Axion effects on the non-radial oscillations of neutron stars

Deepak Kumar; Hiranmaya Mishra

REVIEW 3 major objections 6 minor 9 references

Axion effects on the non-radial oscillations of neutron stars

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that axionic quark matter enhances the f-mode frequencies of hybrid neutron stars beyond canonical nucleonic predictions, making the mode a probe of non-nucleonic degrees of freedom.

desk verdict The hyperonic extension is a real but incremental step; the axion effect is an assumed θ up to π, not a computed axion potential, so the central prediction is conditional. read the letter →

arxiv 2505.03142 v1 pith:ZN3MUFRS submitted 2025-05-06 hep-ph astro-ph.HE

Deepak Kumar , Hiranmaya Mishra This is my paper · ORCID

classification hep-phastro-ph.HE

keywords axionneutronstaroscillationsf-modehybridNambu–Jona-Lasiniomodelhadron-quarkphasetransitionequationofstateCPviolation

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

This paper argues that axions can leave a measurable imprint in neutron star oscillations. Combining a hadronic equation of state with quark matter described by a three-flavor NJL model that includes a CP-violating axion angle $\theta$, the authors construct hybrid stars with a mixed hadron-quark phase. They claim that the axion lowers the density at which quark matter appears, stabilizes hybrid stars against astrophysical mass constraints, and enhances the quadrupolar f-mode frequencies compared with canonical nucleonic stars. If correct, a detected f-mode frequency noticeably above nucleonic predictions would be evidence for non-nucleonic degrees of freedom in the star's core.

What carries the argument

The load-bearing object is the three-flavor NJL Lagrangian with the Kobayashi-Maskawa-'t Hooft determinant term multiplied by $e^{\pm i\theta}$, which lets an axion-induced CP-violating angle shift the quark-matter equation of state. Around it sits a two-layer construction: the relativistic mean-field model supplies the hadronic, hyperon-rich equation of state, and the Gibbs condition joins the hadronic and quark phases into a hybrid-star equation of state. The non-radial oscillation frequencies follow from solving the perturbed Einstein equations for quadrupolar modes, with the Brunt-Väisälä frequency entering through the sound-speed profile. The mechanism that produces the claimed signature is that increasing $\theta$ lowers the density threshold for quark-matter appearance, enlarging the quark core and changing the f-mode frequency.

What would settle it

A single neutron star with a well-measured mass, radius, and f-mode frequency whose measured frequency falls on the canonical nucleonic curve rather than the $\theta>0$ hybrid curves would falsify the claimed enhancement. Equivalently, a tight laboratory or astrophysical bound showing that the effective $\theta$ inside dense matter must be far below $\pi$ would remove the mechanism's quantitative basis.

Watch

Extended reading notes

Core claim

The central claim is that in hybrid neutron stars, the fundamental quadrupolar oscillation (f-mode) frequency is enhanced both by hyperons and by quark matter, and the enhancement is largest when axionic effects are present. The paper models axion effects as a vacuum angle $\theta$ in the flavor-determinant interaction of the three-flavor NJL Lagrangian; at maximum CP violation, $\theta=\pi$, the mixed phase begins at lower density (about $1.9n_0$) than without axions, producing a larger quark core and, once vector repulsion is included, a stiffer overall equation of state. The mass-radius curves satisfy modern astrophysical constraints except for the $\theta=\pi$, zero-vector-coupling case, which fails the maximum-mass bound. The authors conclude that a measured f-mode frequency noticeably above nucleonic-only predictions would indicate quark or hyperonic content in neutron star matter.

Load-bearing premise

The whole enhancement rests on treating the axion effect as a fixed vacuum angle that can be as large as $\theta=\pi$ inside the star; if the strong CP bound or the dynamics of the axion field keep the effective angle minuscule at stellar densities, the predicted f-mode boost disappears.

Editorial extensions

If this is right

Measured f-mode frequencies higher than nucleonic-only predictions become a direct diagnostic for non-nucleonic degrees of freedom rather than just an equation-of-state curiosity.
Gravitational-wave observations of f-modes, for example from post-merger remnants, could place constraints on the axion parameter $\theta$ if the equation of state is otherwise known.
Vector repulsion in quark matter is necessary for the axion-enhanced hybrid stars to satisfy maximum-mass constraints; without it, the $\theta=\pi$ case is ruled out by observations.
Because axions lower the onset density of the hadron-quark mixed phase, precision oscillation measurements could map where the phase transition happens inside the star.

Reading between the lines

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

A dynamical-axion treatment, where $\theta$ evolves with density rather than staying fixed, would be the natural next step; the present constant-$\theta$ scan likely overstates the effect if the axion relaxes toward zero inside the star.
The same $\theta$-enhanced quark matter should also alter tidal deformability and cooling; those independent observables could be cross-checked against the f-mode claim.
The abrupt sound-speed discontinuities at the mixed-phase boundaries would imprint on the gravitational-wave ringdown spectrum, potentially letting detectors hear the phase-transition structure alongside the frequency shift.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing to know: the central qualitative claim—axions enhance f-modes in hybrid stars—was already in the authors' own earlier preprint (arXiv:2411.17828). What's actually new here is the hyperonic hadronic sector and a more complete hybrid star description with a scan over the vector coupling Gv. That's a legitimate but incremental step, not a new phenomenon.

The paper does some things well. The numerical chain is standard: RMF for hadrons, NJL for quark matter, Gibbs construction for the mixed phase, TOV for structure, and the linearized oscillation equations for f-modes. The mass-radius curves and f-mode frequencies behave as expected—hyperons soften, Gv stiffens, and the θ=π, Gv=0.1Gs case satisfies the maximum-mass constraint. The problem is the presentation: Eqs. (2)–(3) are garbled, there are typos throughout, and there are no numerical tables, derivation details, or code, so the numbers can't be checked directly.

The soft spot is the axion sector, and it is load-bearing. The paper treats θ as a free parameter and scans up to π, calling π the 'maximum CP violation effect.' But for a QCD axion, θ is a dynamical field, not a constant. Its expectation value minimizes the full thermodynamic potential, and in vacuum the strong CP bound forces |θ|≲10^-10. Dense matter can in principle shift that minimum, but the paper never computes the axion potential—it just varies θ by hand. So the f-mode enhancement is a sensitivity to an assumed large CP-violating angle, not a demonstrated consequence of axions. The stress-test note is right about this. The authors cite their previous paper for the full derivation, but this manuscript is not self-contained on the point that matters most. If the dense-matter axion potential really does relax to near θ=π, the claim would be interesting; without computing it, the prediction is conditional on an unexamined premise.

The novelty question also needs addressing. The title is nearly identical to the earlier preprint, and the abstract doesn't say what is new beyond that work. A referee should ask for a clear statement of the new contribution, not just the hyperon extension.

Who is this for: people working on hybrid star phenomenology who want a parameter study of how f-modes shift with θ, hyperons, and vector coupling. It is a useful data point, but not a first-principles axion prediction.

Recommendation: send it to peer review, but conditional on revisions. Require a justification of the θ range—ideally a computation of the axion effective potential in dense matter—plus a statement of novelty and either clean equations or a public dataset. I wouldn't desk-reject, but I also wouldn't accept as-is.

Referee Report

3 major / 6 minor

Summary. The manuscript studies the effect of axions on the equation of state (EOS) of quark matter in a three-flavor Nambu–Jona-Lasinio (NJL) model with a theta-term in the 't Hooft determinant, matched to a relativistic mean field (RMF) hadronic EOS via the Gibbs construction. The resulting hybrid EOS is used to solve the TOV equations and the linearized non-radial oscillation equations to obtain neutron star mass-radius relations and quadrupolar f-mode frequencies. The central claim is that a non-zero theta parameter (representing axions) lowers the hadron-quark phase transition density, stabilizes hybrid neutron stars, and substantially enhances f-mode frequencies relative to canonical nucleonic stars, especially when combined with hyperons.

Significance. If the claimed enhancement is physically realized, the result would offer an observable gravitational-wave signature of non-nucleonic degrees of freedom (hyperons and quark matter) and of a CP-violating axion background inside neutron star cores. The paper builds on established machinery (NJL and RMF EOS, Gibbs construction, TOV equations, fluid perturbation theory) and presents parameter-sensitivity plots rather than a first-principles derivation. The significance is strongly tempered by the fact that the axion is modeled as a fixed, externally prescribed vacuum angle without computing the axion field dynamics; the headline prediction is therefore a parameter-sensitivity statement rather than a demonstrated consequence of axions. The manuscript also provides no numerical tables and its central oscillation equations are garbled as typeset, hindering reproducibility and verification.

major comments (3)

[Section 2, Eq. (4)] The axion effect is implemented solely as a static angle theta inserted in the 't Hooft determinant, scanned up to theta = pi ('maximum CP violation effect at theta = pi'). For a QCD axion, theta = a/f_a is a dynamical field whose expectation value minimizes the total thermodynamic potential including the axion potential; in vacuum the strong CP bound requires |theta| less than about 10^-10, and at neutron star core densities the paper does not compute the effective theta. Consequently, the statement that the f-mode enhancement is 'particularly large in the presence of axions' is not established: the enhancement is produced by an externally imposed large theta, not by a demonstrated axion mechanism. The authors should either compute the density-dependent effective theta from an axion potential or explicitly reframe the paper as a study of an arbitrary CP-violating angle and discuss the conditions under which axions could realize such a value.
[Eqs. (2)-(3)] The linearized pulsation equations are garbled as typeset, with misplaced superscripts, exponents, and fractions (e.g., 'omega^2 r^2 e^{Lambda - 2 Phi} Z + Phi' Q' and the expression containing 'Phi' e^{-Lambda + 2 Phi} Q / (omega^2 r^2)'). As printed they are neither readable nor verifiable, and these equations are central to the computed f-mode frequencies. The paper should display the oscillation equations in a correct, standard form and define all symbols (Phi, Lambda, Q, Z, omega_BV, c_s^2) with consistent notation.
[Figure 2 and Section 2] The claimed f-mode enhancement is presented as a single family of curves for one choice of hadronic and quark parameters. The paper does not demonstrate robustness of the enhancement to variations of other NJL couplings (G_s, K), the RMF coupling set, or the matching procedure. Given that the EOS and hence the f-mode frequency depend on several parameters that are only partially varied (theta and G_v), the paper should include at least a sensitivity check with respect to another parameter (e.g., G_s or K) or provide a quantitative discussion of the parameter dependence, otherwise the enhancement may be a fine-tuned result rather than a robust prediction.

minor comments (6)

[Abstract and throughout] There are numerous typographical errors, including 'on on' in the abstract, 'frquencies', 'e ffects', 'V olkoff', 'Nambu--Jona-Lasino' (inconsistent spelling), 'charge neural and beta equlibriataed', and 'frequncies' in the conclusions; the manuscript should be carefully proofread.
[Eq. (1)] The TOV equations are written in a single line with 'dp/dr = ... , dm/dr = ...'; these should be displayed as two separate equations for clarity.
[Section 1, after Eq. (3)] The symbols Phi, Lambda, and c_s^2 that appear in the oscillation equations are not defined in the text; their definitions (relating to the equilibrium metric and the adiabatic sound speed) should be provided.
[Section 2, Fig. 2] The text says 'when theta = pi and G_v = 0.1 gives larger enhancement' but the sentence is grammatically incomplete; it should specify that this case gives the largest enhancement among the models that satisfy the astrophysical constraints.
[Section 2, Fig. 1] The figure caption refers to 'axion parameter theta' but the figure legend and text do not state the values of theta and G_v clearly for each curve; a table or explicit legend would improve readability.
[Introduction] The phrase 'second densest object in the universe after a black hole' is imprecise, as black holes are not characterized by a density in the same way as neutron stars; the sentence should be rephrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the f-mode enhancement is a computed consequence of the stated theta-dependent EOS, not a fitted or self-defined prediction.

full rationale

Walking the derivation chain, Eq. (4) defines the NJL model with an explicit theta-dependent 't Hooft determinant; the EOS is obtained by mean-field thermodynamics, the mixed phase by Gibbs construction, and the f-modes by solving the TOV and linearized pulsation equations (Eqs. (1)-(3)). The f-mode frequencies in Fig. 2 are outputs of this chain; no observed f-mode datum is used to fix theta, G_v, or any EOS parameter. The choice theta = pi, G_v = 0.1 G_s is selected to satisfy maximum-mass constraints, and the f-mode enhancement is then computed, which is a parameter-sensitivity statement rather than a fit relabeled as a prediction. The self-citations [1] and [2] are to the authors' own earlier work, but [2] supplies the standard pulsation equations and [1] is cited for the occurrence of the hadron-quark phase transition; neither is a uniqueness theorem or an unverified premise that by itself forces the f-mode conclusion. Although the title overlaps with the authors' [1], the present paper's quantitative f-mode results are not obtained by importing its conclusion as an input. The physical concern that theta up to pi is imposed without a dynamical axion potential or the strong-CP bound is a model-validity risk, not a circular reduction, because the paper's equations explicitly contain theta as an input and the f-mode frequencies do not reduce to that input by construction. No step satisfies the required standard of exhibiting Eq. X = Eq. Y by definition or a fitted parameter renamed as a prediction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper's results rest on a chain of model assumptions taken from the authors' prior work: the NJL EOS with θ, the RMF hyperonic EOS, the Gibbs construction, and the standard pulsation equations. The only parameters varied by hand are θ and Gv; all other couplings are from earlier literature. No new entities are introduced. The main concern is that the θ dependence is imposed rather than derived from axion field dynamics.

free parameters (2)

θ (axion parameter) = varied from 0 to π (e.g., θ=π for maximum CP violation)
Chosen by hand to explore strong CP violation effects; the physical value of θ is not derived from axion dynamics.
Gv (vector coupling in NJL model) = 0 or 0.1 Gs
Chosen by hand; increasing Gv stiffens the EOS to satisfy maximum mass constraints.

assumptions (4)

domain assumption The three-flavor NJL Lagrangian (Eq. 4) with the given four-quark, 't Hooft determinant, and vector couplings describes quark matter EOS in mean-field approximation.
This is a standard model choice but its quantitative reliability at several times nuclear saturation density is not established.
domain assumption The Gibbs construction with equal pressures and chemical potentials correctly describes the hadron-quark mixed phase.
Standard for hybrid star EOS but known to be one of several possible constructions.
standard math The perturbed Einstein equations (Eqs. 2-3) with the given boundary conditions correctly give the f-mode frequencies.
Taken from prior literature; the paper does not derive them.
ad hoc to paper The axion effect can be represented by a static, spatially uniform θ angle in the NJL model, independent of the axion field dynamics.
The paper uses θ as a free knob without deriving it from the axion potential; this is the main physical approximation.

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Cite this review

Pith. "Pith review of Axion effects on the non-radial oscillations of neutron stars." pith.science (2026). https://pith.science/paper/ZN3MUFRS

@misc{pith2026250503142,
  author       = {Pith},
  title        = {Pith review of: Axion effects on the non-radial oscillations of neutron stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZN3MUFRS}},
  note         = {Machine review of arXiv:2505.03142}
}

abstract

The effects of axions on quark matter equation of state (EOS) is studied within the three flavor Nambu--Jona-Lasinio model and its effects on on the non-radial oscillations of neutron stars is investigated. Using such an EOS for quark matter with axions and a EOS for hadronic matter within the relativistic mean field (RMF) theory, we discuss the hadron-quark phase transition (HQPT) using the Gibbs construction. The EOS so obtained is used to investigate the structure of hybrid neutron star (NS)s. It is found that the presence of axions in the core of compact stars stabilizes hybrid NSs in agreement with modern astrophysical constraints. It is further observed that the quadrupolar fundamental modes ($f$-modes) for such hybrid NSs get substantial enhancements both due to a larger quark core in the presence of axions and from the hyperons as compared to a canonical nucleonic neutron stars.

Figures

Figures reproduced from arXiv: 2505.03142 by the authors.

**Figure 1.** Equation of state of neutron/hybrid stars (left) and the variation of the equilibrium speed of sound in the charge neutral matter as a function of baryon number density (right) with axion parameter θ and vector coupling in the NJL model, Gv. 2. Results and discussions In [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Left:Mass-radius relation for various cases. Right: the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 5 linked inside Pith

[1]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...
[2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...
[3]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...
[4]

Kumar , author H

author D. Kumar , author H. Mishra , title CP violation in cold dense quark matter and axion effects on the non-radial oscillations of neutron stars ( year 2024 ). http://arxiv.org/abs/2411.17828 arXiv:2411.17828

arXiv 2024
[5]

Kumar , author H

author D. Kumar , author H. Mishra , author T. Malik , title Non-radial oscillation modes in hybrid stars: consequences of a mixed phase , journal JCAP volume 02 ( year 2023 ) pages 015 . :10.1088/1475-7516/2023/02/015. http://arxiv.org/abs/2110.00324 arXiv:2110.00324

arXiv 2023
[6]

Sakai , author H

author Y. Sakai , author H. Kouno , author T. Sasaki , author M. Yahiro , title Theta vacuum effects on QCD phase diagram , journal Phys. Lett. B volume 705 ( year 2011 ) pages 349--355 . :10.1016/j.physletb.2011.10.032. http://arxiv.org/abs/1105.0413 arXiv:1105.0413

arXiv 2011
[7]

Chatterjee , author H

author B. Chatterjee , author H. Mishra , author A. Mishra , title CP violation and chiral symmetry breaking in hot and dense quark matter in the presence of a magnetic field , journal Phys. Rev. D volume 91 ( year 2015 ) pages 034031 . :10.1103/PhysRevD.91.034031. http://arxiv.org/abs/1409.3454 arXiv:1409.3454

arXiv 2015
[8]

Mishra , author P

author A. Mishra , author P. K. Panda , author W. Greiner , title Vacuum polarization effects in hyperon rich dense matter: A Nonperturbative treatment , journal J. Phys. G volume 28 ( year 2002 ) pages 67--83 . :10.1088/0954-3899/28/1/305. http://arxiv.org/abs/nucl-th/0101006 arXiv:nucl-th/0101006

arXiv 2002

Show all 9 references

[9]

Leslie Lamport, : a document preparation system, Addison Wesley, Massachusetts, 2nd edition, 1994

1994

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...

[3] [3]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...

[4] [4]

Kumar , author H

author D. Kumar , author H. Mishra , title CP violation in cold dense quark matter and axion effects on the non-radial oscillations of neutron stars ( year 2024 ). http://arxiv.org/abs/2411.17828 arXiv:2411.17828

arXiv 2024

[5] [5]

Kumar , author H

author D. Kumar , author H. Mishra , author T. Malik , title Non-radial oscillation modes in hybrid stars: consequences of a mixed phase , journal JCAP volume 02 ( year 2023 ) pages 015 . :10.1088/1475-7516/2023/02/015. http://arxiv.org/abs/2110.00324 arXiv:2110.00324

arXiv 2023

[6] [6]

Sakai , author H

author Y. Sakai , author H. Kouno , author T. Sasaki , author M. Yahiro , title Theta vacuum effects on QCD phase diagram , journal Phys. Lett. B volume 705 ( year 2011 ) pages 349--355 . :10.1016/j.physletb.2011.10.032. http://arxiv.org/abs/1105.0413 arXiv:1105.0413

arXiv 2011

[7] [7]

Chatterjee , author H

author B. Chatterjee , author H. Mishra , author A. Mishra , title CP violation and chiral symmetry breaking in hot and dense quark matter in the presence of a magnetic field , journal Phys. Rev. D volume 91 ( year 2015 ) pages 034031 . :10.1103/PhysRevD.91.034031. http://arxiv.org/abs/1409.3454 arXiv:1409.3454

arXiv 2015

[8] [8]

Mishra , author P

author A. Mishra , author P. K. Panda , author W. Greiner , title Vacuum polarization effects in hyperon rich dense matter: A Nonperturbative treatment , journal J. Phys. G volume 28 ( year 2002 ) pages 67--83 . :10.1088/0954-3899/28/1/305. http://arxiv.org/abs/nucl-th/0101006 arXiv:nucl-th/0101006

arXiv 2002