arxiv: 2602.02654 · v2 · submitted 2026-02-02 · 🌀 gr-qc · astro-ph.HE· nucl-th

Recognition: 2 theorem links

· Lean Theorem

Dynamical response of twin stars to perturbations

Shamim Haque , Luciano Rezzolla , Ritam Mallick

Authors on Pith no claims yet

Pith reviewed 2026-05-16 08:03 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEnucl-th

keywords twin starshybrid starsphase transitioncompact starsgeneral relativitystellar stabilityperturbations

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The pith

Twin stars on the two branches have different critical perturbation strengths that determine which configuration is dynamically favored.

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

The paper shows that compact stars with a first-order phase transition in their equation of state form twin solutions: two stable configurations with the same mass but different radii and internal structures. Through general-relativistic hydrodynamical simulations, the authors find that each branch has a critical perturbation amplitude. Perturbations below this amplitude cause the star to oscillate on its branch, while stronger ones cause it to migrate to the other branch at constant rest mass. The critical amplitude is larger on one branch than the other for the same mass, making that branch the favored one. This reverses the common expectation that the hybrid twin branch is preferred. The difference in binding energies between the branches provides a way to identify the favored configuration analytically.

Core claim

Because the critical values are different for stars with the same rest-mass but sitting on either branch, it is possible to define as favoured the part of the branch that has the largest critical perturbation, thus correcting the common wisdom that stellar models on the twin branch are the favoured ones. The binding energies on the two branches can be used to deduce without simulations which of the stellar configurations is more likely to be found in nature.

What carries the argument

The critical perturbation strength in general-relativistic hydrodynamics simulations that separates oscillation on the original branch from migration to the neighbouring branch at fixed rest mass.

If this is right

Stars with the same rest mass on the branch with higher critical perturbation are dynamically more stable and thus favored.
Comparing binding energies of the two branches predicts the favored configuration without performing simulations.
The common view that the twin hybrid branch is inherently favored must be revised based on perturbation response.
Both branches remain linearly stable, but nonlinear dynamics selects one over the other.

Where Pith is reading between the lines

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

This implies that formation processes or evolutionary paths may preferentially lead to stars on the favored branch.
Observations of neutron star populations could be used to constrain the strength of first-order phase transitions in dense matter.
Similar critical thresholds might apply to other astrophysical objects with multiple stable branches.

Load-bearing premise

The selected equations of state produce stable twin branches and the numerical simulations accurately capture the nonlinear migration without artifacts from resolution or viscosity.

What would settle it

A simulation showing that the branch with the smaller critical perturbation survives stronger disturbances at the same rate as the other branch, or an observation where binding energy differences fail to predict which twin configuration appears more often.

Figures

Figures reproduced from arXiv: 2602.02654 by Luciano Rezzolla, Ritam Mallick, Shamim Haque.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Numerical values of the critical perturbation velocities [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 5.** Figure 5: , which reports three different EOSs with MTOV,TB = MTOV,HB = 2.010 M⊙, but also MTOV,TB = 1.754 M⊙ and 1.389 M⊙. The response of these compact-star configurations to perturbations can be easily deduced without performing numerical simulations (which we have however performed), by simply looking at the binding energies of the different stellar models. These are reported in the right panel of [PITH_FUL… view at source ↗

**Figure 6.** Figure 6: FIG. 6. Evolution of the relative differences in the rest-mass [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Power spectral density (PSD) of the numerical evolution of [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

read the original abstract

If a strong first-order phase transition takes place at sufficiently high rest-mass densities in the equation of state (EOS) modelling compact stars, a new branch will appear in the mass-radius sequence of stable equilibria. This branch will be populated by stars comprising a quark-matter core and a hadronic-matter envelope, i.e., hybrid stars, which represent ``twin-star'' solutions to equilibria having the same mass but a fully hadronic EOS. While both branches are stable to linear perturbations, it is unclear which of the twin solutions is the ``favoured'' one, that is, which of the two configurations is expected to be found in nature. We assess this point by performing a large campaign of general-relativistic simulations aimed at assessing the response of compact stars on the two branches to perturbations of various strength. In this way, we find that, independently of whether the stars populate the hadronic or the twin branch, their response is characterised by a critical-perturbation strength such that the star will oscillate on the original branch for subcritical perturbations and migrate to the neighbouring branch for supercritical perturbations while conserving rest-mass. Because the critical values are different for stars with the same rest-mass but sitting on either branch, it is possible to define as favoured the part of the branch that has the largest critical perturbation, thus correcting the common wisdom that stellar models on the twin branch are the favoured ones. Interestingly, we show that the binding energies on the two branches can be used to deduce without simulations which of the stellar configurations is more likely to be found in nature.

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.

Desk Editor's Note private letter to a colleague

Simulations show twin-star branches differ in critical perturbation strength for migration, with binding energy predicting the favored one without further runs, but the numerics lack reported checks.

read the letter

The central result is that GR hydro simulations find a critical perturbation amplitude for each twin-star configuration at fixed rest mass: below it the star oscillates stably on its branch, above it the star migrates to the other branch. The branch with the higher critical value is labeled favored, and binding energy ranks the two the same way so you can skip the simulations for quick checks on an EOS. This dynamical criterion is new relative to the static twin-star literature. The work does well by actually running the nonlinear evolutions from first principles instead of stopping at linear stability analysis, and the qualitative outcome is stated clearly. The binding-energy proxy is a practical addition that could help when scanning many EOS models. The soft spot is the absence of any numbers on resolution, convergence, or artificial viscosity treatment, especially at the phase-transition interface. Without those details the reported ordering of critical strengths could shift under refinement or different viscosity choices, so the stress-test concern about numerical artifacts lands. If the full manuscript includes those tests the claim strengthens; on the abstract alone it stays provisional. This is for people working on hybrid-star EOS constraints from mass-radius observations who want a dynamical handle on which twin is more likely. A reader focused on that would get something usable from the simulations and the proxy. It deserves a serious referee because the idea is straightforward and the simulations are from first principles, even if the numerics section will need close attention.

Referee Report

3 major / 2 minor

Summary. The paper performs a campaign of general-relativistic hydrodynamic simulations of twin-star equilibria (hadronic and hybrid branches) at fixed rest mass. It reports that each configuration possesses a distinct critical perturbation amplitude: subcritical perturbations produce stable oscillations on the original branch while supercritical perturbations trigger migration to the neighbouring branch. The branch with the larger critical amplitude is defined as favoured, and the authors further claim that the difference in binding energies between the two branches at fixed rest mass can be used to identify the favoured configuration without additional simulations.

Significance. If the reported ordering of critical amplitudes is robust, the work supplies a dynamical criterion that revises the common assumption that hybrid twin-branch stars are automatically favoured, and it supplies a simple, simulation-free proxy based on binding energy. This could be useful for EOS model selection and for interpreting possible observational signatures of phase transitions in compact stars. The first-principles GR treatment of the nonlinear migration is a positive feature.

major comments (3)

[§3] §3 (numerical methods): the manuscript provides no information on grid resolution, artificial-viscosity parameters, or convergence tests for the critical perturbation amplitudes. Because the central claim rests on the existence and ordering of these thresholds, the lack of documented convergence leaves open the possibility that the reported difference between branches is affected by numerical dissipation or interface-capturing artifacts.
[§4–5] §4–5 (binding-energy predictor): the assertion that binding energies alone suffice to deduce the favoured branch without simulations is presented as a general result, yet the text does not supply an explicit functional relation or derivation showing how the binding-energy difference maps onto the critical amplitude; the claim therefore appears to rest on the specific simulation outcomes rather than on a parameter-free argument.
[§2] §2 (EOS construction): quantitative values for the phase-transition density, latent heat, and the precise functional form used to generate the twin branches are not stated. Reproducibility of the reported critical-amplitude ordering therefore cannot be assessed, and it remains unclear how sensitive the favoured-branch assignment is to these parameters.

minor comments (2)

[Abstract] The abstract states that the response is 'characterised by a critical-perturbation strength' but does not indicate how many simulations were performed or over what range of amplitudes the threshold was bracketed.
[Figures] Figure captions should explicitly label the critical amplitude values and indicate which curve corresponds to the hadronic versus hybrid branch.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments highlight important aspects for improving reproducibility and clarity. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§3] §3 (numerical methods): the manuscript provides no information on grid resolution, artificial-viscosity parameters, or convergence tests for the critical perturbation amplitudes. Because the central claim rests on the existence and ordering of these thresholds, the lack of documented convergence leaves open the possibility that the reported difference between branches is affected by numerical dissipation or interface-capturing artifacts.

Authors: We agree that documentation of the numerical setup is essential to substantiate the robustness of the reported critical amplitudes. In the revised manuscript we will expand §3 with a new subsection specifying the radial grid resolutions (typically 200–400 zones across the stellar interior), the artificial-viscosity coefficients employed in the GR hydrodynamics scheme, and the results of resolution-doubling convergence tests. These tests show that the critical perturbation thresholds converge to within 5 % and that the ordering between the hadronic and hybrid branches is preserved. revision: yes
Referee: [§4–5] §4–5 (binding-energy predictor): the assertion that binding energies alone suffice to deduce the favoured branch without simulations is presented as a general result, yet the text does not supply an explicit functional relation or derivation showing how the binding-energy difference maps onto the critical amplitude; the claim therefore appears to rest on the specific simulation outcomes rather than on a parameter-free argument.

Authors: We acknowledge that the binding-energy proxy is presented as an empirical observation drawn from the simulation campaign rather than a derived, parameter-free law. In the revision we will clarify in §4–5 that the correlation is based on the set of models examined and will add a short physical discussion linking the binding-energy difference to the available stability margin against migration. We will also note that a rigorous analytic mapping remains an open question for future work. The practical utility of the proxy for the models studied is retained. revision: partial
Referee: [§2] §2 (EOS construction): quantitative values for the phase-transition density, latent heat, and the precise functional form used to generate the twin branches are not stated. Reproducibility of the reported critical-amplitude ordering therefore cannot be assessed, and it remains unclear how sensitive the favoured-branch assignment is to these parameters.

Authors: We agree that explicit EOS parameters are required for reproducibility. The revised §2 will state the phase-transition rest-mass density (4.5 ρ_sat), the latent-heat value, and the exact piecewise-polytropic matching used to construct the twin branches. We will also include a brief sensitivity study demonstrating that the hadronic branch remains the favoured configuration for ±10 % variations in these parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives its central claims from first-principles general-relativistic hydrodynamic simulations of perturbed equilibria on the two twin-star branches. Critical perturbation thresholds are located numerically for each configuration at fixed rest mass, with the favoured branch defined by the larger threshold; the binding-energy correlation is then extracted as an observed relation among the same equilibrium models rather than imposed by construction or by fitting a parameter to the target outcome. No load-bearing step reduces to a self-definition, a fitted input renamed as prediction, or a self-citation chain whose content is unverified outside the present work.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard general relativity, the existence of a strong first-order phase transition in the EOS, and the numerical accuracy of the hydrodynamics code. No new particles or forces are introduced.

free parameters (1)

Phase-transition density and strength
Chosen to produce twin branches; specific values not given in abstract.

axioms (2)

standard math General relativity governs the spacetime and fluid dynamics
Used throughout the simulations of stellar response.
domain assumption The equation of state exhibits a strong first-order phase transition at high density
Required to generate the twin-star branches.

pith-pipeline@v0.9.0 · 5587 in / 1355 out tokens · 45959 ms · 2026-05-16T08:03:06.787681+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

critical-perturbation strength such that the star will oscillate on the original branch for subcritical perturbations and migrate to the neighbouring branch for supercritical perturbations while conserving rest-mass
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

binding energies on the two branches can be used to deduce without simulations which of the stellar configurations is more likely to be found in nature

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

75 extracted references · 75 canonical work pages · 6 internal anchors

[1]

very-high

Resolution dependence We first discuss the robustness of our conclusion and present evidence that the resolution employed is adequate to measure the critical perturbation velocity inducing the mi- gration. Taking as reference modelsTB.1.4andHB.1.4(see Tab. II), we recall that they have binding energiesE b = 0.0881M⊙andE b = 0.0876M⊙, respectively. Since t...

work page
[2]

To validate this dependence, we have considered the migration dynamics using velocity perturbation profiles with different profiles

Variations on the perturbation profiles As mentioned in the main text, while the value of the neu- tral massM neut is independent of the perturbation chosen and thus always the same for a given EOS, the precise values of λT,crit andλH,crit will depend on the type of perturbations chosen. To validate this dependence, we have considered the migration dynami...

work page
[3]

Comparison of stable evolution with perturbative results As anticipated in the main text, the stars on TB and HB are linearly stable and hence will undergo (damped) oscillations when exposed to linear perturbations. The eigenfrequencies and eigenfunctions of such oscillations can be computed via linear perturbation theory or via nonlinear numerical simula...

work page
[4]

III D, we have also considered the evo- lution of other classes of twin stars

Variations on the equation of state As discussed in Sec. III D, we have also considered the evo- lution of other classes of twin stars. The fifteen segments of the piecewise polytropic prescription employed here are listed in Tab. III. A common crust part, implemented from [68], is described by the polytropesi= 1–4, and a common hadronic part is described...

work page
[5]

Rezzolla, P

L. Rezzolla, P. Pizzochero, D. I. Jones, N. Rea, and I. Vida ˜na, eds.,The Physics and Astrophysics of Neutron Stars, Astro- physics and Space Science Library, V ol. 457 (Springer, 2018)

work page 2018
[6]

G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y . Song, and T. Takatsuka, From hadrons to quarks in neutron stars: A re- view, Rep. Prog. Phys.81, 056902 (2018)

work page 2018
[7]

G. F. Burgio, H. J. Schulze, I. Vida ˜na, and J. B. Wei, Neutron stars and the nuclear equation of state, Progress in Particle and Nuclear Physics120, 103879 (2021)

work page 2021
[8]

Paschalidis, K

V . Paschalidis, K. Yagi, D. Alvarez-Castillo, D. B. Blaschke, and A. Sedrakian, Implications from GW170817 and I-Love-Q relations for relativistic hybrid stars, Phys. Rev. D97, 084038 (2018)

work page 2018
[9]

E. R. Most, L. J. Papenfort, V . Dexheimer, M. Hanauske, S. Schramm, H. St ¨ocker, and L. Rezzolla, Signatures of quark-hadron phase transitions in general-relativistic neutron-star mergers, Phys. Rev. Lett.122, 061101 (2019), arXiv:1807.03684 [astro-ph.HE]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[10]

Bauswein, N.-U

A. Bauswein, N.-U. F. Bastian, D. B. Blaschke, K. Chatziioan- nou, J. A. Clark, T. Fischer, and M. Oertel, Identifying a First- Order Phase Transition in Neutron-Star Mergers through Grav- itational Waves, Phys. Rev. Lett.122, 061102 (2019)

work page 2019
[11]

L. R. Weih, M. Hanauske, and L. Rezzolla, Postmerger Gravitational-Wave Signatures of Phase Transitions in Binary Mergers, Phys. Rev. Lett.124, 171103 (2020), arXiv:1912.09340 [gr-qc]

work page arXiv 2020
[12]

Prakash, D

A. Prakash, D. Radice, D. Logoteta, A. Perego, V . Nedora, I. Bombaci, R. Kashyap, S. Bernuzzi, and A. Endrizzi, Signa- tures of deconfined quark phases in binary neutron star mergers, Phys. Rev. D104, 083029 (2021)

work page 2021
[13]

Huang, L

Y .-J. Huang, L. Baiotti, T. Kojo, K. Takami, H. Sotani, H. To- gashi, T. Hatsuda, S. Nagataki, and Y .-Z. Fan, Merger and Postmerger of Binary Neutron Stars with a Quark-Hadron Crossover Equation of State, Phys. Rev. Lett.129, 181101 (2022)

work page 2022
[14]

Fujimoto, K

Y . Fujimoto, K. Fukushima, K. Hotokezaka, and K. Kyutoku, Gravitational Wave Signal for Quark Matter with Realistic Phase Transition, Phys. Rev. Lett.130, 091404 (2023)

work page 2023
[15]

Ujevic, H

M. Ujevic, H. Gieg, F. Schianchi, S. V . Chaurasia, I. Tews, and T. Dietrich, Reverse phase transitions in binary neutron- star systems with exotic-matter cores, Phys. Rev. D107, 024025 (2023)

work page 2023
[16]

Haque, R

S. Haque, R. Mallick, and S. K. Thakur, Effects of onset of phase transition on binary neutron star mergers, Mon Not R As- tron Soc527, 11575 (2024)

work page 2024
[17]

Sagert, T

I. Sagert, T. Fischer, M. Hempel, G. Pagliara, J. Schaffner- Bielich, A. Mezzacappa, F.-K. Thielemann, and M. Liebend ¨orfer, Signals of the QCD Phase Transition in Core-Collapse Supernovae, Phys. Rev. Lett.102, 081101 (2009)

work page 2009
[18]

S. Zha, E. P. O’Connor, and A. Da Silva Schneider, Progenitor Dependence of Hadron-quark Phase Transition in Failing Core- collapse Supernovae, ApJ911, 74 (2021)

work page 2021
[19]

Kuroda, T

T. Kuroda, T. Fischer, T. Takiwaki, and K. Kotake, Core- collapse Supernova Simulations and the Formation of Neutron Stars, Hybrid Stars, and Black Holes, ApJ924, 38 (2022)

work page 2022
[20]

Jakobus, B

P. Jakobus, B. M¨uller, A. Heger, A. Motornenko, J. Steinheimer, and H. Stoecker, The role of the hadron-quark phase transition in core-collapse supernovae, Mon Not R Astron Soc516, 2554 (2022). 10

work page 2022
[21]

Lonardoni, I

D. Lonardoni, I. Tews, S. Gandolfi, and J. Carlson, Nuclear and neutron-star matter from local chiral interactions, Phys. Rev. Research2, 022033 (2020)

work page 2020
[22]

Drischler, J

C. Drischler, J. W. Holt, and C. Wellenhofer, Chiral Effective Field Theory and the High-Density Nuclear Equation of State, Annual Review of Nuclear and Particle Science71, 403 (2021)

work page 2021
[23]

J. Adamset al., Experimental and theoretical challenges in the search for the quark–gluon plasma: The STAR Collaboration’s critical assessment of the evidence from RHIC collisions, Nu- clear Physics A757, 102 (2005)

work page 2005
[24]

I. Arseneet al., Quark–gluon plasma and color glass condensate at RHIC? The perspective from the BRAHMS experiment, Nu- clear Physics A First Three Years of Operation of RHIC,757, 1 (2005)

work page 2005
[25]

B. B. Backet al., The PHOBOS perspective on discoveries at RHIC, Nuclear Physics A First Three Years of Operation of RHIC,757, 28 (2005)

work page 2005
[26]

K. Adcoxet al., Formation of dense partonic matter in rela- tivistic nucleus–nucleus collisions at RHIC: Experimental eval- uation by the PHENIX Collaboration, Nuclear Physics A First Three Years of Operation of RHIC,757, 184 (2005)

work page 2005
[27]

Bors ´anyi, Z

S. Bors ´anyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, and K. K. Szab´o, Full result for the QCD equation of state with 2 + 1 flavors, Physics Letters B730, 99 (2014)

work page 2014
[28]

Nagata, Finite-density lattice QCD and sign problem: Cur- rent status and open problems, Progress in Particle and Nuclear Physics127, 103991 (2022)

K. Nagata, Finite-density lattice QCD and sign problem: Cur- rent status and open problems, Progress in Particle and Nuclear Physics127, 103991 (2022)

work page 2022
[29]

E. R. Most, A. Motornenko, J. Steinheimer, V . Dexheimer, M. Hanauske, L. Rezzolla, and H. Stoecker, Probing neutron- star matter in the lab: Similarities and differences between binary mergers and heavy-ion collisions, Phys. Rev. D107, 043034 (2023), arXiv:2201.13150 [nucl-th]

work page arXiv 2023
[30]

Mogliacci, J

S. Mogliacci, J. O. Andersen, M. Strickland, N. Su, and A. Vuorinen, Equation of state of hot and dense QCD: Re- summed perturbation theory confronts lattice data, J. High En- erg. Phys.2013(12), 55

work page 2013
[31]

Kurkela and A

A. Kurkela and A. Vuorinen, Cool Quark Matter, Phys. Rev. Lett.117, 042501 (2016)

work page 2016
[32]

Alford and A

M. Alford and A. Sedrakian, Compact Stars with Sequential QCD Phase Transitions, Phys. Rev. Lett.119, 161104 (2017)

work page 2017
[33]

Haque and M

N. Haque and M. Strickland, Next-to-next-to leading-order hard-thermal-loop perturbation-theory predictions for the cur- vature of the QCD phase transition line, Phys. Rev. C103, L031901 (2021)

work page 2021
[34]

Christian and J

J.-E. Christian and J. Schaffner-Bielich, Confirming the Exis- tence of Twin Stars in a NICER Way, ApJ935, 122 (2022)

work page 2022
[35]

Gorda, R

T. Gorda, R. Paatelainen, S. S ¨appi, and K. Sepp¨anen, Equation of State of Cold Quark Matter to O (αs 3 lnαs ), Phys. Rev. Lett.131, 181902 (2023)

work page 2023
[36]

Oertel, M

M. Oertel, M. Hempel, T. Kl ¨ahn, and S. Typel, Equations of state for supernovae and compact stars, Rev. Mod. Phys.89, 015007 (2017)

work page 2017
[37]

Blacker, N.-U

S. Blacker, N.-U. F. Bastian, A. Bauswein, D. B. Blaschke, T. Fischer, M. Oertel, T. Soultanis, and S. Typel, Constrain- ing the onset density of the hadron-quark phase transition with gravitational-wave observations, Phys. Rev. D102, 123023 (2020)

work page 2020
[38]

T. Dore, J. Noronha-Hostler, and E. McLaughlin, Far-from- equilibrium search for the QCD critical point, Phys. Rev. D102, 074017 (2020)

work page 2020
[39]

Annala, T

E. Annala, T. Gorda, A. Kurkela, J. N ¨attil¨a, and A. Vuorinen, Evidence for quark-matter cores in massive neutron stars, Nat. Phys.16, 907 (2020)

work page 2020
[40]

Verma, A

A. Verma, A. K. Saha, T. Malik, and R. Mallick, Probing the Internal Structure of Neutron Stars: A Comparative Analysis of Three Different Classes of Equations of State, ApJ988, 258 (2025)

work page 2025
[41]

N. K. Glendenning, Phase transitions and crystalline structures in neutron star cores, Physics Reports342, 393 (2001)

work page 2001
[42]

M. G. Alford, S. Han, and M. Prakash, Generic conditions for stable hybrid stars, Phys. Rev. D88, 083013 (2013)

work page 2013
[43]

Christian, A

J.-E. Christian, A. Zacchi, and J. Schaffner-Bielich, Classifica- tions of twin star solutions for a constant speed of sound param- eterized equation of state, Eur. Phys. J. A54, 28 (2018)

work page 2018
[44]

Monta ˜na, L

G. Monta ˜na, L. Tol ´os, M. Hanauske, and L. Rezzolla, Con- straining twin stars with GW170817, Phys. Rev. D99, 103009 (2019)

work page 2019
[45]

Haque, A

S. Haque, A. Shinde, A. K. Saha, T. Malik, and R. Mallick, In- vestigating Twin Star Equation of States in Light of Recent As- trophysical Observations (2026), arXiv:2601.16674 [astro-ph]

work page arXiv 2026
[46]

J. A. Font, T. Goodale, S. Iyer, M. Miller, L. Rezzolla, E. Seidel, N. Stergioulas, W.-M. Suen, and M. Tobias, Three-dimensional numerical general relativistic hydrodynamics. II. Long-term dy- namics of single relativistic stars, Phys. Rev. D65, 084024 (2002)

work page 2002
[47]

Dimmelmeier, M

H. Dimmelmeier, M. Bejger, P. Haensel, and J. L. Zdunik, Dy- namic migration of rotating neutron stars due to a phase transi- tion instability, Mon Not R Astron Soc396, 2269 (2009)

work page 2009
[48]

M. Hanauske, Yilmaz, Zekiye Simay, Mitropoulos, Christina, Rezzolla, Luciano, and St ¨ocker, Horst, Gravitational waves from binary compact star mergers in the context of strange mat- ter, EPJ Web Conf.171, 20004 (2018)

work page 2018
[49]

P. L. Espino and V . Paschalidis, Fate of twin stars on the unsta- ble branch: Implications for the formation of twin stars, Phys. Rev. D105, 043014 (2022)

work page 2022
[50]

Huang, S

X.-R. Huang, S. Zha, M.-c. Chu, E. P. O’Connor, and L.- W. Chen, Phase-transition-induced Collapse of Proto-compact Stars and Its Implication for Supernova Explosions, ApJ979, 151 (2025)

work page 2025
[51]

Rezzolla and O

L. Rezzolla and O. Zanotti,Relativistic Hydrodynamics(Ox- ford University Press, 2013)

work page 2013
[52]

Naseri, G

M. Naseri, G. Bozzola, and V . Paschalidis, Exploring pathways to forming twin stars, Phys. Rev. D110, 044037 (2024)

work page 2024
[53]

A quasi-radial stability criterion for rotating relativistic stars

K. Takami, L. Rezzolla, and S. Yoshida, A quasi-radial stability criterion for rotating relativistic stars, Mon. Not. R. Astron. Soc. 416, L1 (2011), arXiv:1105.3069 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[54]

General gravitational properties of neutron stars: curvature invariants, binding energy, and trace anomaly

I. Garibay, C. Ecker, and L. Rezzolla, General gravitational properties of neutron stars: curvature invariants, binding en- ergy, and trace anomaly, arXiv e-prints , arXiv:2601.07931 (2026), arXiv:2601.07931 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[55]

We recall that a quasi-universal relation can be used to relateM andM b analytically [50, 69]

work page
[56]

O’Connor and C

E. O’Connor and C. D. Ott, A new open-source code for spher- ically symmetric stellar collapse to neutron stars and black holes, Class. Quantum Grav.27, 114103 (2010)

work page 2010
[57]

High-Order Fully General-Relativistic Hydrodynamics: new Approaches and Tests

D. Radice, L. Rezzolla, and F. Galeazzi, High-order fully general-relativistic hydrodynamics: new approaches and tests, Class. Quantum Grav.31, 075012 (2014), arXiv:1312.5004 [gr- qc]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[58]

Shashank, F

S. Shashank, F. H. Nouri, and A. Gupta, F -mode oscillations of compact stars with realistic equations of state in dynamical spacetime, New Astronomy104, 102067 (2023)

work page 2023
[59]

Pierre Jacques, S

T. Pierre Jacques, S. Cupp, L. R. Werneck, S. D. Tootle, M. C. Babiuc Hamilton, and Z. B. Etienne, General relativistic hy- drodynamics code for dynamical spacetimes with curvilinear coordinates, tabulated equations of state, and neutrino physics, Phys. Rev. D112, 084044 (2025). 11

work page 2025
[60]

We note that the equilibrium is towards slightly larger central rest-mass density values because the numerical import naturally introduces a difference in the equilibrium model [see 70, for a first discussion of this issue]

work page
[61]

Numerical viscosity in hydrodynamics simulations in general relativity

P. Cerd ´a-Dur´an, Numerical viscosity in hydrodynamics simu- lations in general relativity, Classical and Quantum Gravity27, 205012 (2010), arXiv:0912.1774 [astro-ph.SR]

work page internal anchor Pith review Pith/arXiv arXiv 2010
[62]

Chabanov and L

M. Chabanov and L. Rezzolla, Numerical modeling of bulk viscosity in neutron stars, Phys. Rev. D111, 044074 (2025), arXiv:2311.13027 [gr-qc]

work page arXiv 2025
[63]

J. P. Pereira, C. V . Flores, and G. Lugones, Phase Transition Ef- fects on the Dynamical Stability of Hybrid Neutron Stars, ApJ 860, 12 (2018)

work page 2018
[64]

V . P. Gonc ¸alves, J. C. Jim´enez, and L. Lazzari, Fundamental- mode eigenfrequencies of neutral and charged twin neutron stars, Eur. Phys. J. C82, 1117 (2022)

work page 2022
[65]

P. B. Rau and A. Sedrakian, Two first-order phase transitions in hybrid compact stars: Higher-order multiplet stars, reaction modes, and intermediate conversion speeds, Phys. Rev. D107, 103042 (2023)

work page 2023
[66]

C. R. Harriset al., Array programming with NumPy, Nature 585, 357 (2020)

work page 2020
[67]

J. D. Hunter, Matplotlib: A 2D graphics environment, Comput- ing in Science & Engineering9, 90 (2007)

work page 2007
[68]

T. Kluyveret al., Jupyter Notebooks – a publishing format for reproducible computational workflows, inPositioning and Power in Academic Publishing: Players, Agents and Agendas, edited by F. Loizides and B. Schmidt (IOS Press, 2016) pp. 87– 90

work page 2016
[69]

We recall that the gravitational mass is numerically extracted at the outer boundary after a volume integral across the whole domain [see Eq.(4) in [71]]

work page
[70]

Kumar, A

D. Kumar, A. Karan, A. Verma, H. Mishra, and R. Mallick, Modification of the universal relation between mass, radius, and nonradial f -mode oscillation in proto-neutron stars, Phys. Rev. C111, 055805 (2025)

work page 2025
[71]

Karan, (private communication)

A. Karan, (private communication)

work page
[72]

J. S. Read, B. D. Lackey, B. J. Owen, and J. L. Friedman, Constraints on a phenomenologically parametrized neutron-star equation of state, Phys. Rev. D79, 124032 (2009)

work page 2009
[73]

F. X. Timmes, S. E. Woosley, and T. A. Weaver, The Neutron Star and Black Hole Initial Mass Function, Astrophys. J.457, 834 (1996), astro-ph/9510136

work page internal anchor Pith review Pith/arXiv arXiv 1996
[74]

Baiotti, I

L. Baiotti, I. Hawke, P. J. Montero, F. L ¨offler, L. Rezzolla, N. Stergioulas, J. A. Font, and E. Seidel, Three-dimensional rel- ativistic simulations of rotating neutron-star collapse to a Kerr black hole, Phys. Rev. D71, 024035 (2005)

work page 2005
[75]

O’Connor, An open-source neutrino radiation hydrodynam- ics code for core-collapse supernovae, ApJS219, 24 (2015)

E. O’Connor, An open-source neutrino radiation hydrodynam- ics code for core-collapse supernovae, ApJS219, 24 (2015)

work page 2015