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Axisymmetric stability of neutron stars as extreme rotators in massive scalar-tensor theory

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Pith's one-line read Differentially rotating scalarized neutron stars, which can carry an angular momentum roughly an order of magnitude above the general-relativistic limit, become axisymmetrically unstable at, within numerical error, the turning point of…

desk verdict First nonlinear test of the turning-point criterion for ultra-rotating scalarized neutron stars; the result is plausible, but the stable/unstable boundary rests on a short, noise-seeded evolution window. read the letter →

arxiv 2502.03973 v1 pith:DXGJVWFO submitted 2025-02-06 gr-qc astro-ph.HE

Alan Tsz-Lok Lam , Kalin V. Staykov , Hao-Jui Kuan , Daniela D. Doneva , Stoytcho S. Yazadjiev This is my paper · ORCID
classification gr-qcastro-ph.HE PACS 04.25.D04.40.Dg04.50.Kd97.60.Jd
keywords scalar-tensorgravityneutronstarsdifferentialrotationaxisymmetricstabilityturning-pointcriterionnumericalrelativityscalarizationhypermassive
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 asks whether scalarized neutron stars that spin far faster than any general-relativistic neutron star—differentially rotating objects built to mimic binary-merger remnants—are dynamically stable. It answers by generating sequences of such equilibria and evolving them with a fully relativistic axisymmetric code, relying on the small numerical perturbation present in the simulation to excite any unstable axisymmetric mode. The central finding is that the turning-point criterion survives: the onset of instability coincides, within numerical error, with the maximum of gravitational mass along a fixed-angular-momentum sequence, i.e. $(\partial M_G/\partial\epsilon_c)|_J = 0$. Models below that point relax back to the initial profile; models above it collapse to a black hole in a few milliseconds; sequences that never reach a turning point are stable throughout. Because these ultra-rotators have no counterparts in general relativity, the result means the most extreme scalarized remnants could actually form in a binary coalescence, at least briefly.

What carries the argument

The load-bearing object is the turning-point criterion for axisymmetric stability: for a one-parameter family of equilibria at fixed angular momentum $J$, the configuration where the gravitational mass $M_G$ is extremal as a function of central energy density, $(\partial M_G/\partial\epsilon_c)|_J = 0$, marks where stability changes. Around this sits three-layered machinery: an equilibrium solver that generates scalarized, differentially rotating stars with the four-parameter Uryu rotation law (here $\lambda_1 = 1.5$, $\lambda_2 = 0.5$, giving quasi-toroidal shapes); an axisymmetric BSSN/Z4c evolution code that uses the cartoon method and fixed mesh refinement to follow nonlinear collapse; and Fourier analysis of central density and scalar-field oscillations that extracts the axisymmetric mode frequencies. The criterion does the cheap predictive work, and the evolutions test it against full nonlinear collapse.

What would settle it

Evolve the massive $m_\phi = 0.01$, $B=12$, $J=8$ model closest to the turning point (the one labeled stable near $\epsilon_c \simeq 1.15\times 10^{15}\,\mathrm{g/cm^3}$) for at least 100 ms with an explicit seeded axisymmetric density perturbation, and compute the complex frequencies of the axisymmetric quasi-normal modes. If the model collapses on a longer timescale, or if an unstable mode exists with growth time longer than the 15 ms window, the claim that the turning-point criterion is sufficient would be falsified.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that a criterion proven for rigidly rotating stars in general relativity—the turning-point theorem—continues to be a sufficient condition for axisymmetric instability for differentially rotating, scalarized stars in DEF scalar-tensor theory. Along fixed-$J$ sequences with scalar mass $m_\phi = 0$ and $m_\phi = 0.01$ (about $1.33\times 10^{-12}$ eV), the marginally stable model sits at the turning point $(\partial M_G/\partial\epsilon_c)|_J = 0$; models slightly below it damp their perturbations within a few milliseconds, models slightly above it collapse to a black hole, and sequences with no turning point are stable at every sampled density. In the massive case, Fourier analysis of the central rest-mass density and scalar field shows the scalar ($\phi$) mode frequency decreasing toward the Yukawa cutoff frequency as the turning point is approached, suggesting that the instability is triggered by the $\phi$-mode reaching the cutoff rather than by the fundamental fluid mode reaching zero frequency, as in general relativity.

Load-bearing premise

The paper's stable/unstable classification assumes that the small random numerical perturbation present in the simulations is enough to excite any existing axisymmetric instability, and that a model which has not collapsed within about 15 ms is dynamically stable; a slower-growing instability near the massive-sequence turning point could therefore be mislabeled as stable.

Editorial extensions

If this is right

  • The turning-point condition $(\partial M_G/\partial\epsilon_c)|_J = 0$ can be used as a cheap predictor of the axisymmetric stability boundary for scalarized differentially rotating stars, without running full nonlinear evolutions.
  • Stable ultra-rotating scalarized neutron stars exist at angular momenta well above the maximum sustainable in general relativity, up to the largest sequences sampled; such objects could be merger remnants that persist for many dynamical timescales.
  • Unstable models collapse to a black hole within a few milliseconds, so a scalarized hypermassive remnant that crosses the turning point would be short-lived and would produce a prompt black-hole signal.
  • In the massive scalar theory the approach to instability is accompanied by the scalar-mode frequency decreasing to the Yukawa cutoff $f_c$, a spectral fingerprint distinct from the usual fluid-mode softening in general relativity.
  • The criterion and the simulations only cover axisymmetric modes; nonaxisymmetric instabilities such as bar and one-arm modes are not excluded by these results.

Reading between the lines

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

  • I infer that the massive-case result suggests a new diagnostic: monitoring the $\phi$-mode frequency of a post-merger remnant and flagging instability as it approaches the Yukawa cutoff $f_c$ would give a cheaper, mode-based stability criterion analogous to the $f$-mode zero-crossing in general relativity.
  • I infer that the 15 ms evolution window is likely too short to certify stability of the near-turning-point massive models, because scalar waves near the cutoff propagate slowly and an unstable mode could grow on a longer timescale; longer evolutions or a linear mode-growth calculation would settle this.
  • I infer that whether real mergers produce these ultra-rotators depends on three-dimensional physics the axisymmetric setup omits, notably bar-mode and one-arm instabilities and gravitational-wave angular-momentum loss, so the paper's stability result is necessary but not sufficient for their astrophysical realization.
  • I infer that the turning-point behavior may be equation-of-state dependent; the paper only uses MPA1, and testing a broader set of nuclear equations of state would show whether the criterion remains sufficient for realistic remnants.
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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

3 major / 5 minor

Summary. The paper investigates the axisymmetric dynamical stability of extremely rapidly and differentially rotating neutron stars in Damour-Esposito-Farèse scalar-tensor theory, motivated by possible binary-neutron-star merger remnants. The authors construct fixed-angular-momentum equilibrium sequences with a modified RNS code, then evolve selected models with the axisymmetric SACRA-2D code extended to DEF theory with Z4c constraint propagation. For massless and massive scalar fields with B=12 and the MPA1 EOS, they classify models as stable or unstable depending on whether a random numerical perturbation relaxes or leads to collapse within about 15 ms. The central claim is that the turning-point criterion, Eq. (3), remains largely valid for these differentially rotating scalarized stars: the onset of instability is found near the mass maximum along constant-J sequences, and unstable models collapse to black holes. The paper also presents a mode-frequency analysis for one massive sequence, suggesting that the instability near the turning point is related to the scalar mode approaching the Yukawa cutoff frequency.

Significance. If the result holds, it is significant: it extends a classic GR stability criterion to a regime—super-rotating scalarized neutron stars—that has no GR counterpart, and it sharpens the astrophysical question of whether binary-merger remnants can be stably supported with enormous angular momenta. The paper is a well-targeted numerical study that directly confronts nonlinear evolution with equilibrium predictions, and it is honest about several limitations, including restriction to axisymmetric perturbations and the premature termination of some equilibrium sequences. The main positive contribution is the construction of stable and unstable branches for ultra-rotating scalarized stars and the demonstration, for the tested cases, that the turning point separates them. However, the finite simulation window and uncontrolled perturbation mean that the boundary between stable and unstable models is not yet established with the precision claimed in the text.

major comments (3)
  1. [Sec. III A and Figs. 1-4] The central claim rests on the binary stable/unstable classification, but the classification is based on a single 'random numerical perturbation' and a simulation time of about 15 ms. Since the growth rate of the unstable mode vanishes at the turning point, a model on the unstable side arbitrarily close to the onset can have a collapse time exceeding the simulation window, and an uncontrolled random perturbation may not project onto the unstable eigenmode. The observation that the last stable model lies slightly left of the mass maximum in Figs. 1(a)-(c) is exactly the signature one would expect if weakly unstable models near the turning point were misclassified, and resolution convergence does not address the physical growth time. Please quantify the perturbation amplitude, report growth or collapse times for models near the onset, and either evolve longer or seed a known unstable mode so that the stable/unstable boundary is not set by the simulation time.
  2. [Sec. III and Sec. IV] The conclusion that Eq. (3) is 'largely valid' for differentially rotating DEF stars is presented as a general statement, but only one EOS (MPA1), one coupling (B=12), one rotational-law shape (lambda1,lambda2)=(1.5,0.5), and two scalar masses are tested. Scalarization and high-angular-momentum behavior are known to be sensitive to these choices, so the current data support a statement restricted to the tested parameter set. Either add cases (for example a second EOS or a different coupling) or explicitly narrow the Abstract and Discussion claims to the parameter space actually simulated.
  3. [Figs. 1 and 5, Sec. IV] The statement that the observed onset of instability 'agrees within the numerical error' is not quantified. No convergence study is shown, no error bar is given for the threshold energy density epsilon_thre in Table I, and the distance between the last stable model and the mass maximum is not measured. The authors should either define the numerical error quantitatively (for example from model bracketing or resolution testing) or soften the claim to an approximate agreement without the phrase 'within the numerical error'.
minor comments (5)
  1. [Abstract and Table I caption] The word 'asymmetric' appears where 'axisymmetric' is meant, both in the Abstract and in the caption of Table I; this should be corrected throughout.
  2. [Sec. I, around Eq. (3)] The text contains the typo 'the tuning-point' instead of 'the turning-point'.
  3. [Sec. II A, Eq. (6)] The typesetting of Eq. (6) appears to be missing parentheses: the intended relation is presumably 1/(omega(phi)+3/2) = B ln phi; please fix the display.
  4. [Fig. 3 caption and axis label] The caption describes the bottom panel as the 'absolute value of central scalar field |phi_c|', while the vertical-axis label and the text refer to |phi(67.7 km)|, the scalar field extracted at a finite radius inside the star; this inconsistency should be resolved.
  5. [Sec. III B, Fig. 6] The mode identification is described as 'speculated to be the quasi-radial m=0 fundamental mode and phi-mode'; since this identification is used to support the instability mechanism, the authors should either give additional evidence for the mode assignment or present it more cautiously as tentative.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the turning-point criterion is an external benchmark from Friedman et al., and the paper's simulations are compared against it rather than fitted to it. Self-citations are methodological background, not load-bearing evidence for the central claim.

full rationale

The central claim is that the turning-point criterion, Eq. (3), remains a sufficient condition for axisymmetric instability for differentially rotating scalarized neutron stars in the DEF theory. The criterion is taken from the external reference Friedman, Ipser, and Sorkin [46], and the paper's numerical evolutions are used to classify models as stable or unstable by observing whether a random numerical perturbation relaxes or leads to collapse. This classification is not derived from the turning-point criterion itself; instead, the criterion is the benchmark being tested. No parameter is fitted to make the observed onset agree with the turning point, and the paper explicitly notes only approximate agreement 'within the numerical error.' The self-citations in the paper are to (i) the modified RNS equilibrium code from earlier works [31,33,34] used to generate initial data, (ii) the SACRA-2D evolution code [78], and (iii) background results on massive scalar-tensor mergers and scalar potentials [13,23,93]. None of these self-cited items supplies the stability criterion or the classification of stable versus unstable models; they are numerical infrastructure and contextual motivation. The discussion of the phi-mode frequency approaching the Yukawa cutoff is an interpretive observation made after the simulations, not a quantity used to define the onset. The short evolution window and the uncontrolled amplitude of the random perturbation are legitimate limitations on the discriminating power of the test, but they are not a circular reduction: the stability labels are not constructed from the criterion being evaluated. Since the benchmark is external and the comparison is a direct numerical test, the paper is self-contained against an external standard, and the self-citations are not load-bearing. Score 1 reflects only the presence of normal methodological self-citations, not any actual circularity.

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

The paper introduces no new entities or forces. The central claim depends on the DEF theory action, the chosen differential rotation law, the external turning-point theorem, and the fidelity of the numerical evolution. The free parameters (B, m_phi, lambda1, lambda2, EOS) are all inherited from the theory or chosen from prior constraints rather than fitted to the stability outcome.

free parameters (4)
  • scalar coupling B = 12
    Fixed across all sequences; chosen to produce scalarized solutions, not fitted to the stability result.
  • scalar mass m_phi = 0 and 0.01 (code units)
    One massless and one massive case; 0.01 is chosen to match the 1.33e-12 eV lower bound consistent with GW170817, not tuned to the outcome.
  • rotation-law shape parameters (lambda1, lambda2) = 1.5, 0.5
    Chosen to mimic quasi-toroidal post-merger differential rotation profiles from prior work [34,62,63].
  • nuclear EOS = MPA1
    A single EOS is used; the generality of the claim over EOS is untested.
assumptions (4)
  • domain assumption The DEF scalar-tensor action with scalar mass term (Eqs. 5-7) is the correct theory of gravity for the study.
    The entire analysis assumes this theory; no derivation or test of the theory itself is offered.
  • standard math The turning-point theorem of Friedman, Ipser, and Sorkin [46], proven for uniformly rotating barotropic stars in GR, is a valid benchmark to test against.
    Used as the external criterion; the paper does not re-derive it, and its extension to differential rotation is exactly what is being tested.
  • domain assumption The Uryu et al. differential rotation law (Eq. 17) with p=1, q=3 adequately represents the rotation profile of merger remnants.
    Stability conclusions are tied to this particular rotation law and the shape parameters lambda1, lambda2.
  • domain assumption The BSSN/Z4c evolution scheme, moving puncture gauge, and HLLC Riemann solver faithfully capture the nonlinear axisymmetric dynamics of the stars.
    No standalone code test or convergence study is presented; the reliability of the conclusions rests on the accuracy of these numerical methods.

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Cite this review

Pith. "Pith review of Axisymmetric stability of neutron stars as extreme rotators in massive scalar-tensor theory." pith.science (2026). https://pith.science/paper/DXGJVWFO

@misc{pith2026250203973,
  author       = {Pith},
  title        = {Pith review of: Axisymmetric stability of neutron stars as extreme rotators in massive scalar-tensor theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DXGJVWFO}},
  note         = {Machine review of arXiv:2502.03973}
}
read the original abstract

Differentially rotating scalarized neutron stars, mimickers of binary merger remnants, can possess an enormous angular momentum larger than what could possibly be sustained in a neutron star in general relativity by about one order of magnitude. A natural question to ask is whether these solutions are stable and thus can realize in a binary coalescence. With this motivation in mind, we examine the criterion of dynamical stability against axisymmetric perturbations for these ultra-rotators by numerically tracking their nonlinear evolution in an axisymmetric setup. We demonstrate that the turning-point criterion still serves as a sufficient condition for asymmetric (in)stability. Our findings open an interesting question of whether the merger of two scalarized neutron stars can produce (possibly short-lived) ultra-highly rotating merger remnants.

Figures

Figures reproduced from arXiv: 2502.03973 by the authors.

Figure 1
Figure 1. FIG. 1: Dynamical stability of sequences (a) [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Evolution of maximum density [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Evolution of maximum density [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Snapshots for a stable model in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Dynamical stability of sequences (a) [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Spectrum of the axisymmetric oscillations in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

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