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REVIEW 4 major objections 7 minor 47 references

Neutron stars in degenerate higher-order scalar-tensor theories: Axial perturbations

T0 review · 4 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For a one-parameter DHOST subclass, axial neutron-star perturbations obey a Regge-Wheeler-type equation equivalent to general relativity on an effective conformal metric, and the fundamental $l=2$ quasinormal mode deviates from GR in both…

desk verdict A solid analytic derivation of axial DHOST neutron-star perturbations with an effective-metric correspondence, undermined mostly by an unreproducible numerical section and a parameter typo. read the letter →

arxiv 2507.02749 v1 pith:ICLB3UGI submitted 2025-07-03 gr-qc

Hamza Boumaza , David Langlois This is my paper · ORCID
classification gr-qc MSC 83C5583C3583D05 PACS 04.40.Dg04.30.-w04.50.Kd
keywords DHOSTtheoriesaxialperturbationsneutronstarsquasinormalmodesuniversalrelationsRegge-Wheelerequationeffectivemetricmodifiedgravity
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

The paper sets out to establish that, in a one-parameter subclass of degenerate higher-order scalar-tensor (DHOST) theories, the axial (odd-parity) oscillations of neutron stars are governed by a single master equation of Regge-Wheeler type, identical in form to general relativity but written with an effective metric obtained by a conformal rescaling of the background. Because axial perturbations do not couple to the fluid or the scalar field, they leave one propagating degree of freedom, and this equivalence lets the authors compute quasinormal modes (QNMs) directly from the neutron-star backgrounds found in their earlier work. For the model $F = \kappa/2 + \sigma X$, the fundamental $l=2$ mode shifts with the parameter $p = q^2\sigma$ in both real frequency and damping time, relative to GR. The empirical compactness-based universal relations also shift with $p$, with equation-of-state scatter below about five percent, which is what would make the theory distinguishable from GR through neutron-star ringdown observations.

What carries the argument

The load-bearing object is the quadratic axial action (4.6)-(4.7), expanded around the static background and reduced, after eliminating the two metric perturbations $H_0$ and $H_1$ in favor of the combination $\chi = \dot{H}_1 - H_0'$, to a single-field action (4.13). The master equation follows after introducing the tortoise coordinate $dr_* = \sqrt{h/f}\,dr$ and the field redefinition $U = \lambda/(r\sqrt{F})$: a Regge-Wheeler-type equation (4.20) with potential (4.21), equivalent to GR in the conformally rescaled effective metric (4.22). The vanishing coefficient $Q_3$, a consequence of the scalar-field equation of motion, is what decouples axial perturbations from the scalar and fluid sectors.

What would settle it

An independent numerical integration of the master equation (4.20) on the same backgrounds, using a different quasinormal-mode code or publicly released background solutions, would either reproduce the quoted frequencies and damping times or not; equivalently, a future neutron-star ringdown observation with independently measured mass and compactness can be checked against the fitted universal relations (5.16)-(5.19), whose fitted p-dependent coefficients differ by more than the reported equation-of-state scatter.

Watch

Extended reading notes

Core claim

The central claim is that for the shift-symmetric DHOST action (2.3) with arbitrary $F(X)$, axial perturbations around static, spherically symmetric neutron-star backgrounds satisfy the master equation (4.20) with potential (4.21), which is exactly the Regge-Wheeler equation of general relativity evaluated on the effective conformal metric $d\tilde{s}^2 = F(-f\,dt^2 + h\,dr^2 + r^2\,d\Omega^2)$. The quadratic action contains a single degree of freedom, there is no ghost or gradient instability when $F > 0$, and the axial perturbation speed equals the speed of light. For the illustrative one-parameter family $F = \kappa/2 + \sigma X$, the authors compute the fundamental $l=2$ QNM for five realistic equations of state and find that, as $p = q^2\sigma$ increases, the real frequency $\omega_R$ decreases appreciably while the damping time changes more moderately; the shifts exceed the equation-of-state scatter in the fitted universal relations, so the parameter $p$ could in principle be inferred from neutron-star ringdown data.

Load-bearing premise

The results inherit the equilibrium neutron-star solutions from the authors' previous work, which assume the scalar-field ansatz $\phi = q t + \psi(r)$ and a Schwarzschild exterior with $F_{\rm ext} = \kappa/2 - p$; if those background configurations are not the correct static solutions of the theory, the master equation and the quoted quasinormal frequencies do not follow.

Editorial extensions

If this is right

  • Axial QNMs of neutron stars in this DHOST subclass can be computed by solving the standard GR Regge-Wheeler equation on the effective metric, so existing GR codes transfer directly.
  • No ghost or gradient instabilities appear in the axial sector for $F > 0$, and gravitational waves propagate at the speed of light, consistent with current constraints on the gravitational-wave speed.
  • The fundamental $l=2$ mode's frequency is lower for larger $p$, and its damping time is also modified, so the deviation is not merely a renormalization of the star's mass.
  • The universal relations $\omega_R r_s$ versus compactness, $\omega_R M$ versus compactness, and rescaled $\tilde{\omega}_I$ versus $\tilde{\omega}_R$ have $p$-dependent coefficients with scatter below about 5-10%, providing a route to constrain $p$ without knowing the equation of state.

Reading between the lines

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

  • If the effective-metric correspondence holds beyond the axial sector, a similar map for polar perturbations would be much more involved because polar modes couple to the fluid and scalar; until that is computed, the full ringdown template for this theory remains incomplete, and the axial-only shift may understate or overstate the theory's observational signature.
  • The $p$-dependence of the universal relations suggests a concrete observational target: stacking several neutron-star ringdown events with independent mass and compactness measurements could discriminate $p = 0$ from $p = 2 \times 10^{-3}$ even if single events are too noisy, since the predicted separation is systematic rather than equation-of-state scatter.
  • The conformal effective metric also hints that the same axial master equation may hold for wider DHOST subfamilies or for stars with scalar hair, as long as the odd-parity sector decouples; this is a testable extension of the black-hole results the paper cites.
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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

4 major / 7 minor

Summary. The paper studies axial (odd-parity) perturbations of non-rotating neutron stars in a shift-symmetric one-parameter DHOST subclass with F = κ/2 + σX (Eq. (2.3)). The background solutions are taken from the authors' previous work [6], with a scalar-field ansatz φ = qt + ψ(r). For the axial sector, the fluid velocity is slaved to the metric (v = −H0) and δφ = 0, so the authors reduce the quadratic action to a single dynamical field λ and obtain a Regge-Wheeler-like master equation (4.20) with potential (4.21), which they identify with the GR axial equation in the conformally related effective metric (4.22). They then compute the fundamental l = 2 QNM frequencies and damping times for five equations of state (SLy, FPS, BSk19, BSk20, BSk21) and three values of the parameter p ≡ q²σ (0, 10⁻³, 2×10⁻³), using the complex-coordinate integration method of Ref. [35]. They report that increasing p lowers the frequency ω_R and modifies the damping time relative to GR (Figs. 1–2), and they fit quadratic universal relations between rescaled frequencies, damping times, and compactness (Eqs. (5.16)–(5.19)), proposing them as a way to discriminate GR from this DHOST model.

Significance. If the numerical results are correct, the paper's main contributions are (i) a fairly explicit, self-contained derivation showing that axial perturbations of DHOST neutron stars obey the same master equation as GR in an effective conformal metric, extending the black-hole result of Ref. [19] to stars, and (ii) a concrete, falsifiable prediction that the DHOST parameter p shifts both the frequency and the damping time of the fundamental l = 2 axial mode. The analytic part is careful and internally consistent: the quadratic action is given explicitly, the l = 1 residual mode is identified with slow rotation, the Schwarzschild exterior limit of the potential (4.21) reproduces the standard Regge-Wheeler form, and the no-ghost condition is stated. The numerical part, however, is currently not verifiable from the manuscript: no code or data are provided, no validation at p = 0 against known GR results is reported, no convergence tests or error bars are given, and Eq. (5.17) contains an internal inconsistency in the stated p values. Because these issues are concrete and fixable, and the central analytic derivation is sound, the appropriate path is a major revision rather than rejection.

major comments (4)
  1. [§V, Eq. (5.14) and Figs. 1–2] The p = 0 curves are presented as the GR baseline, but the paper reports no validation of the QNM extraction procedure against known results. For the axial (w-)modes of non-rotating relativistic stars, reference values exist in the literature (e.g., Kokkotas and Schutz, MNRAS 255 (1992) 119, and the GR limits of Refs. [39] and [42] cited in this paper), and for the exterior problem the l = 2 Schwarzschild QNM provides a clean target. Since the p = 0 and p ≠ 0 runs share the same integration, matching, and root-finding pipeline, a discrepancy at p = 0 would directly invalidate the claimed DHOST deviations in Figs. 1–2. Please add a quantitative GR-limit check, including a test in which the matching point is moved far outside the star so the mode should converge to a Schwarzschild QNM, and state the achieved accuracy.
  2. [§V, Eq. (5.17)] In Eq. (5.17) the two modified-gravity fits are labeled p = 10⁻² and p = 2 × 10⁻², whereas every other fit in the paper (Eqs. (5.16), (5.18), (5.19)) and all figures use p = 10⁻³ and p = 2 × 10⁻³. As printed, the ω_R M universal relation does not correspond to the parameter values studied anywhere else in the manuscript, and it is unclear which fits were actually computed. This must be corrected and the fits re-verified, because the comparison of universal relations across p values is a central quantitative claim of the paper.
  3. [§IV, between Eqs. (4.7) and (4.8)] The reduction of the quadratic action (4.6) to the single-field action (4.13), and hence to the master equation (4.20), relies on the statement that Q3 = 0 as a consequence of the scalar-field equation of motion (3.11)–(3.12). This cancellation is not demonstrated, and the expression for Q3 in Eq. (4.7) is sufficiently complicated that the claim is not evident by inspection. Because a non-zero Q3 would couple H0 and H1 and change the form of the master equation, please include the explicit derivation, or a supporting calculation in an appendix, showing that the background equations imply Q3 ≡ 0.
  4. [§V, Figs. 1–6 and Eqs. (5.16)–(5.19)] No error bars are given for the individual QNM frequencies ω_R and damping times, and no convergence tests are reported for the numerical procedure: the integration step along the complex contour (5.12), the distance at which the asymptotic form (5.13) is imposed, the matching condition (5.14), or the root-finding tolerance for ω. The quoted uncertainties in Eqs. (5.16)–(5.19) are only fit-coefficient errors and do not include the solver error. Without at least representative convergence data or a release of the code, the magnitude of the reported p-induced shifts in Figs. 1–2 cannot be assessed against numerical noise.
minor comments (7)
  1. [§V, Fig. 2] The axis label '-1/ω_I (s⁻¹)' is dimensionally inconsistent: the inverse of ω_I has units of time, and the plotted range (50–200) needs an explicit unit (ms or μs). Please also state whether ω_R in Figs. 1–2 and Eqs. (5.16)–(5.19) denotes angular or cyclic frequency; the numerical values suggest cyclic frequency in kHz, but the text uses ω in the Fourier convention (4.18) as an angular frequency.
  2. [§V, sentence before Eq. (5.12)] The sentence preceding Eq. (5.12) states that the outgoing wave grows exponentially at spatial infinity while the ingoing wave is small. With the e^{iωt} convention of (4.18) and ω_I < 0 (as implied by the damped modes in Fig. 2), the term e^{iωr*} grows and e^{−iωr*} decays; as written, the outgoing and ingoing behavior appears to be swapped.
  3. [§V, Eq. (5.16)] The linear-in-C coefficients carry uncertainties larger than the central values (6±10 for GR, 7±8 for p = 10⁻³). Since the text argues these coefficients are consistent with zero, consider re-fitting with the linear term fixed to zero in those cases and report a goodness-of-fit statistic (R² or χ² or binned scatter) to support the stated 'less than 5%' deviation.
  4. [§V, central density range] The value of ρ0 = m_n n0 is printed as 1.675 × 10⁻¹⁴ g·cm⁻³; with n0 = 0.1 fm⁻³ and m_n = 1.675 × 10⁻²⁴ g, the nuclear saturation density is 1.675 × 10¹⁴ g·cm⁻³. Please correct the exponent.
  5. [Throughout, §V figures] The manuscript contains typos ('asssociated' in Sec. II, 'modess' in Sec. V) and the arXiv rendering of Figs. 3–6 has corrupted axis labels, legends, and multi-line tick marks (e.g., the caption of Fig. 2 and the y-axis of Fig. 5). Please check the compiled version and provide clean figures.
  6. [§V, mode identification] The modes are identified only as the 'fundamental l = 2 QNM.' Since axial perturbations of a non-rotating star do not support the polar f-mode, the computed modes should be the spacetime (w-)modes. Stating this explicitly, and noting where the results sit relative to the w_I/w_II families, would help the comparison with the GR literature.
  7. [§IV, after Eq. (4.15)] The no-ghost/no-gradient statement requires F > 0 everywhere. Given that F = κ/2 + σX with p/κ of order a few percent, it would be useful to state explicitly that all background solutions used in §V indeed satisfy F > 0 throughout the star.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the axial master equation is derived from the action, the QNMs are genuine solutions of the resulting ODE, and the universal relations are explicit fits to computed data.

full rationale

The paper's central derivation is self-contained: the axial perturbation action (4.6)-(4.9) is obtained by expanding the DHOST plus fluid action to second order, and the master equation (4.20)-(4.21) follows by variation and field redefinitions (U = lambda/(r sqrt(F)), dr* = sqrt(h/f) dr) without importing the GR Regge-Wheeler result as an input. The claimed equivalence to GR in the effective metric (4.22) is an interpretation of the already-derived equation, not a step that supplies the equation. The QNM frequencies and damping times are genuine solutions of the second-order ODE (5.9) with regularity and outgoing-wave boundary conditions; they are not fitted to target values. The background neutron-star solutions are taken from the authors' prior paper [6]; although load-bearing, that is an independent input (separate modified-TOV solutions with stated assumptions), not equivalent to the perturbation spectrum claimed here, so it does not make the QNM claim circular. The universal relations (5.16)-(5.19) are explicitly phenomenological fits to the computed QNM data; the paper does not present the fitted coefficients as independent predictions, so there is no fitted-input-called-prediction step. The apparent inconsistency of p-values in Eq. (5.17) (10^-2 versus 10^-3 elsewhere) and the absence of a GR-limit validation of the pipeline are robustness/correctness concerns, not circularity. Consequently no step reduces by construction or by self-citation to its own input.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on the background solutions from the authors' previous work [6] and on the effective-metric correspondence from [19]; the only numbers tuned to this paper's target output are the universal-relation polynomial coefficients, which are fits to the numerical data.

free parameters (2)
  • p = q^2 sigma (modified gravity parameter) = 10^-3 and 2x10^-3 (and, in Eq 5.17 only, 10^-2 and 2x10^-2)
    Chosen by hand to illustrate deviations from GR; not fitted to QNM data, but it controls all beyond-GR effects.
  • Quadratic universal-relation coefficients (a0, a1, a2) in Eqs (5.16)-(5.19) = e.g., a0=105.1, a1=6, a2=-369 for p=0 in Eq (5.16); see equations for all values
    Fitted to 115 compactness samples per EOS; used to define the empirical universal relations.
assumptions (6)
  • domain assumption The DHOST action (2.3) with A1=A2=0 represents a degenerate single-scalar theory with gravitational wave speed equal to light.
    Invoked in Section II; ensures the perturbation framework is valid.
  • domain assumption The Schutz variational principle (2.5) correctly describes a perfect fluid and yields the standard fluid equations.
    Used in Section II and Appendix A to derive the fluid perturbation action and the relation v = -H0.
  • domain assumption The background neutron star solutions of [6], based on the scalar field ansatz (3.2) and exterior Schwarzschild matching with F_ext = kappa/2 - p, are valid equilibrium configurations.
    All QNM results in Section V are computed on these backgrounds; the paper does not re-derive them.
  • domain assumption The effective-metric equivalence for axial perturbations, proven for DHOST black holes in [19], extends to neutron stars via the conformal factor F(X).
    Used in Section IV to identify the effective metric (4.22); the paper notes the analogy but does not give a full proof for stars.
  • domain assumption The analytical EOS parametrizations of [33,34] accurately represent the five realistic equations of state.
    Used as numerical input in Section V.
  • domain assumption The complex-coordinate rotation method of [35] correctly selects quasinormal mode boundary conditions for this system.
    Used in Section V for QNM extraction; no convergence test is reported.

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Pith. "Pith review of Neutron stars in degenerate higher-order scalar-tensor theories: Axial perturbations." pith.science (2026). https://pith.science/paper/ICLB3UGI

@misc{pith2026250702749,
  author       = {Pith},
  title        = {Pith review of: Neutron stars in degenerate higher-order scalar-tensor theories: Axial perturbations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ICLB3UGI}},
  note         = {Machine review of arXiv:2507.02749}
}
read the original abstract

We study the axial (or odd-parity) perturbations of neutron stars in a one-parameter subclass of Degenerate Higher-Order Scalar-tensor (DHOST) theories. After recalling the equilibrium neutron star configurations obtained in a previous work by solving the generalised Tolman-Oppenheimer-Volkoff equations in DHOST theories, we derive the action at quadratic order in linear perturbations of axial type. We then compute the quasi-normal modes (QNMs) for several values of the modified gravity parameter and various equations of state, observing deviations of both frequencies and damping times with respect to general relativity. We also analyze the impact of our modified gravity parameter on the universal relations relating the (rescaled) frequencies and damping times to the compactness of the neutron star.

Figures

Figures reproduced from arXiv: 2507.02749 by the authors.

Figure 1
Figure 1. FIG. 1: Real part [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Damping time of the fundamental [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Compactness- [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: ˜ω [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

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