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arxiv: 2606.30264 · v1 · pith:W2TPBRRXnew · submitted 2026-06-29 · 🌌 astro-ph.HE · hep-ph

Systematic study of the morphology and length of slow stable hybrid star branches

Mauro Mariani , Milva G. Orsaria , Germ\'an Lugones , Ignacio F. Ranea-Sandoval This is my paper

Pith reviewed 2026-06-30 05:10 UTC · model grok-4.3

classification 🌌 astro-ph.HE hep-ph
keywords hybrid starsneutron starsphase transitionequation of stateslow conversionmass-radius relationhybrid branchstability
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The pith

When the hadron-quark phase conversion is slow, hybrid stars develop extended stable branches in the mass-radius plane that are inaccessible under rapid conversion.

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

The paper defines the length of the slow stable hybrid star branch as a measure of the extended stability region arising when the hadron-quark phase conversion timescale exceeds the radial oscillation period. It classifies these branches into four morphological types—waterfall, bridge, tail, and tail-bridge—based on their shape in the mass-radius diagram. Using a large set of hybrid equations of state, the study shows that viable long branches under current constraints are mostly waterfall type. Stiff hadronic equations of state, often excluded in rapid-conversion models, fit all constraints in the slow-conversion framework. This opens a new region of viable parameter space in the transition density versus density jump plane.

Core claim

The length of the slow stable hybrid star branch is introduced as a quantitative measure of the extended stability region that arises in hybrid neutron stars when the hadron-quark phase conversion is slow compared to the radial oscillation timescale. Combining generalized piecewise-polytropic hadronic equations of state of varying stiffness with a constant-speed-of-sound quark-matter model, four morphological types for the slow stable branch are identified: waterfall branches that descend monotonically from the hadronic maximum mass, bridges that connect the hadronic branch to a second unconditionally stable hybrid branch, tails that extend briefly beyond the maximum mass of an unconditional

What carries the argument

The slow stable hybrid star branch and its length as a quantitative measure of the extended stability region under slow phase conversion.

Load-bearing premise

The hadron-quark phase conversion is slow compared to the radial oscillation timescale.

What would settle it

A precise mass and radius measurement placing a hybrid star inside the slow-conversion extended branch but outside all rapid-conversion stable regions, combined with independent confirmation of hybrid composition, would support the claim; the opposite placement would challenge it.

Figures

Figures reproduced from arXiv: 2606.30264 by Germ\'an Lugones, Ignacio F. Ranea-Sandoval, Mauro Mariani, Milva G. Orsaria.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Pressure as a function of energy density for the three hadronic EoSs ( [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic mass-radius diagrams illustrating the four [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic mass-radius relationships illustrating the role of the CSS parameters on the SSHS branch morphology, for [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Mass-radius relationships for all hybrid EoSs constructed in this work, separated by the underlying hadronic model: [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Same data as Fig [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Color map of the SSHS branch length [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Hadronic baryon number density at the phase transition, [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
read the original abstract

We introduce and systematically study the length of the slow stable hybrid star branch as a quantitative measure of the extended stability region that arises in hybrid neutron stars when the hadron-quark phase conversion is slow compared to the radial oscillation timescale. Combining generalized piecewise-polytropic hadronic equations of state of varying stiffness with a constant-speed-of-sound quark-matter model, we construct a large set of hybrid equations of state spanning a broad range of transition pressures, energy-density jumps, and quark-matter speeds of sound. We identify four morphological types for the slow stable branch in the mass-radius plane: waterfall branches that descend monotonically from the hadronic maximum mass, bridges that connect the hadronic branch to a second unconditionally stable hybrid branch, tails that extend briefly beyond the maximum mass of an unconditionally stable hybrid branch, and tail-bridges that combine features of the latter two. Their prevalence is governed primarily by the transition pressure and the energy-density jump, while the branch length is also significantly influenced by the stiffness of the hadronic sector and the quark-matter speed of sound. Imposing current astrophysical and microphysical constraints shows that viable long branches are predominantly of waterfall type, and that stiff hadronic equations of state -- strongly disfavored under the rapid-conversion assumption -- remain compatible with all current constraints within the slow-conversion framework. In the plane of transition baryon density versus density jump, slow stable configurations open a new region of viable parameter space inaccessible under rapid conversions.

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 / 1 minor

Summary. The paper introduces the length of the slow stable hybrid star branch as a quantitative measure of the extended stability region in hybrid neutron stars when the hadron-quark phase conversion is slow compared to the radial oscillation timescale. Combining generalized piecewise-polytropic hadronic EOS of varying stiffness with a constant-speed-of-sound quark-matter model, the authors construct a large set of hybrid EOS spanning ranges of transition pressures, energy-density jumps, and quark-matter speeds of sound. They identify four morphological types for the slow stable branch in the mass-radius plane (waterfall, bridges, tails, tail-bridges), map their prevalence and lengths as functions of the input parameters, and apply current astrophysical and microphysical constraints to conclude that viable long branches are predominantly waterfall type while stiff hadronic EOS remain compatible, thereby opening a new region of viable parameter space in the transition baryon density versus density jump plane.

Significance. If the slow-conversion assumption holds over the explored parameter ranges, the work demonstrates that slow stable configurations open a previously inaccessible region of viable hybrid-star parameter space, potentially reconciling stiff hadronic EOS with observations. The large-scale numerical survey of EOS models and the morphological classification provide a concrete, reproducible framework for assessing phase-transition effects on stability; these are strengths that would remain useful even if the assumption is later refined.

major comments (2)
  1. [Abstract] Abstract: The central results on extended slow-stable branches and new viable parameter space rest on the assumption that hadron-quark conversion is slow relative to radial oscillation timescales. The abstract states the premise explicitly but supplies no microphysical bounds on conversion rate versus the ~ms dynamical time; if the assumption fails for the explored transition densities and jumps, the morphologies collapse to the standard rapid-conversion case and the new parameter-space region disappears. A concrete test would be to compare the assumed regime against specific conversion timescale calculations for the transition densities and jumps used in the survey.
  2. [Abstract] Abstract (constraints paragraph): The claim that viable long branches are predominantly of waterfall type and that stiff hadronic EOS remain compatible with all current constraints is load-bearing for the reopened-parameter-space conclusion, yet the manuscript provides no tabulated fractions, specific constraint values applied, or figures quantifying the fraction of models that survive versus those excluded under rapid versus slow conversion.
minor comments (1)
  1. The four morphological types are named but the precise criteria separating bridges from tail-bridges (or tails from waterfall) are not stated as explicit conditions on the mass-radius curve; adding a short definitional paragraph or equation would aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each major comment below and indicate the revisions we will make to improve the clarity and completeness of the paper.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central results on extended slow-stable branches and new viable parameter space rest on the assumption that hadron-quark conversion is slow relative to radial oscillation timescales. The abstract states the premise explicitly but supplies no microphysical bounds on conversion rate versus the ~ms dynamical time; if the assumption fails for the explored transition densities and jumps, the morphologies collapse to the standard rapid-conversion case and the new parameter-space region disappears. A concrete test would be to compare the assumed regime against specific conversion timescale calculations for the transition densities and jumps used in the survey.

    Authors: We agree that providing context on the validity of the slow-conversion assumption would strengthen the abstract. While a comprehensive microphysical calculation of conversion timescales for every parameter combination in our survey is beyond the scope of this work, we will revise the abstract to explicitly note that the results are conditional on the slow-conversion regime and add a short discussion in the introduction referencing literature on hadron-quark phase conversion rates (such as those based on nucleation theory). This will clarify the regime of applicability without performing new calculations. revision: partial

  2. Referee: [Abstract] Abstract (constraints paragraph): The claim that viable long branches are predominantly of waterfall type and that stiff hadronic EOS remain compatible with all current constraints is load-bearing for the reopened-parameter-space conclusion, yet the manuscript provides no tabulated fractions, specific constraint values applied, or figures quantifying the fraction of models that survive versus those excluded under rapid versus slow conversion.

    Authors: We acknowledge that the manuscript would benefit from quantitative support for these claims. In the revised version, we will add a table listing the specific astrophysical and microphysical constraints applied (e.g., maximum mass, radius limits from NICER and gravitational wave observations), along with the fractions of models in each morphological category that remain viable under both rapid and slow conversion assumptions. We will also include a brief description or additional panel in a figure to illustrate the survival rates in the transition density versus density jump plane. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from numerical EOS construction

full rationale

The paper constructs hybrid EOS models numerically by varying hadronic stiffness, transition pressure, density jump, and quark speed of sound, then computes branch morphologies and lengths as direct outputs of the Tolman-Oppenheimer-Volkoff solutions under the stated slow-conversion premise. No quantity is defined in terms of itself, no fitted parameter is relabeled as a prediction, and no load-bearing step reduces to a self-citation or imported uniqueness theorem. The slow-conversion assumption is an explicit modeling choice that enlarges the stability region; the four morphological types and the claim of new viable parameter space follow from the resulting mass-radius curves rather than by construction from the inputs.

Axiom & Free-Parameter Ledger

4 free parameters · 1 axioms · 0 invented entities

The central claim depends on the slow conversion assumption and the specific EOS parametrizations used to construct the hybrid models; free parameters are the varied inputs in the polytropic and quark matter models.

free parameters (4)
  • transition pressure
    Varied across a broad range to study effects on branch morphology and prevalence.
  • energy-density jump
    Varied to determine prevalence of branch types and length.
  • quark-matter speed of sound
    Varied as it significantly influences branch length.
  • hadronic stiffness parameters
    Generalized piecewise-polytropic parameters varied for different stiffness levels.
axioms (1)
  • domain assumption The phase conversion between hadronic and quark matter is slow compared to the radial oscillation timescale.
    This is the key assumption enabling the slow stable branch, stated in the abstract.

pith-pipeline@v0.9.1-grok · 5805 in / 1551 out tokens · 44274 ms · 2026-06-30T05:10:38.516063+00:00 · methodology

discussion (0)

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Reference graph

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