{"id":"1ebcb5be-b9e6-4ffd-9eee-dde6beda4733","arxiv_id":"2507.02749","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Axial quasinormal modes of neutron stars in a one-parameter DHOST gravity family deviate from general relativity and obey a Regge-Wheeler-like equation in an effective conformal metric.","lead":"This paper computes how neutron stars vibrate in a modified theory of gravity, and finds the vibration frequencies and decay times differ from Einstein's general relativity. The differences could serve as a gravitational-wave test of gravity using neutron star merger signals.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Numerical QNM results lack a GR-limit validation; if the p=0 pipeline does not reproduce known GR neutron-star axial frequencies, the claimed DHOST deviations in Figs. 1-2 are unsubstantiated.","rationale":"The paper's analytic derivation—from the quadratic axial action (4.6) to the single master equation (4.20) with potential (4.21) and the effective-metric interpretation—is internally consistent and follows standard methods. The reduction is plausible: the l≥2 action contains one degree of freedom, the transformation U = λ/(r√F) removes the first-derivative term, and the resulting Schrödinger form is what one would expect for axial perturbations decoupled from the fluid. Credit is due for this part. However, the paper's headline claim is not just the equation but the numerical statement that QNMs deviate from GR. That statement is supported only by an unverified computational pipeline. The reader's weakest assumption pointed to the backgrounds from [6]; my concern is broader and more directly testable: the p=0 limit should reproduce standard GR neutron-star axial QNMs, and the paper does not show that it does. This is the single most load-bearing gap because every p≠0 curve is computed with the same code and the same shooting/matching method; any systematic error in the GR limit would invalidate the claimed deviations. The concrete test is cheap and decisive. I therefore keep the reader's CONDITIONAL verdict unchanged, adding this reproducibility condition as the primary requirement for acceptance.","tokens_in":19153,"tokens_out":24106,"duration_ms":262345,"concrete_test":"Compare the p=0 (GR) curves in Figs. 1 and 2 for SLy and FPS against published GR axial QNM results for the same equations of state and mass range (e.g., Blázquez-Salcedo et al. 2018, or an independent Lindblom-Detweiler integration). Require agreement in ω_R and |ω_I| to within 1%. Then, for one p≠0 point, recompute the QNM with an independent solver that applies purely outgoing-wave boundary conditions at infinity instead of the complex-coordinate rotation; agreement would separate physical deviations from numerical artifacts.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim is that the DHOST parameter p shifts the fundamental l=2 QNM frequency and damping time relative to GR. This claim rests entirely on a numerical pipeline that (i) imports background solutions from a previous paper [6] without reproducing them and (ii) solves the second-order ODE (5.9) with a complex-coordinate boundary-condition method. The paper provides no code, data, or comparison against any known solution. The most load-bearing soft spot is the missing validation of the p=0 limit, which should reduce exactly to GR for the same equations of state. If this limit does not match established GR results, the same pipeline errors propagate to p≠0, so the reported deviations would not be evidence of DHOST physics. The background concern is related but secondary: a GR-limit check tests the entire chain because p=0 uses the same integration, matching, and QNM extraction procedure. The presence of inconsistent parameter values in Eq. (5.17) (10^-2/2×10^-2 versus 10^-3 elsewhere) further indicates that the numerical and fitting details are not tightly controlled.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19309,"tokens_out":34015,"duration_ms":352467,"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":[{"comment":"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.","section":"§V, Eq. (5.14) and Figs. 1–2"},{"comment":"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.","section":"§V, Eq. (5.17)"},{"comment":"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.","section":"§IV, between Eqs. (4.7) and (4.8)"},{"comment":"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.","section":"§V, Figs. 1–6 and Eqs. (5.16)–(5.19)"}],"minor_comments":[{"comment":"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.","section":"§V, Fig. 2"},{"comment":"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.","section":"§V, sentence before Eq. (5.12)"},{"comment":"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.","section":"§V, Eq. (5.16)"},{"comment":"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.","section":"§V, central density range"},{"comment":"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.","section":"Throughout, §V figures"},{"comment":"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.","section":"§V, mode identification"},{"comment":"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.","section":"§IV, after Eq. (4.15)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a direct follow-up to the authors' PRD 106 (2022) paper [6], and the new contributions (axial quadratic action, master equation, effective-metric correspondence, QNM survey, and universal relations) are clearly delineated. The main gate for publication is the numerical validation: the internal inconsistency in Eq. (5.17) and the complete absence of a GR-limit test of the QNM pipeline prevent me from accepting the quantitative claims as they stand. If the authors supply the GR validation, convergence information, and the corrections, I expect the paper to be publishable. I would also encourage the authors to make the QNM code or data available, given that the central results are numerical. Finally, the figure rendering in the arXiv source is badly corrupted in §V and needs to be fixed in the journal version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is the extension of the effective-metric correspondence from DHOST black holes to neutron stars, for axial modes. The paper shows that in the F = kappa/2 + sigma X subclass, axial perturbations obey a Regge-Wheeler-like equation that is exactly GR in a conformally rescaled metric. That is a clean, non-obvious result, and it is derived carefully via quadratic action methods. The l=1 slowly-rotating solution is also a nice sanity check. I was a bit skeptical of the effective-metric claim, but the derivation in Section IV holds together, and the decoupling of axial modes from the fluid makes the correspondence plausible. Credit where due: the perturbation derivation is thorough, the paper is honest that the backgrounds come from previous work, and the universal-relation analysis is a reasonable attempt to separate EOS effects from modified-gravity effects.\n\nThe soft spots are mostly in Section V. The numerical QNM pipeline has no code, no data, no error bars, and no explicit validation against known GR neutron-star axial frequencies for p = 0. The stress-test note is right: if the p=0 limit does not reproduce established results, the deviations at p != 0 are meaningless. This is the single most load-bearing omission. I would not call it fatal because the effective-metric result is analytic and the p=0 curves look qualitatively like GR, but the paper as written is not reproducible. The typo in Eq. (5.17) is minor but telling: it lists p = 10^-2 and 2x10^-2 while the rest of the paper uses 10^-3 and 2x10^-3. That should be fixed. The universal relations are empirical quadratic fits with ~5% scatter, and the claim that they can \"discriminate\" GR from DHOST is overstated given the scatter and the absence of any statistical analysis. Also, because axial modes do not couple to the fluid, the observational impact is limited compared to polar modes; the paper notes this only in passing.\n\nThe citation pattern looks fine, with appropriate reference to prior Horndeski and nonminimal-derivative-coupling work. This is not a case of inflated novelty: the specific subclass and the neutron-star application are new, and the effective-metric extension is a real step.\n\nFor a reader in modified-gravity neutron-star phenomenology, this is worth engaging with. The analytic part will survive, and the numerical part can be checked if the authors release code or data. I would send it to peer review, but I would ask for a GR-limit validation and the typo fix before acceptance.","headline":"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.","tokens_in":19902,"tokens_out":1079,"would_cite":true,"duration_ms":14530,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C55","83C35","83D05"],"pacs":["04.40.Dg","04.30.-w","04.50.Kd"],"model":"deepseek-v4-flash","headline":"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…","keywords":["DHOST theories","axial perturbations","neutron stars","quasinormal modes","universal relations","Regge-Wheeler equation","effective metric","modified gravity"],"falsifier":"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.","tokens_in":18884,"feed_emoji":"🌊","tokens_out":13600,"duration_ms":132170,"temperature":0.7,"pith_summary":"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.","feed_headline":"DHOST gravity shifts neutron-star ringdown modes from GR","feed_subtitle":"The l=2 frequency and damping time shift with p by more than equation-of-state scatter.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It supplies the degeneracy conditions that define DHOST theories and guarantee a single propagating scalar degree of freedom.","marker":"[1]"},{"why":"It provides the equilibrium neutron-star configurations and generalized TOV solutions on which all perturbation calculations here are built.","marker":"[6]"},{"why":"It establishes the effective-metric correspondence for axial black-hole perturbations in DHOST gravity, which the paper extends to neutron stars.","marker":"[19]"},{"why":"It shows that the subclass with $A_1 = A_2 = 0$ has gravitational-wave speed equal to light, tying the model to observational constraints.","marker":"[23]"},{"why":"It provides the perfect-fluid variational principle used to write the matter action and its axial perturbations.","marker":"[25]"},{"why":"It introduces the Regge-Wheeler gauge and the axial perturbation framework that the master equation generalizes.","marker":"[30]"},{"why":"It supplies the analytic parametrization of the SLy and FPS equations of state used in the numerical analysis.","marker":"[33]"},{"why":"It supplies the analytic parametrization of the BSk19, BSk20, and BSk21 equations of state used in the numerical analysis.","marker":"[34]"},{"why":"It provides the complex-coordinate numerical method for imposing outgoing-wave boundary conditions and finding quasinormal modes.","marker":"[35]"},{"why":"It gives the universal-relations framework for quasinormal modes in modified gravity that the paper uses as a comparison for its fits.","marker":"[37]"}],"fun_headline_variants":["DHOST ringdowns: frequency and damping deviate from GR","Neutron-star axial QNMs in DHOST show GR deviations","DHOST gravity imprints on neutron-star ringdown modes","Axial perturbations in DHOST: neutron-star modes shift"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["DHOST ringdowns: frequency and damping deviate from GR","Neutron-star axial QNMs in DHOST show GR deviations","DHOST gravity imprints on neutron-star ringdown modes","Axial perturbations in DHOST: neutron-star modes shift"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000412,"raw_usage":{"total_tokens":2113,"prompt_tokens":906,"completion_tokens":1207,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":1136}},"tokens_in":522,"tokens_out":1207,"duration_ms":13842,"temperature":1.0,"reasoning_tokens":1136,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:22:03.537891+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the degeneracy conditions that define DHOST theories and guarantee a single propagating scalar degree of freedom."},{"cited_title":"Second-order scalar-tensor field equations in a four-dimensional space,","cited_arxiv_id":null,"evidence_quote":"It shows that the subclass with $A_1 = A_2 = 0$ has gravitational-wave speed equal to light, tying the model to observational constraints."},{"cited_title":"Variational aspects of relativistic field theories, with application to perfect fluids,","cited_arxiv_id":null,"evidence_quote":"It introduces the Regge-Wheeler gauge and the axial perturbation framework that the master equation generalizes."},{"cited_title":"Stability of a Schwarzschild singularity,","cited_arxiv_id":null,"evidence_quote":"It supplies the analytic parametrization of the SLy and FPS equations of state used in the numerical analysis."}],"review_version":1}