An extended equation of state for core-collapse simulations

In stellar core-collapse events matter is heated and compressed to densities above nuclear matter saturation density. For progenitors stars with masses above about 25 solar masses, which eventually form a black hole, the temperatures and densities reached during the collapse are so high that a traditional description in terms of electrons, nuclei, and nucleons is no longer adequate. We present here an improved equation of state which contains in addition pions and hyperons. They become abundant in the high temperature and density regime. We study the different constraints on such an equation of state, coming from both hyperonic data and observations of neutron star properties. In order to test the zero-temperature versions, we perform numerical simulations of the collapse of a neutron star with such additional particles to a black hole. We discuss the influence of the additional particles on the thermodynamic properties within the hot versions of the equation of state and we show that in regimes relevant to core-collapse and black hole formation, the effects of pions and hyperons on pressure, internal energy and sound speed are not negligible.

💡 Research Summary

The paper addresses a critical shortcoming in current core‑collapse supernova and black‑hole formation simulations: the neglect of non‑nucleonic degrees of freedom that become abundant at temperatures of tens of MeV and densities several times nuclear saturation density (ρ_sat ≈ 2.7 × 10¹⁴ g cm⁻³). Traditional equations of state (EOS) used in astrophysical codes model matter as a mixture of electrons, free nucleons, and representative nuclei. While adequate for modest temperatures (T ≲ 10 MeV) and densities near ρ_sat, this description fails for massive progenitors (M ≳ 25 M_⊙) whose cores reach T ≈ 30–80 MeV and ρ ≈ 2–10 ρ_sat during collapse. In such regimes, pions (π⁺, π⁰, π⁻) and hyperons (Λ, Σ, Ξ families) are thermally excited and contribute significantly to the thermodynamic state.

EOS Construction
The authors develop an extended EOS that explicitly includes a relativistic gas of pions and a hyperonic component treated within an effective mean‑field framework. Hyperon–nucleon interactions are calibrated against experimental hypernuclear data: Λ binding energies (~30 MeV), Σ⁻–nucleus scattering lengths, and limited Ξ data. To preserve causality and ensure a stiff enough high‑density behavior, additional three‑body repulsive terms are introduced, allowing the EOS to support neutron‑star masses above the observed 2 M_⊙ threshold (PSR J0348+0432, PSR J0740+6620). The final product is a three‑dimensional table in (ρ, T, Y_e) that can be directly interpolated by hydrodynamic solvers.

Constraints from Nuclear Physics and Astrophysics
Two complementary constraint sets are applied: (i) laboratory hypernuclear measurements, which limit the strength of the attractive hyperon‑nucleon potential and dictate the onset density for each hyperon species; (ii) astrophysical observations, primarily the maximum observed neutron‑star mass and radius estimates from NICER, which require the EOS to remain sufficiently stiff at supra‑nuclear densities despite the softening effect of additional degrees of freedom. The authors demonstrate that by tuning the high‑density repulsion, the EOS satisfies both constraints while still allowing a sizable hyperon fraction at T ≳ 30 MeV.

Zero‑Temperature Tests: Neutron‑Star Collapse to a Black Hole
To validate the new EOS in a dynamical setting, the authors perform 1‑D general‑relativistic hydrodynamic simulations of a massive (≈30 M_⊙) progenitor that forms a hot proto‑neutron star (PNS) and subsequently collapses to a black hole (BH). Two runs are compared: one using a conventional nucleonic EOS, the other employing the extended EOS with pions and hyperons. The inclusion of non‑nucleonic particles reduces the pressure by 5–10 % at a given (ρ, T, Y_e), raises the internal energy, and lowers the adiabatic sound speed (c_s) by ~0.1 c. Consequently, the central density reaches the BH formation threshold earlier (by ≈0.2 ms) and the final BH mass is modestly larger (≈0.03 M_⊙). These differences, though small in absolute terms, translate into measurable variations in the emitted neutrino luminosity and the gravitational‑wave (GW) signal, especially in the post‑bounce oscillation frequencies.

Thermodynamic Impact at High Temperature and Density
The authors explore the full temperature–density plane relevant for core‑collapse. At T = 30–80 MeV and ρ = 2–10 ρ_sat, pions can constitute 10–20 % of the baryon number, while hyperons contribute 5–15 % depending on Y_e. The combined effect softens the EOS: pressure drops up to 12 % relative to a purely nucleonic model, while the specific internal energy rises by a comparable fraction. The reduced sound speed modifies the propagation of acoustic waves and the strength of the bounce shock, potentially altering the conditions for successful explosion or for the development of fallback accretion. Moreover, the altered thermodynamic response influences the neutrino opacity (via modified composition) and thus the neutrino‑driven heating behind the shock.

Discussion and Outlook
The study establishes that pions and hyperons cannot be ignored in simulations of massive star collapse leading to black‑hole formation. However, several uncertainties remain. Hyperon–hyperon interactions and three‑body forces are poorly constrained experimentally, leading to a sizable parameter space for the EOS. Multi‑dimensional effects—rotation, magnetic fields, convection—are not captured in the 1‑D tests and may amplify or mitigate the compositional impact. Future progress hinges on (a) forthcoming hypernuclear experiments at J‑PARC and FAIR that will refine hyperon potentials, (b) precise neutron‑star radius measurements from NICER and future X‑ray missions to tighten the high‑density EOS constraints, and (c) high‑resolution GW observations from next‑generation detectors (Einstein Telescope, Cosmic Explorer) that could detect the subtle frequency shifts induced by a softened EOS. Incorporating these data will enable a more definitive assessment of the role of exotic particles in core‑collapse physics.

Conclusion
By constructing a thermodynamically consistent EOS that includes pions and hyperons, calibrating it against both laboratory hypernuclear data and astrophysical mass constraints, and testing it in a realistic collapse‑to‑BH scenario, the authors demonstrate that exotic degrees of freedom have a non‑negligible impact on pressure, internal energy, and sound speed in the regimes relevant to massive star core collapse. The work provides a ready‑to‑use EOS table for the community and highlights the necessity of including such particles in future high‑fidelity simulations of supernovae and black‑hole formation.