Radial Oscillations in Hybrid Stars with Slow Quark Phase Transition

Bet\^ania C. Backes; Grigoris Panotopoulos; Il\'idio Lopes; Ishfaq Ahmad Rather; Kauan D. Marquez

REVIEW 2 major objections 6 minor 18 references

Radial Oscillations in Hybrid Stars with Slow Quark Phase Transition

T0 review · 2 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Slow quark phase transitions create a stable branch of hybrid stars beyond the maximum mass.

desk verdict Slow-stable hybrid stars are plausible physics, but this proceedings paper is a condensed summary of the authors' own JCAP work and never states the junction conditions that carry the central claim. read the letter →

arxiv 2412.04007 v1 pith:CTELE5SB submitted 2024-12-05 nucl-th astro-ph.HEastro-ph.SR

Ishfaq Ahmad Rather , Kauan D. Marquez , Betânia C. Backes , Grigoris Panotopoulos , Ilídio Lopes This is my paper · ORCID

classification nucl-thastro-ph.HEastro-ph.SR

keywords neutronstarshybridquarkmatterslowphasetransitionradialoscillationsstellarstabilityhyperonsrelativisticmean-fieldmodel

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 studies radial oscillations of hybrid neutron stars—stars with a hadronic outer layer and a quark-matter core—and asks when such stars are dynamically stable. The authors solve the relativistic radial-oscillation equations for four hadronic compositions (nucleons only, nucleons with deltas, nucleons with hyperons, and both) matched to quark matter by a Maxwell construction at equal pressure and chemical potential. The central claim is that when the hadron-quark interface converts phases slowly, the fundamental mode frequency does not vanish at the maximum-mass configuration, so stars with central densities above the maximum mass remain stable against small radial perturbations. They call these configurations Slow Stable Hybrid Stars (SSHSs), and the size of the stable branch depends on the energy-density jump at the transition and the stiffness of the quark equation of state. If correct, the usual rule that the maximum mass is also the stability limit fails for hybrid stars with slow conversion.

What carries the argument

The central object is the Sturm-Liouville system for radial perturbations, Eqs. (5)-(6), with the dimensionless displacement $\xi=\Delta r/r$ and pressure perturbation $\eta=\Delta P/P$, solved by a shooting method that selects eigenfrequencies satisfying center and surface boundary conditions. The load-bearing ingredient is the treatment of the hadron-quark interface under the assumption of slow conversion: the interface does not restore chemical equilibrium during the oscillation, and the junction conditions of Refs. [11,12] are used. This slow-conversion treatment is what keeps $\omega_0^2$ real even where $dM/d\varepsilon_c<0$, giving the SSHS branch.

What would settle it

A direct comparison of the hadron-quark conversion timescale at the interface with the fundamental period of an SSHS would settle the claim; if the timescale is shorter than the period, the slow-conversion treatment fails and $\omega_0^2$ should vanish at the maximum mass. A non-linear simulation of radial pulsations of a hybrid star with a finite conversion rate would provide the definitive test.

Watch

Extended reading notes

Core claim

The paper finds that, for slow phase conversion at the hadron-quark interface, the fundamental radial mode squared $\omega_0^2$ stays positive for central densities beyond the point where $dM/d\varepsilon_c<0$, so the maximum-mass configuration is not the last stable point. The fundamental mode frequency $\nu_0$ reaches zero only at a higher central density, and the stable configurations in between are the Slow Stable Hybrid Stars (SSHSs). For rapid conversion, by contrast, $\nu_0$ drops to zero exactly at the maximum mass, recovering the standard stability criterion. The length of the SSHS branch depends on the energy-density jump at the transition and on the stiffness of the quark equation of state; for the N+H composition with the smallest jump (38 MeV/fm$^3$) and a stiff quark EoS, the stable branch extends 0.88 km beyond the maximum-mass radius.

Load-bearing premise

The slow-conversion assumption: phase conversion at the hadron-quark interface is slow enough that chemical equilibrium is not restored during a radial oscillation; if the conversion timescale is comparable to or shorter than the oscillation period, the SSHS branch would not exist.

Editorial extensions

If this is right

For slow phase transitions, the maximum mass is not the dynamical stability limit; the f-mode frequency vanishes only at a higher central density, so an SSHS branch of stable configurations exists.
The length of the SSHS branch is controlled by the energy-density jump at the hadron-quark transition and the stiffness of the quark EoS; the longest branch found is 0.88 km for the N+H EoS with the smallest jump (38 MeV/fm$^3$).
The fundamental-mode frequency of SSHSs is lower than that of ordinary neutron stars of similar mass, and the radial eigenfunctions have reduced amplitudes, so SSHSs would look like quieter versions of ordinary neutron stars in radial pulsation.
All four equations of state considered satisfy current pulsar mass-radius constraints, so the existence of the SSHS branch is not excluded by present observations.

Reading between the lines

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

If the slow-conversion assumption is right, the common practice of reading the maximum mass of a hybrid equation of state as its dynamical stability limit should be revised; the true limit is the vanishing of the f-mode frequency.
The slow/fast distinction is likely generic to first-order phase transitions in compact stars, so similar stable branches could appear in other contexts such as proto-neutron stars or stars with exotic dark-matter cores.
A concrete testable extension would compute the complex (damped) frequencies of SSHS radial modes; if damping is weak, SSHSs might be observable through quasi-periodic oscillations, and if strong, they would be silent.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Kauan, quick read of the Rather et al. proceedings. The core physics is solid: for a slow hadron-quark phase transition, radial stability can persist past the maximum-mass configuration (the SSHS branch). That behavior was established by Pereira, Flores, Lugones and by Lugones, and the present authors apply it to a broader set of EoS with hyperons and deltas. The parameter scan and the eigenfunction plots are useful, and the qualitative agreement with prior results is a good consistency check. The paper deserves credit for that.

The soft spot is exactly what the stress-test flags: equations (5) and (6) are written for a continuous star, but the Maxwell construction produces a sharp density discontinuity. To stitch the Sturm-Liouville problem across that interface you need junction conditions, and the paper just says "numerical calculations have shown" and cites Refs. [11,12]. Since the entire SSHS phenomenon is carried by those conditions, omitting them from a paper that claims to demonstrate the effect is a genuine gap. The timescale point is also fair: no estimate of the conversion timescale tau_conv versus the fundamental-mode period is given, so the slow limit is an assumption, not a demonstrated regime. That said, this is a proceedings contribution and the authors point to their JCAP paper (Ref. [18]) for full details, so the omission is more about self-containedness than correctness. I don't think the underlying physics is wrong, but the paper as written does not stand alone.

The overlap with Ref. [18] is substantial but not a flaw for a conference summary — they are explicit about it. No code or data, but that's normal for a proceedings.

Who is this for? People working on hybrid-star phenomenology who want a quick map of which EoSs admit an SSHS branch. Not the place to learn the slow-conversion formalism. If this crossed my desk as a regular journal submission, I would send it out — the referee could check the junction-condition treatment and the timescale — but I'd expect heavy revision to include the missing material. As a proceedings, it's fine. Would I cite it? Only together with Ref. [18]. Bring it to reading group only if someone is working on oscillation stability of hybrid stars; otherwise skip.

Referee Report

2 major / 6 minor

Summary. This proceedings paper studies radial oscillations of hybrid neutron stars built from DD-RMF hadronic matter (with nucleons, hyperons, and Delta baryons) and a density-dependent quark model, joined by a Maxwell phase transition. Solving the standard relativistic radial pulsation equations for the lowest ten modes, the authors find that for slow hadron-quark conversion at the interface the fundamental mode remains real beyond the maximum-mass configuration, producing a branch of 'Slow Stable Hybrid Stars' (SSHSs). The length of this branch is reported to depend on the energy-density jump and quark EoS stiffness. The paper also presents eigenfunctions and mass-radius relations for four compositions.

Significance. If the SSHS branch is real, the paper would identify a physically interesting exception to the usual turning-point stability criterion: a sequence of hybrid stars with dM/depsilon_c < 0 but omega_0^2 > 0, stable against radial perturbations. The numerical setup is standard, the EoS models are established, and the qualitative behavior is consistent with earlier work by Pereira et al. and Lugones. The paper fits no new parameters to the target result, which is a positive feature for reproducibility. The main weakness is that the crucial slow-conversion interface conditions are not stated or justified in the manuscript, so the central result cannot be independently checked from the text.

major comments (2)

[Sec. 2.3, Eqs. (5)-(6)] The oscillation equations are written for a continuously stratified star, whereas the Maxwell construction of Sec. 2.2 produces a sharp hadron-quark interface with a discontinuous energy density. Solving the Sturm-Liouville problem across that interface requires explicit junction conditions for xi and eta (or the Lagrangian pressure perturbation). The manuscript neither writes these conditions nor explains how the shooting method enforces them; the sentence 'Numerical calculations have shown...' delegates the central effect to Refs [11,12]. Because the existence of the SSHS branch is entirely carried by those junction conditions, this omission is load-bearing for the main claim.
[Sec. 2.3] The slow-conversion limit is assumed without any estimate of the conversion timescale tau_conv relative to the fundamental-mode period P ~ 2*pi/omega_0, which is sub-millisecond for these models. If tau_conv is comparable to or shorter than P, the rapid-conversion boundary condition is the relevant one, omega_0^2 passes through zero at the maximum mass, and the SSHS branch disappears. The paper should provide a microphysical estimate of tau_conv for the DD-RMF/DDQM EoSs, or at least a parametric discussion of the range of tau_conv/P for which the SSHS branch exists.

minor comments (6)

[Figure 2 caption] The left panel is labeled as eta(r) = Delta r/r, but Sec. 2.3 defines eta = Delta P/P; correct the typo.
[Eq. (7)] The dimensionless variable s is not defined; specify that s = omega/omega_0 and clarify the units in the expression for nu.
[Figure 1 caption] The caption says the solid symbol marks 'the last stable point which is the maximum mass configuration' while the SSHS branch is drawn between the solid and star symbols; if the SSHS configurations are stable, the solid symbol is not the last stable point, so the labeling should be revised.
[Sec. 3] The statement that all EoSs satisfy astrophysical constraints needs a confidence level, since the N+H and N+H+Delta sequences have M_max = 1.97-1.98 M_sun, below the central value of PSR J0740+6620.
[Sec. 2.2 and figure legends] The parameters C and D^(1/2) appear only in figure legends (e.g., (0.90,125), (0.65,133)); define them in the text and list the values used, along with the resulting coexistence pressure and energy-density jump for each model.
[Sec. 2.3, Eq. (5)] The adiabatic index gamma is not defined; specify its expression in terms of the EoS and state how it is evaluated at the phase-transition discontinuity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SSHS branch is a computed outcome of external slow-conversion junction conditions and fixed EoS parameters, not a fitted or redefined input.

full rationale

The paper's central claim—that slow phase transitions allow stable hybrid configurations beyond the maximum-mass point—is obtained by solving the standard Sturm-Liouville radial-oscillation problem, Eqs. (5)-(6), with the slow-conversion junction conditions taken from the external references [11,12]. The eigenvalues are computed by the shooting method, and no parameter is fitted to the SSHS outcome; the hadronic and quark-model parameters are fixed by previous work. The stated dependence of the SSHS branch length on the energy-density jump and quark EoS stiffness is a parametric consequence of scanning the input EoSs, not a quantity used to construct the EoSs. The only self-citation, Ref. [18], is a pointer to the authors' fuller companion paper for details on higher modes and EoS behavior; it is not used as a load-bearing premise, uniqueness theorem, or source of the slow-conversion ansatz. The skeptical concern about unstated junction conditions and conversion timescales is a legitimate physical-robustness issue, but it is not circularity: the manuscript openly attributes the slow/fast dichotomy to Refs. [11,12] rather than redefining the target result as an input. No equation reduces to another by construction, and no fitted parameter is renamed as a prediction, so no circular step can be exhibited.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on: (1) the assumed slow conversion timescale at the interface, which is imported from prior literature and is the physical input that creates the SSHS branch; (2) the Maxwell construction, which selects a coexistence pressure; (3) the specific hadronic and quark model parameter sets, including C and D^(1/2) values. None of these are derived in this paper. The paper introduces no new particles or forces. The free parameters are all carried over from earlier work, but they still control the outcome.

free parameters (3)

DDME2 nuclear interaction parameters = Standard DDME2 set from Lalazissis et al. 2005
Parameters of the hadronic Lagrangian, taken from prior literature; not fitted in this paper.
Hyperon and delta coupling ratios = SU(3)/SU(6) scheme from Lopes et al. 2023
Choice of meson-baryon couplings for hyperons and deltas; from Ref [8].
Quark model parameters C and D^(1/2) = (0.90, 125 MeV) and (0.65, 133 MeV) as labeled in Fig. 1
Density-dependent quark mass model parameters; selected by hand or from Backes et al.; they set the coexistence point and influence the SSHS branch length.

assumptions (5)

standard math TOV equations for hydrostatic equilibrium
Used to construct mass-radius relations in Section 3.
domain assumption Maxwell construction with equal pressure and chemical potential at the hadron-quark interface
Section 2.2, Eq. (4); assumes a sharp interface and no mixed phase.
domain assumption Slow phase conversion at the interface (conversion timescale much longer than oscillation period)
Section 2.3, following Refs [11,12]; this is the load-bearing assumption that produces the SSHS branch.
domain assumption Zero temperature, spherical symmetry, no rotation or magnetic field
Standard for radial oscillation studies of cold neutron stars; listed as future extensions in Section 4.
domain assumption Density-dependent quark mass model remains valid at the densities of the quark core
Section 2.2; model parameters extrapolated from lower-density deconfinement studies.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Radial Oscillations in Hybrid Stars with Slow Quark Phase Transition." pith.science (2026). https://pith.science/paper/CTELE5SB

@misc{pith2026241204007,
  author       = {Pith},
  title        = {Pith review of: Radial Oscillations in Hybrid Stars with Slow Quark Phase Transition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CTELE5SB}},
  note         = {Machine review of arXiv:2412.04007}
}

abstract

This study investigates the radial oscillations of hybrid neutron stars, characterized by a composition of hadronic external layers and a quark matter core. Utilizing a density-dependent relativistic mean-field model that incorporates hyperons and baryons for describing hadronic matter, and a density-dependent quark model for quark matter, we analyze the ten lowest eigenfrequencies and their corresponding oscillation functions. Our focus lies on neutron stars with equations-of-state involving N, N + $\Delta$, N + H, and N + H + $\Delta$, featuring a phase transition to quark matter. Emphasizing the effects of a slow phase transition at the hadron-quark interface, we observe that the maximum mass is attained before the fundamental mode's frequency decreases for slow phase transitions. This observation implies the stability of stellar configurations with higher central densities than the maximum mass, called Slow Stable Hybrid Stars (SSHSs), even under small radial perturbations. The length of these SSHS branch depends upon the energy density jump between two phases and the stiffness of the quark EoS.

Figures

Figures reproduced from arXiv: 2412.04007 by the authors.

**Figure 1.** Energy density and pressure variation (left) and Mass-Radius relations (right) for different model compositions with a phase transition to the quark matter at different parameter values (𝐶, 𝐷1/2 ). In the right panel, the solid symbol marks the last stable point which is the maximum mass configuration. The line from solid to the star symbol marks the SSHS branch. The several credible regions for mass and radius are … view at source ↗

**Figure 2.** The radial displacement perturbation 𝜉 (𝑟) = Δ𝑟/𝑟 (right panels) and the radial pressure perturbation 𝜂(𝑟) = Δ𝑟/𝑟 (left panels) as a function of dimensionless radius distance 𝑟/𝑅 for lower 𝑓 -mode (n = 0), lower order 𝑝-modes (n = 1, 2, 3). 0 0.5 1 1.5 2 2.5 M (MO ) 0 0.5 1 1.5 2 2.5 3 3.5 4 ν 0 (kHz) N (0.90, 125) N+ ∆ (0.90, 125) N + H (0.65, 133) N + H + ∆ (0.65, 133) 0 0.5 1 1.5 2 2.5 M (MO ) 0 0.5 1 1.5 2 2.5 3… view at source ↗

**Figure 3.** Fundamental frequency as a function of the mass sequence for slow conversion (left plot) and rapid conversion (right plot) for different compositions of EoS with phase transition. frequencies due to their higher mass and unique composition, particularly influenced by strange quark matter. These profiles indicate structural similarities between SSHS and NSs, with subtle differences reflecting the distinct nature of S… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 16 canonical work pages

[18]

Rather, K.D

I.A. Rather, K.D. Marquez, B.C. Backes, G. Panotopoulos and I. Lopes,Radial oscillations of hybrid stars and neutron stars including delta baryons: the effect of a slow quark phase transition,JCAP 05(2024) 130 [2401.07789]. 7

arXiv 2024
[1]

Glendenning,Compact stars: Nuclear physics, particle physics, and general relativity (1997)

N.K. Glendenning,Compact stars: Nuclear physics, particle physics, and general relativity (1997)

work page 1997
[2]

Marquez, D.P

K.D. Marquez, D.P. Menezes, H. Pais and C.m.c. Providência,Δ baryons in neutron stars, Phys. Rev. C106(2022) 055801

work page 2022
[3]

Abbott and R

B.P. Abbott and R. Abbottet al.Phys. Rev. Lett.121 (2018) 161101

work page 2018
[4]

Chandrasekhar,The Dynamical Instability of Gaseous Masses Approaching the Schwarzschild Limit in General Relativity., Astrophys

S. Chandrasekhar,The Dynamical Instability of Gaseous Masses Approaching the Schwarzschild Limit in General Relativity., Astrophys. J.140 (1964) 417

work page 1964
[5]

Thorne and A

K.S. Thorne and A. CampolattaroAstrophysical Journal149(1967) 591

work page 1967
[6]

Rather, K.D

I.A. Rather, K.D. Marquez, G. Panotopoulos and I. Lopes,Radial oscillations in neutron stars with delta baryons, Phys. Rev. D107 (2023) 123022

work page 2023
[7]

Lalazissis, T

G.A. Lalazissis, T. Nikšić, D. Vretenar and P. Ring,New relativistic mean-field interaction with density-dependent meson-nucleon couplings,Phys. Rev. C71 (2005) 024312

work page 2005

Show all 18 references

[8]

Lopes, K.D

L.L. Lopes, K.D. Marquez and D.P. Menezes,Baryon coupling scheme in a unified su (3) and su (6) symmetry formalism,Physical Review D107 (2023) 036011

2023
[9]

Backes, K.D

B.C. Backes, K.D. Marquez and D.P. Menezes,Effects of strong magnetic fields on the hadron-quark deconfinement transition,The European Physical Journal A57(2021) 1. 6 Radial Oscillations in Hybrid Stars with Slow Quark Phase Transition Ishfaq Ahmad Rather

2021
[10]

Gondek, P

D. Gondek, P. Haensel and J.L. Zdunik,Radial pulsations and stability of protoneutron stars.,A&A 325 (1997) 217

1997
[11]

Pereira, C.V

J.P. Pereira, C.V. Flores and G. Lugones,Phase transition effects on the dynamical stability of hybrid neutron stars,Astrophys. J.860 (2018) 12

2018
[12]

Lugones,Stellar stability in the presence of reacting interfaces: consequences for hybrid stars,J

G. Lugones,Stellar stability in the presence of reacting interfaces: consequences for hybrid stars,J. Phys. Conf. Ser.2536 (2023) 012008

2023
[13]

Miller et al.,The radius of PSR J0740+6620 from nicer and xmm-newton data, The Astrophysical Journal Letters918 (2021) L28

M.C. Miller et al.,The radius of PSR J0740+6620 from nicer and xmm-newton data, The Astrophysical Journal Letters918 (2021) L28

2021
[14]

Riley et al.,A nicer view of the massive pulsar PSR J0740+6620 informed by radio timing and XMM-Newton spectroscopy, The Astrophysical Journal Letters918 (2021) L27

T.E. Riley et al.,A nicer view of the massive pulsar PSR J0740+6620 informed by radio timing and XMM-Newton spectroscopy, The Astrophysical Journal Letters918 (2021) L27

2021
[15]

Milleret al.,PSR J0030+0451 mass and radius from NICER data and implications for the properties of neutron star matter,Astrophys

M.C. Milleret al.,PSR J0030+0451 mass and radius from NICER data and implications for the properties of neutron star matter,Astrophys. J.887 (2019) L24

2019
[16]

Rileyet al., A NICER view of PSR J0030+0451: Millisecond pulsar parameter estimation, Astrophys

T.E. Rileyet al., A NICER view of PSR J0030+0451: Millisecond pulsar parameter estimation, Astrophys. J.887(2019) L21

2019
[17]

Choudhury et al.,A NICER View of the Nearest and Brightest Millisecond Pulsar: PSR J0437-4715, 2407.06789

D. Choudhury et al.,A NICER View of the Nearest and Brightest Millisecond Pulsar: PSR J0437-4715, 2407.06789

arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [18]

Rather, K.D

I.A. Rather, K.D. Marquez, B.C. Backes, G. Panotopoulos and I. Lopes,Radial oscillations of hybrid stars and neutron stars including delta baryons: the effect of a slow quark phase transition,JCAP 05(2024) 130 [2401.07789]. 7

arXiv 2024

[2] [1]

Glendenning,Compact stars: Nuclear physics, particle physics, and general relativity (1997)

N.K. Glendenning,Compact stars: Nuclear physics, particle physics, and general relativity (1997)

work page 1997

[3] [2]

Marquez, D.P

K.D. Marquez, D.P. Menezes, H. Pais and C.m.c. Providência,Δ baryons in neutron stars, Phys. Rev. C106(2022) 055801

work page 2022

[4] [3]

Abbott and R

B.P. Abbott and R. Abbottet al.Phys. Rev. Lett.121 (2018) 161101

work page 2018

[5] [4]

Chandrasekhar,The Dynamical Instability of Gaseous Masses Approaching the Schwarzschild Limit in General Relativity., Astrophys

S. Chandrasekhar,The Dynamical Instability of Gaseous Masses Approaching the Schwarzschild Limit in General Relativity., Astrophys. J.140 (1964) 417

work page 1964

[6] [5]

Thorne and A

K.S. Thorne and A. CampolattaroAstrophysical Journal149(1967) 591

work page 1967

[7] [6]

Rather, K.D

I.A. Rather, K.D. Marquez, G. Panotopoulos and I. Lopes,Radial oscillations in neutron stars with delta baryons, Phys. Rev. D107 (2023) 123022

work page 2023

[8] [7]

Lalazissis, T

G.A. Lalazissis, T. Nikšić, D. Vretenar and P. Ring,New relativistic mean-field interaction with density-dependent meson-nucleon couplings,Phys. Rev. C71 (2005) 024312

work page 2005