Rotational enhancement and stability of protoquark stars during thermal evolution

This study presents the first systematic analysis of rigidly rotating protoquark stars within the density-dependent quark mass framework, revealing that thermal evolution and rapid rotation significantly enhance stellar stability and deformation, thereby creating distinct observational signatures that future multimessenger data must account for to robustly identify quark matter in compact stars.

The Gist

Imagine a cosmic construction site where the universe is building the densest, most extreme objects possible: Quark Stars. These are the "super-cousins" of neutron stars, made not of neutrons, but of a soup of fundamental particles called quarks.

This paper is like a detailed architectural blueprint for a very specific, short-lived phase of these stars' lives: their infancy. Just after a star is born, it is incredibly hot, spinning wildly, and full of trapped particles called leptons (like neutrinos). The authors, Adamu Issifu and his team, wanted to understand how rotation (spinning) and heat affect these baby stars as they grow up and cool down.

Here is the story of their findings, explained simply:

1. The Spinning Top Effect

Imagine a figure skater. When they pull their arms in, they spin faster. But if they are made of a special, stretchy material, spinning them out actually makes them heavier (in terms of how much mass they can hold before collapsing).

The paper finds that for these baby quark stars, spinning is a superpower.

• The Claim: If a quark star spins fast enough (approaching the speed where it would fly apart), it can support 40% more mass than if it were standing still.
• The Analogy: Think of a spinning pizza dough. The centrifugal force pushes the dough outward, making it wider and flatter. This "outward push" acts like a safety net, holding up more weight than a stationary dough could. For these stars, that safety net is so strong it lets them carry nearly half again as much mass as a non-spinning twin.

2. The "Hot and Leaky" Phase

When these stars are first born, they are like a pressure cooker full of hot steam and trapped particles.

• The Claim: As the star cools down and lets these particles escape (a process called "deleptonization"), it shrinks.
• The Analogy: Imagine a giant, hot, fluffy cloud. As the sun comes out and the cloud cools, the water droplets condense, and the cloud shrinks into a smaller, denser ball.
  • Hot Baby Star: Big, puffy, and can hold a lot of mass because it's "inflated" by heat and trapped particles.
  • Cold Adult Star: Compact, dense, and smaller.
  • The Twist: The authors found that the "hot" version of the star is actually larger and less dense than the "cold" version, which is the opposite of what happens with regular neutron stars (which get bigger as they cool).

3. The Danger Zone (Wobbling and Waves)

Because these stars are spinning so fast and are so "squishy" (deformable), they are unstable.

• The Claim: The energy of their spin is nearly 20% of the energy holding them together. This is a very high ratio.
• The Analogy: Imagine a spinning top that is wobbling so violently it's about to fly apart. The paper suggests these stars are in a "wobbly" state where they are very likely to emit gravitational waves (ripples in space-time). They are essentially screaming at the universe, "Look at me!" through these ripples, especially when they are young and hot.

4. The Two Blueprints

The researchers didn't just use one set of rules; they tested two different "recipes" (models) for how quarks interact, based on real data from telescopes and gravitational wave detectors.

• Recipe A (Stiffer): Makes the star harder to squish. It supports more mass but is a bit more rigid.
• Recipe B (Softer): Makes the star easier to squish. It supports slightly less mass but allows the star to spin faster and deform more easily.
• The Result: Both recipes agree on the main story: Spinning makes the star bigger and heavier, and cooling makes it shrink. However, the exact numbers (like how big the star is or how fast it spins) depend on which recipe you use.

5. Why This Matters for Detecting Them

The authors argue that if we want to find these quark stars in the future, we can't just look at their size or mass. We have to look at the whole picture:

• How fast are they spinning?
• How hot are they?
• How much do they wobble?

If we see a star that is huge, spinning incredibly fast, and wobbly, it might be a "baby" quark star. If we see a small, cold, slow spinner, it might be a "grown-up" one. The paper concludes that to identify these mysterious objects, astronomers need to combine data about their heat, spin, and size all at once.

Summary

In short, this paper says: Baby quark stars are like giant, hot, spinning balloons. Spinning them makes them huge and able to hold more weight. As they cool down, they shrink and tighten up. Because they spin so fast while they are young, they are very likely to send out detectable ripples in space-time, giving us a unique way to spot them before they cool down and become harder to distinguish from other stars.

Technical Summary

Technical Summary: Rotational Enhancement and Stability of Protoquark Stars During Thermal Evolution

Problem Statement
The theoretical existence of compact objects composed of self-bound up, down, and strange quarks (quark stars, or QSs) remains a subject of significant interest in nuclear astrophysics. While static models of quark matter (QM) exist, there is a lack of systematic understanding regarding the rotational properties of proto-quark stars (proto-QSs) during their early thermal evolution. Specifically, the interplay between rapid rigid rotation, thermal effects (finite temperature and lepton fraction), and the density-dependent quark mass (DDQM) equation of state (EOS) has not been fully explored. Distinguishing QSs from hadronic neutron stars (NSs) requires identifying unique signatures in macroscopic observables such as mass, radius, moment of inertia, and quadrupole moments, particularly in the context of the "quark-nova" scenario where an NS transitions to deconfined QM.

Methodology
The authors present the first systematic study of rigidly rotating proto-QSs using a quasi-static equilibrium approach. The methodology involves the following components:

• Equation of State (EOS): The study utilizes the Density-Dependent Quark Mass (DDQM) framework. Two specific parameter sets, derived via Bayesian inference from recent astrophysical data in a prior work [27], are employed:
  1. A "stiffer" set ($C=0.8, \sqrt{D}=127.4$ MeV) yielding higher maximum masses.
  2. A "softer" set ($C=0.65, \sqrt{D}=133.2$ MeV) yielding maximum masses slightly above the $2 M_\odot$ threshold.
     The EOS incorporates finite-temperature effects and lepton fractions ($Y_l$) using a formalism that accounts for thermal screening and the excitation of quark-antiquark pairs.
• Evolutionary Stages: The thermal evolution is modeled as a sequence of four quasi-static snapshots representing distinct stages:
  1. Hot, lepton-rich matter ($s_B=1, Y_l=0.4$).
  2. Intermediate deleptonization ($s_B=2, Y_l=0.2$).
  3. Neutrino-transparent, lepton-poor stage ($s_B=2, Y_{\nu_e}=0$).
  4. Cold, catalyzed quark star ($T=0$).
• Rotational Modeling: Equilibrium configurations are constructed using the open-source rns code, which solves Einstein's field equations for stationary, axisymmetric spacetimes. The study assumes rigid rotation and explores configurations from non-rotating ($r_p/r_e = 1.0$) to the Keplerian mass-shedding limit ($r_p/r_e \approx 0.5$).
• Observational Constraints: The model parameters are tested against current multimessenger data, including mass-radius measurements from HESS J1731–347, PSR J0030+0451, PSR J0740+6620, and constraints from GW170817.

Key Contributions

• First Systematic Isentropic Study: This work provides the first comprehensive analysis of rotating proto-QSs based on isentropic EOS within the DDQM framework, bridging the gap between hot, lepton-rich formation stages and cold, catalyzed configurations.
• Complete Energy Decomposition: The authors perform a full energy decomposition of rotating proto-QSs, calculating gravitational potential energy ($|W|$), rotational kinetic energy ($T_{kin}$), binding energy ($E_{bind}$), and internal energy ($U$) across the evolutionary sequence.
• Thermal-Rotational Ordering: The study establishes a clear ordering of stellar properties driven by thermal evolution, demonstrating how deleptonization and cooling systematically alter rotational observables.
• Differentiation from Hadronic Stars: The paper explicitly contrasts the behavior of rotating proto-QSs with hadronic protoneutron stars, highlighting qualitative differences in mass-radius relations, moments of inertia, and quadrupole deformations.

Results

• Rotational Enhancement: Rotation substantially enhances the maximum stable mass of proto-QSs. At the Keplerian limit, the maximum mass increases by up to $\sim 40\%$ compared to the non-rotating case, a figure significantly higher than the typical $\sim 20\%$ enhancement observed in hadronic stars.
• Thermal Evolution Trends:
  • Mass and Radius: All stellar properties (mass, equatorial radius, angular momentum, moment of inertia, and quadrupole moment) peak during the hot, lepton-rich stages and decrease monotonically as the star cools and deleptonizes.
  • Temperature: Core temperatures are generally lower in QM than in hadronic matter at fixed entropy due to the higher effective degeneracy of quark degrees of freedom (color and spin).
  • Keplerian Frequency: Hot, lepton-rich configurations approach the Keplerian limit at lower masses compared to cold configurations. As the star cools and the EOS stiffens, the maximum mass and Keplerian frequency increase, allowing cold QSs to sustain higher absolute spin frequencies.
• Stability and Instabilities: Near the Keplerian limit, the ratio of rotational kinetic energy to gravitational potential energy ($T_{kin}/|W|$) reaches values of $0.18–0.19$. This indicates a heightened susceptibility to non-axisymmetric instabilities (such as the Chandrasekhar–Friedman–Schutz instability), suggesting that rapidly rotating proto-QSs are promising sources of continuous gravitational waves.
• Observational Consistency:
  • The "softer" EOS parameterization is consistent with a broader set of observational constraints, including GW170817, PSR J0030+0451, PSR J0740+6620, and HESS J1731–347.
  • The "stiffer" parameterization satisfies constraints from HESS J1731–347 and PSR J0740+6620 but does not align with the full set of constraints (e.g., GW170817) in the static limit.
  • The study notes that no single static EOS may reproduce all observed compact object candidates, suggesting a population coexistence of hadronic and quark stars.

Significance and Claims
The paper claims that the rotational and thermal properties of proto-QSs offer a distinct signature in the mass–radius–spin plane that differentiates them from hadronic configurations. Specifically, rotating proto-QSs exhibit larger radii, higher moments of inertia, and stronger quadrupolar deformations compared to hadronic protoneutron stars of similar mass.

The authors emphasize that future multimessenger observations must account for both the thermal history and the rotational state of compact stars to robustly identify quark matter. The results suggest that the combination of mass, radius, spin, and gravitational-wave data (specifically regarding quadrupole moments and $T_{kin}/|W|$ ratios) is essential for constraining the underlying EOS and distinguishing between hadronic and quark interiors. The work establishes a framework for future extensions, including differential rotation, magnetic fields, and dynamical stability studies, but does not propose specific new experimental proposals beyond the analysis of existing and future multimessenger data.