Powell and Müller (Phys. Rev. D 114, 023003, 2026) showed with 93 two-dimensional core-collapse supernova simulations that the high-density nuclear equation of state (EoS) remains a first-order uncertainty for the explosion outcome: with the nucleonic SFHo and SFHx EoS most of 15 progenitors explode, whereas with the CMF EoS, which contains a smooth crossover to deconfined quark matter at high density, only 2 of 15 explode. The authors explicitly flag what their demonstration leaves open: the result may change in three dimensions (3D), the neutrino interaction rates used were not consistent with each EoS, and a systematic scan over the full range of EoS allowed by current observational and theoretical constraints is still missing. The open problem is to convert this 2D sensitivity demonstration into a robust, predictive mapping from progenitor to explosion outcome: determine, with controlled microphysics, whether the pronounced CMF suppression of neutrino-driven explosions persists in 3D, and quantify the ensemble-level impact of allowed EoS uncertainties on explosion statistics and neutron-star/black-hole mass distributions. As of August 2026, no 3D simulations with a quark-crossover EoS, no simulations with EoS-consistent neutrino opacities, and no constrained-space EoS scan exist; a May 2026 review states verbatim that systematic 3D studies of the EoS-dependence of supernova explosions for many progenitor models are yet to be conducted.

Background

A core-collapse supernova (CCSN) is the terminal explosion of a massive star whose iron core, supported by electron degeneracy pressure, collapses to a proto-neutron star (PNS). In the standard neutrino-driven delayed-explosion mechanism, the bounce shock stalls within milliseconds and is revived only if neutrinos emitted from the hot PNS deposit enough energy in the semi-transparent 'gain region' behind the shock before the star collapses to a black hole. Whether this happens depends on a competition between the advection timescale of matter through the gain region and the neutrino heating timescale, and it is sensitive to the nuclear equation of state (EoS), the thermodynamic relation between density, temperature, composition, and pressure of matter above nuclear saturation density (≈2.7×1014 g cm−3\approx 2.7\times10^{14}\,\mathrm{g\,cm^{-3}}). The EoS sets the radius and contraction of the warm PNS, and thereby the luminosities and mean energies of the emitted neutrinos, which in turn control the heating conditions. Modern constraints from measured neutron-star masses and radii, heavy-ion collisions, and chiral effective field theory have narrowed the allowed EoS space considerably; widely used families include the nucleonic SFHo and SFHx EoS of Steiner, Hempel, and Fischer (2013), which differ mainly in their symmetry energy, the energy cost of neutron-to-proton number asymmetry in nuclear matter, and the CMF EoS of Motornenko et al. (2020), a chiral mean-field model (an effective relativistic model of dense hadronic matter built on approximate chiral symmetry) that includes a smooth crossover from nucleonic matter to deconfined quark matter at high density and is very stiff up to 55–6×1014 g cm−36\times10^{14}\,\mathrm{g\,cm^{-3}}. A separate line of work on first-order hadron–quark phase transitions (Sagert et al. 2009; Fischer et al. 2011; Jakobus et al. 2022) asked whether a second collapse triggered by deconfinement can launch an explosion; that 'phase-transition mechanism' remains unconfirmed, and for smooth crossovers such as CMF no second bounce occurs at all, so the CMF EoS enters the explosion problem only through the ordinary neutrino-driven channel. Powell and Müller (2026) ran 93 two-dimensional (2D) simulations with the CoCoNuT-FMT neutrino-hydrodynamics code (fast multigroup neutrino transport) for 15 solar-metallicity progenitors with zero-age main-sequence masses between 9.71 M⊙9.71\,M_\odot and 36.61 M⊙36.61\,M_\odot, each with all three EoS. They found explosions for almost all progenitors with SFHo and SFHx but only for two progenitors with CMF, tracing the difference to lower neutrino luminosities and mean energies and slightly weaker PNS contraction under CMF, and also found EoS sensitivity in the black-hole-formation channel, including quiet black-hole formation without shock revival. However, the neutrino interaction rates (opacities) employed were taken from a common set rather than computed consistently with each EoS, all simulations were 2D (hence lacking, e.g., spiral modes of the standing accretion shock instability (SASI), a global sloshing-and-spiralling instability of the stalled accretion shock, and genuine 3D turbulence), rotation and magnetic fields were omitted, and only three EoS were sampled. The authors therefore state that their almost-complete lack of CMF explosions and robust SFHo/SFHx explosions 'are not firm predictions', that a wider range of outcomes for both EoS classes remains plausible, and that it 'will be important to scan the range of allowed EoS more thoroughly and to also use neutrino opacities fully consistent with the EoS, which the current study cannot yet accomplish'.

Problem Statement

Determine, with controlled and EoS-consistent microphysics, how the high-density nuclear equation of state decides the fate of massive stars — the robust, predictive version of the sensitivity demonstrated by Powell & Müller (2026). The common objective uniting the required results is a quantitative mapping from progenitor to explosion outcome whose EoS uncertainty is known, and it comprises four jointly necessary, scientifically linked strands that the source work explicitly leaves open: (i) extend the EoS–explodability comparison to fully three-dimensional simulations for the same progenitor series, settling whether the near-complete suppression of neutrino-driven explosions for the quark-crossover CMF EoS (2 of 15 progenitors, versus explosions for almost all progenitors with the nucleonic SFHo and SFHx EoS) persists once 3D effects — spiral standing-accretion-shock-instability modes, genuine turbulence, rotation, and magnetic fields — are included, or whether 3D dynamics widens the range of successful explosions for CMF and of black-hole formation for SFHo/SFHx; (ii) redo the comparison with neutrino interaction rates (opacities) computed self-consistently with each EoS rather than from a common set, since EoS-dependent luminosities and mean energies are precisely the channel through which the EoS acts; (iii) quantify how additional micro- and macrophysics — neutrino flavour conversion, realistic 3D progenitor structure, magnetic fields — and numerical accuracy modify the outcomes and their EoS sensitivity; and (iv) scan the full range of EoS that satisfy current neutron-star mass, radius, and nuclear-theory constraints — including nucleonic and quark-crossover/hybrid families — systematically enough to quantify the ensemble-level impact of EoS uncertainties on explosion statistics (fractions of successful explosions, explosion energies, black-hole vs neutron-star formation channels, including fallback and quiet black-hole formation) and on the resulting neutron-star and black-hole birth-mass distributions. An answer should establish, for each EoS family and progenitor range, whether the explosion outcome is predictable from first principles at all, and conversely what constraints on the EoS could be inferred from observed explodability and compact-remnant demographics.

References

  1. Jade Powell and Bernhard Müller, 'Impact of the nuclear equation of state on the explodability of massive stars', Physical Review D 114, 023003 (2026), DOI: 10.1103/xlzf-k5wg, arXiv:2510.20076, https://arxiv.org/abs/2510.20076
  2. Bernhard Müller and Bailey Sykes, 'Core-collapse supernovae and supernova neutrinos', preprint of a chapter for the Encyclopedia of Nuclear Physics, section Nuclear Astrophysics (2026), arXiv:2605.25879, https://arxiv.org/abs/2605.25879
  3. Aleksandr Rusakov, Adam S. Burrows, Tianshu Wang, and David Vartanyan, 'An Exploration of the Equation of State Dependence of Core-Collapse Supernova Explosion Outcomes and Signatures' (2026), arXiv:2602.09025, https://arxiv.org/abs/2602.09025
  4. Mariam Gogilashvili and Irene Tamborra, 'Flavor Conversion Enhances or Suppresses Supernova Explodability Independent of the Progenitor Mass', Physical Review D 114, 043054 (2026), DOI: 10.1103/j5b2-5q2p, arXiv:2605.18972, https://arxiv.org/abs/2605.18972
  5. Kohsuke Sumiyoshi, Hajime Togashi, Shun Furusawa, Shuying Li, Hong Shen, Ken'ichiro Nakazato, and Hideyuki Suzuki, 'Influence of effective mass of the relativistic mean field theory on core collapse supernovae and compact objects' (2026), arXiv:2608.05582, https://arxiv.org/abs/2608.05582
  6. Andrew W. Steiner, Matthias Hempel, and Tobias Fischer, 'Core-collapse supernova equations of state based on neutron star observations', The Astrophysical Journal 774, 17 (2013), DOI: 10.1088/0004-637X/774/1/17, arXiv:1207.2184, https://arxiv.org/abs/1207.2184
  7. Anton Motornenko, Jan Steinheimer, Volodymyr Vovchenko, Stefan Schramm, and Horst Stoecker, 'Equation of state for hot QCD and compact stars from a mean field approach', Physical Review C 101, 034904 (2020), DOI: 10.1103/PhysRevC.101.034904, arXiv:1905.00866, https://arxiv.org/abs/1905.00866
  8. Pia Jakobus, Bernhard Müller, Alexander Heger, Anton Motornenko, Jan Steinheimer, and Horst Stoecker, 'The Role of the Hadron-Quark Phase Transition in Core-Collapse Supernovae', Monthly Notices of the Royal Astronomical Society 516, 2554 (2022), DOI: 10.1093/mnras/stac2352, arXiv:2204.10397, https://arxiv.org/abs/2204.10397