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arxiv: 2605.01737 · v1 · submitted 2026-05-03 · 🌀 gr-qc

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Axial w-modes of anisotropic neutron stars

Sushovan Mondal
Authors on Pith no claims yet

Pith reviewed 2026-05-09 17:18 UTC · model grok-4.3

classification 🌀 gr-qc
keywords axial w-modesanisotropic neutron starscompactnessdamping timesequations of stategravitational wavespressure anisotropyperturbation equations
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The pith

Axial w-mode frequencies of anisotropic neutron stars depend approximately linearly on compactness, with anisotropy modifying the slope and intercept.

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

The paper calculates the frequencies and damping times of axial w-modes for neutron stars whose radial and tangential pressures differ. Models are built from the BSk21 and SLy4 equations of state together with the Horvat and Bowers-Liang prescriptions for anisotropy. The linearized perturbation equations are solved with a continued-fraction technique, revealing that frequency falls steadily with increasing mass along each stable sequence. The frequency shows a roughly linear rise with stellar compactness, and both the slope and intercept shift when the strength or type of anisotropy changes. Empirical fitting formulas are supplied that express frequency and damping time directly in terms of compactness and anisotropy parameters.

Core claim

The axial w-modes of anisotropic neutron stars exhibit an approximately linear dependence of their frequencies on stellar compactness. For fixed anisotropy strength the frequency decreases monotonically with mass along the stable branch. Configurations with dominant radial pressure have higher frequencies than those with dominant tangential pressure at low mass, but the ordering reverses near the upper end of the stable branch. Damping times increase with mass and become shorter when tangential pressure dominates at fixed mass. The Bowers-Liang ansatz produces a wider spread in both frequencies and damping times than the Horvat ansatz. Empirical expressions are given for frequency and for ln

What carries the argument

The continued-fraction method for solving the linearized axial perturbation equations on static anisotropic neutron-star backgrounds constructed with the Horvat and Bowers-Liang pressure-anisotropy ansatzes.

If this is right

  • For any fixed compactness the axial w-mode frequency can be estimated from the supplied empirical fit once the anisotropy parameter is specified.
  • Damping times shorten when tangential pressure exceeds radial pressure at the same mass, with the effect strongest near the maximum stable mass.
  • The Bowers-Liang ansatz generates a broader range of frequencies and longer damping times than the Horvat ansatz at equal compactness.
  • At low masses, stars with radial pressure larger than tangential pressure oscillate at higher frequencies, while the reverse holds near the maximum mass.
  • The sensitivity of damping time to anisotropy grows markedly for more compact configurations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If axial w-modes are detected in gravitational waves from neutron-star events, the reported relations could be inverted to constrain the level of pressure anisotropy inside the star.
  • The dominance of compactness in the linear fit suggests that anisotropy enters mainly as a correction term that could be calibrated against future observations.
  • The same modeling approach could be applied to polar w-modes or to rotating anisotropic stars to check whether the linear compactness trend persists.
  • Direct comparison of these empirical formulas against full numerical evolutions of oscillating anisotropic stars would test the accuracy of the continued-fraction results.

Load-bearing premise

The chosen functional forms for pressure anisotropy together with the BSk21 and SLy4 equations of state produce stable stellar configurations whose axial w-mode spectra are reliably obtained by the continued-fraction technique.

What would settle it

A numerical computation or gravitational-wave observation of an axial w-mode frequency for a known compactness and anisotropy strength that lies significantly off the reported linear frequency-compactness line would falsify the central relation.

Figures

Figures reproduced from arXiv: 2605.01737 by Sushovan Mondal.

Figure 1
Figure 1. Figure 1: FIG. 1: Mass–radius relations for anisotropic neutron-star configurations constructed using the BSk21 and SLy4 view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Frequency-compactness and damping time-compactness relations of the fundamental axial view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Frequency-mass relations of axial view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Damping time-mass relations of axial view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Frequency-compactness relations of axial view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Damping time-compactness relations of axial view at source ↗
read the original abstract

We investigate the axial $w$-mode oscillations of anisotropic neutron stars. Stellar configurations are constructed using two realistic equations of state, BSk21 and SLy4, with two prescriptions for pressure anisotropy, the Horvat ansatz and the Bowers-Liang ansatz. The axial $w$-mode frequencies are computed by solving the linearized perturbation equations using a continued-fraction method. For each fixed anisotropy strength, the axial $w$-mode frequency decreases monotonically with increasing stellar mass along the stable branch, with its magnitude depending on both the equation of state and the nature of the anisotropy. At low stellar masses, configurations with dominant radial pressure ($p_r>p_t$) exhibit higher frequencies than those with dominant tangential pressure, whereas toward the upper end of the stable branch this ordering is reversed, and configurations with $p_t>p_r$ attain higher frequencies at the same mass. The axial $w$-mode frequency displays an approximately linear dependence on compactness, with anisotropy modifying both the slope and the intercept. The Bowers-Liang ansatz produces a wider spread in the frequency values compared to the Horvat ansatz. We also analyze the damping times associated with the axial $w$-modes and find that they increase with stellar mass, with a rapid rise toward the upper end of the stable branch. At a fixed mass, increasing the tangential pressure relative to the radial pressure leads to shorter damping times, while configurations with dominant radial pressure exhibit longer damping times. The sensitivity of the damping time to anisotropy is more pronounced for more compact stars, and the Bowers-Liang ansatz yields systematically larger damping times than the Horvat ansatz. Finally, we provide empirical expressions for the axial $w$-mode frequency and damping time as functions of stellar compactness and anisotropy strength.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript investigates axial w-modes of anisotropic neutron stars constructed with BSk21 and SLy4 equations of state and Horvat/Bowers-Liang anisotropy prescriptions. It solves the linearized axial perturbation equations via the continued-fraction method, reporting that w-mode frequency decreases monotonically with mass on the stable branch, exhibits approximate linear dependence on compactness (with anisotropy modifying slope and intercept), and provides empirical fitting expressions for frequency and damping time as functions of compactness and anisotropy strength. Damping times increase with mass and show greater sensitivity to anisotropy at higher compactness, with Bowers-Liang yielding larger spreads than Horvat.

Significance. If the numerical trends hold, the work extends asteroseismology to anisotropic neutron-star models, potentially aiding interpretation of gravitational-wave signals from oscillating or merging compact objects. The use of realistic EOS, standard methods, and provision of empirical relations are positive features, though the results remain specific to the chosen ansatzes and may require further validation for broader applicability.

major comments (2)
  1. The empirical fitting expressions for frequency and damping time (described in the abstract and results) are constructed from the computed data points; the manuscript should detail the functional form chosen, the fitting procedure, the exact ranges of compactness and anisotropy strength, and quantitative fit metrics (e.g., residuals or R²) to substantiate their utility beyond interpolation.
  2. Application of the continued-fraction method is load-bearing for all reported frequencies, damping times, and linearity claims, yet the text provides limited information on convergence criteria, truncation errors, or direct validation against the isotropic (zero-anisotropy) limit; explicit error budgets or comparison tables would be required to confirm the monotonic trends and ordering reversals with anisotropy.
minor comments (3)
  1. The abstract could briefly indicate the range of anisotropy parameters explored to contextualize the reported ordering reversals between radial- and tangential-dominant cases.
  2. Figures displaying frequency versus mass or compactness should label the stable branch explicitly and, if possible, indicate numerical resolution or precision.
  3. Ensure the introduction cites key prior isotropic w-mode studies to clearly frame the extension to anisotropic configurations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and constructive feedback on our manuscript. The suggestions will help strengthen the presentation of the numerical results and empirical relations. We address each major comment below.

read point-by-point responses

  1. Referee: The empirical fitting expressions for frequency and damping time (described in the abstract and results) are constructed from the computed data points; the manuscript should detail the functional form chosen, the fitting procedure, the exact ranges of compactness and anisotropy strength, and quantitative fit metrics (e.g., residuals or R²) to substantiate their utility beyond interpolation.

    Authors: We agree that additional details on the empirical fits are warranted to allow readers to assess their robustness. In the revised manuscript we will add a dedicated paragraph (or short subsection) in the results section that specifies: (i) the exact functional forms adopted (linear in compactness with additive and multiplicative terms linear in the anisotropy parameter), (ii) the least-squares fitting procedure, (iii) the precise ranges of compactness (approximately 0.10–0.32) and anisotropy strengths explored for each EOS and ansatz, and (iv) quantitative goodness-of-fit measures including R² values and maximum relative residuals. These additions will be placed immediately after the presentation of the fitting expressions. revision: yes

  2. Referee: Application of the continued-fraction method is load-bearing for all reported frequencies, damping times, and linearity claims, yet the text provides limited information on convergence criteria, truncation errors, or direct validation against the isotropic (zero-anisotropy) limit; explicit error budgets or comparison tables would be required to confirm the monotonic trends and ordering reversals with anisotropy.

    Authors: We acknowledge that the current description of the continued-fraction implementation is concise and would benefit from more explicit numerical validation. In the revised version we will expand the methods section to include: (i) the convergence criteria employed (number of continued-fraction terms and tolerance on the frequency), (ii) estimates of truncation error obtained by increasing the truncation order, and (iii) a direct comparison table (or figure) of our zero-anisotropy results against published isotropic w-mode frequencies for BSk21 and SLy4. This will provide an explicit error budget supporting the reported monotonic trends and anisotropy-induced ordering reversals. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs stellar models from two standard EOS (BSk21, SLy4) and two established anisotropy ansatzes (Horvat, Bowers-Liang), then solves the linearized axial perturbation equations via the continued-fraction method to obtain w-mode frequencies and damping times. All reported trends (monotonic decrease with mass, approximate linearity with compactness, ordering reversals, sensitivity to anisotropy) and differences between ansatzes are direct numerical outputs of this procedure. The empirical expressions are explicitly described as fits to the computed data rather than first-principles derivations or predictions, and the central claims do not depend on these fits for their validity. No step reduces by construction to its inputs, no self-citation is load-bearing, and the numerical chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central results rest on standard general-relativistic stellar-structure equations modified by phenomenological anisotropy, plus the assumption that the continued-fraction technique converges for the chosen boundary conditions; no new particles or forces are introduced.

free parameters (1)
  • anisotropy strength parameter
    Fixed value chosen independently for each model run to scan different levels of radial versus tangential pressure difference.
axioms (2)
  • domain assumption The background metric and hydrostatic equilibrium are described by the anisotropic Tolman-Oppenheimer-Volkoff equations.
    Invoked to construct the unperturbed stellar models before linearizing the axial perturbations.
  • domain assumption The continued-fraction method accurately solves the linearized axial perturbation equations with appropriate boundary conditions at the center and at infinity.
    Standard technique in the field; its validity is assumed rather than re-derived.

pith-pipeline@v0.9.0 · 5611 in / 1491 out tokens · 56181 ms · 2026-05-09T17:18:19.205412+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On the non-radial oscillations of realistic anisotropic neutron stars: Axial modes

    gr-qc 2026-05 unverdicted novelty 5.0

    Axial modes of anisotropic neutron stars show mass-scaled frequency and damping time with nearly universal quadratic dependence on compactness, insensitive to EOS and anisotropy model.

Reference graph

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