Proto-Strange Quark Stars

• Proto-strange quark stars are compact objects composed entirely of deconfined up, down, and strange quarks, exhibiting high temperatures and neutrino trapping immediately after core collapse.
• Theoretical models, such as the MIT bag model and density-dependent mass schemes, produce stiff equations of state that support masses above 2 M☉ with a unique M ∝ R³ scaling relation.
• Their evolution from hot, lepton-rich conditions to cold, mature states features delayed cooling, distinctive neutrino signals, and magnetic stabilization, offering observable signatures for next-generation detectors.

A proto-strange quark star (proto-SQS) is a newly formed compact stellar object hypothesized to be composed entirely of self-bound strange quark matter (SQM)—deconfined up, down, and strange quarks—under the Bodmer–Witten conjecture for absolutely stable SQM. The proto-SQS phase encompasses the first tens of seconds after core collapse, in which the matter is hot, lepton-rich, and neutrino-trapped, progressing to a cold, mature SQS after deleptonization and cooling. Theoretical treatments model the thermodynamic, transport, and structural properties using frameworks such as the MIT bag model (with fixed or density-dependent bag constant), density-dependent quark mass schemes, and isospin/temperature dependent mass models. These models yield stiff equations of state (EOSs) able to support gravitational masses exceeding $2\,M_\odot$, with unique scaling relations and thermal/chemical evolution distinct from proto-neutron stars and other compact objects (Bordbar et al., 2020, Torres et al., 2013, Issifu et al., 2023, Chen et al., 16 Jan 2026).

1. Physical Model and Thermodynamical Structure

Proto-strange quark star matter is assumed to be globally charge-neutral, $\beta$-equilibrated, and composed purely of quarks ($u$, $d$, $s$) and leptons (electrons, muons, trapped neutrinos). The MIT bag model posits quarks as an ideal Fermi gas confined to a volume with energy $B_\text{bag}$, either fixed (e.g., $B_\text{bag}=90\,\mathrm{MeV/fm}^3$) or density-dependent to capture medium effects:

$$B(n) = B_\infty + (B_0 - B_\infty)\exp\left[-\beta\left(\frac{n}{n_0}\right)^2\right]$$

Number, entropy, and energy densities are computed self-consistently via Fermi–Dirac distributions, with entropy per baryon $S$ treated as constant during the neutrino-trapping epoch. The total energy density,

$$\epsilon_\text{tot} = \sum_{i=u,d,s} \epsilon_i + B,$$

and pressure,

$$P = n_B \frac{d\epsilon_\text{tot}}{dn_B} - \epsilon_\text{tot},$$

are combined to yield structural properties via the Tolman–Oppenheimer–Volkoff (TOV) equations. Density-dependent mass models generalize this by introducing quark masses of the form $m_i(n_B,T)$, with confinement and perturbative correction terms, leading to thermodynamic potentials and EOSs that depend on both density and temperature (Issifu et al., 2023, Chen et al., 2021, Chen et al., 16 Jan 2026). The sound velocity $c_s^2$ approaches the conformal limit $1/3$ only at high densities where interactions are negligible.

2. Evolutionary Phases: Neutrino-Trapped to Cold SQS

The proto-SQS evolution proceeds through several quasi-equilibrium snapshots:

• Birth (~0–1 s): High entropy per baryon ($S/n_B=1$), electron lepton fraction $Y_{L,e}\simeq0.4$, trapped neutrinos, central temperatures $\sim9\,\mathrm{MeV}$.
• Heating/Deleptonization (~1–10 s): Entropy rises to $S/n_B\sim2$, $Y_{L,e}$ decreases, central $T$ increases ($15$–$21\,\mathrm{MeV}$), $s$-quark fraction grows, muons appear as $e^\pm$ chemical potential rises.
• Neutrino-transparent phase (~10–50 s): $Y_{\nu_e}\to0$, maximum core $T\sim34\,\mathrm{MeV}$, EOS softens, onset densities for $s$-quarks and muons shift down.
• Cold SQS ($T\to0$): Lepton-poor, entropy minimized, central densities maximize, mass-radius relation shifts downward with $M_\text{max}$ dropping by $\sim0.14\,M_\odot$ from birth to cold state; radii decrease by several percent (Chen et al., 16 Jan 2026, Issifu et al., 2023).

Stellar dynamics during these phases are computed using isentropic models and the TOV equations, with typical evolutionary tracks for fixed baryon mass showing monotonic decreases in gravitational mass and radius as deleptonization/cooling proceed.

3. Equation of State, Stability Criteria, and Maximum Mass

The EOS of proto-SQS is governed by the interplay of confinement (often modeled via $B$ or density-dependent masses), perturbative corrections, and temperature-dependent terms. Stability criteria are set by the free energy per baryon $F/A$; absolute stability under the Bodmer–Witten hypothesis requires

$(F/A)_\text{SM} < 930\,\mathrm{MeV}$

for three-flavor matter, with two-flavor ($u$, $d$) matter unbound at all relevant densities. Finite temperature widens the stability window, allowing higher $B$ or mass parameters for stable proto-SQSs at $T\sim$ $20$–$40\,\mathrm{MeV}$ (Torres et al., 2013, Dexheimer et al., 2013).

Maximum mass predictions strongly depend on the EOS:

Model / Stage	$M_\text{max}/M_\odot$	$R_\text{max}$ (km)
Isentropic MIT bag, $S=1.5$	$1.95$–$2.15$	$11$–$12$
Density-dependent mass model	$2.07$–$2.21$	$13$–$14$
CIDDM, finite $T$	$1.65$–$2.05$	$9.6$–$11.1$

Thermal pressure supports larger masses and radii at early times, which decrease upon cooling. The M–R relation for self-bound proto-SQSs typically obeys $M \propto R^3$, unlike neutron stars. Inclusion of strong magnetic fields ($\sim10^{18}$–$10^{19}$ G) further stabilizes SQM by widening stability windows and increasing $M_\text{max}$ (Dexheimer et al., 2013).

4. Transport, Cooling, and Neutrino Signal

Neutrino transport in proto-SQSs is governed by charged- and neutral-current quark-lepton reactions, calculated via explicit cross-section integrals over Fermi-Dirac distributions (Benvenuto et al., 2013). In the diffusion approximation, lepton-number and energy fluxes drive deleptonization and cooling:

• Deleptonization timescale: $\tau_\text{del} \sim 10$–$15$ s (longer than proto-neutron stars).
• Cooling timescale: $\tau_\text{cool} \sim 40$–$50$ s (twice as long as proto-neutron stars).
• Radial profiles: Initially, core temperatures are $\sim30$–$40$ MeV and $Y_{L,c}\sim0.3$, falling to $T\sim20$ MeV and $Y_{L,c}\sim0$ over tens of seconds.

Observable neutrino luminosities begin at $L_\nu\sim3\times10^{52}$ erg/s, decaying exponentially. Mean neutrino energies drop from $\sim20$ MeV to $\lesssim5$ MeV within $\sim10$ s, generating a longer-lived, softer burst than in standard PNSs—a distinctive prediction for next-generation neutrino detectors (Benvenuto et al., 2013).

5. Astrophysical Formation Pathways and Observational Constraints

Formation of proto-SQSs requires either core-collapse SNe at sufficient mass/density or conversion of proto-neutron stars via capture of strangelets (dark-matter nuggets), as in scenarios for objects such as HESS J1731–347 or compact binaries undergoing hypercritical fallback accretion (Clemente et al., 2022, Becerra et al., 29 Jul 2025). Fallback rates in compact binaries can be $\dot{M}\sim0.1\,M_\odot/\mathrm{s}$, raising central densities beyond the deconfinement threshold ($n_c\sim5\,n_0$), rapidly converting the core to SQM within $\lesssim10$ ms.

After conversion, conservation of baryon number and angular momentum yields slightly reduced gravitational mass and increased radius (e.g., $$M^\text{grav}_\text{NS}\rightarrow M^\text{grav}_\text{SQS}$$, $R_\text{NS}<R_\text{SQS}$). The released energy $(\Delta M)c^2$ can reach $10^{52}$–$10^{53}$ erg, producing a neutrino burst. Long-term cooling is slowed by color superconductivity, prolonging surface X-ray emission.

Observational constraints include mass–radius measurements (pulsars with $M\gtrsim2\,M_\odot$, radii $12$–$14$ km), thermal light curves (slow cooling in HESS J1731–347), and possible secondary gravitational-wave peaks following core collapse. SQSs with gravitational mass $<1.0\,M_\odot$ remain a robust possibility in low-metallicity, dark-matter-rich environments (Clemente et al., 2022, Becerra et al., 29 Jul 2025).

6. Key Theoretical Implications and Distinguishing Features

Proto-strange quark stars exhibit several distinguishing physical and observational signatures:

• Stiff EOS and high mass support: $M_\text{max}\sim2$–$2.3\,M_\odot$ is achievable, including high compactness and gravitational redshift, with $z_s$ increasing for more compact, hot configurations (Bordbar et al., 2020).
• Unique scaling: $M\propto R^3$ for self-bound SQSs, contrasting $M\propto R^{-3}$ for gravitationally bound neutron stars.
• Delayed cooling and softening neutrino signal: Longer emission times, lower mean neutrino energies, and secondary neutrino/thermal X-ray plateau(s).
• Magnetic stabilization: Strong fields widen model stability domains, enhancing both maximum mass and binding energy per baryon.
• Evolutionary trajectory: Transition from neutrino-trapped, hot proto-SQS to cold, mature SQS entails decreasing mass and radius and increasing central density/temperature.
• Chemical composition evolution: Deleptonization delays the onset of $s$-quark abundance, with muons emerging in later, lepton-poor stages (Chen et al., 16 Jan 2026, Issifu et al., 2023).

These features are testable by combining TOV-based mass–radius predictions, neutrino light-curve timing, and surface thermal evolution with ongoing pulsar and compact-object observations. Direct searches for macroscopic strangelets, gravitational-wave post-merger “knees”, and X-ray timing of central compact objects offer promising future diagnostics.