid for ringdown
    Standard assumption in QNM analysis; perturbations are small in the post-merger regime.
  • domain assumption Background is Kerr (no hair at zeroth order)
    Stated in §II.B; the perturbative expansion is built around the GR solution.

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Cite this review

Pith. "Pith review of Beyond black hole spectroscopy: Quasinormal mode contamination by massless scalars." pith.science (2026). https://pith.science/paper/V2SFLOXZ

@misc{pith2026260708619,
  author       = {Pith},
  title        = {Pith review of: Beyond black hole spectroscopy: Quasinormal mode contamination by massless scalars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V2SFLOXZ}},
  note         = {Machine review of arXiv:2607.08619}
}
abstract

Testing General Relativity (GR) with black hole ringdowns has conventionally focused on attempting to detect shifts away from the quasinormal mode (QNM) frequencies of the Kerr metric. It has recently been argued, however, that the ringdown signal will also be contaminated with the QNM frequencies of any new fields that are present in a beyond-GR scenario, provided that they couple nonminimally to gravity. We study black hole perturbations for the shift-symmetric Horndeski action, which includes all interactions between a massless scalar and gravity that lead to second order equations upon variation. We perturb linearly in the field and also employ a perturbative expansion in the scalar charge per unit black hole mass, $q$. Assuming that the scalar amplitude is suppressed by $q$, we demonstrate that, to order $q^2$, the coupling between the scalar and the Gauss-Bonnet invariant is the only term that contributes to both frequency shifts and contamination, and that the two effects appear at the same perturbative order. If the assumption about the suppression of the scalar amplitude is relaxed, contamination can appear at leading order in $q$, and hence dominate over frequency shifts. In this case, contamination also receives subleading corrections from an additional coupling

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