Toward a Unified Understanding of the Dense Matter Equation of State

TL;DR

The paper surveys how to constrain the dense matter EOS at supra-saturation densities by unifying information from heavy-ion collisions and multi-messenger astrophysics using three complementary frameworks (NMMA, BAND, MUSES). It details methods to extract EOS from HIC (observables like flow and sub-threshold kaons) and MMA (neutron star masses, radii, tidal deformabilities), and articulates Bayesian strategies to fuse these diverse data streams. It introduces NMMA for joint GW and EM analyses with nuclear priors, MUSES as a modular, open-source platform for assembling EOS modules across regimes, and BAND as a Bayesian toolkit for calibration, emulation, and model mixing to propagate uncertainties. The authors also outline actionable opportunities—benchmarking, emulators, and community tools—to push toward a fully unified, predictive description of dense nuclear matter across the QCD phase diagram with upcoming experimental and observational advances.

Abstract

Efforts to understand the equation of state (EOS) of dense nuclear matter at supra-saturation densities have grown more sophisticated over the past decade, driven by a surge in high-precision data from both terrestrial experiments and astrophysical observations. While for the former, heavy-ion collisions (HIC) represent a unique opportunity to constraint the EOS in a controlled laboratory setting, the latter can be precisely probed thanks to the advent of multi-messenger astronomy (MMA). However, as we move away from our understanding drawn from individual sources and limited statistics to the era of precision physics with improved datasets, the need for a systematic way to combine them becomes clear. In this article, we trace the individual methods for extracting the EOS both for HIC and MMA. We then review the current state-of-the-art efforts to combine these individual information sources from Bayesian multi-source analysis, e.g., the Nuclear Physics and Multi-Messenger Astrophysics (NMMA) and Bayesian Analysis of Nuclear Dynamics (BAND) frameworks, and fully integrated EOS frameworks, i.e., the Modular Unified Solver for the Equation of State (MUSES) calculation engine. We highlight the scientific advances made possible by each step and outline the remaining challenges that must be addressed to build a coherent, predictive picture of dense nuclear matter across all relevant regimes.

Figures (17)

• Figure 1: Schematic of the three-dimensional QCD phase diagram, represented as temperature ($T$), baryonic ($\mu_B$) and isospin ($\mu_I$) densities. A critical endpoint is conjectured to separate a smooth cross-over from a first-order phase transition, accompanied by a chiral phase transition at high densities. Exotic states such as quarkyonic matter or colour-superconducting phases could emerge at high baryon densities. Proton fractions for astrophysical systems evolve from 0.4 for supernovae down to 0.1 for cold neutron stars, where complementary measurements can be performed with heavy-ion collisions at FAIR, RHIC and NICA energies. Figure from H.R. Schmidt (EKU Tübingen and GSI Darmstadt) and taken from NupeccLRP2017.
• Figure 2: Snapshots of simulations showing the density evolution of a binary neutron star merger (top row) and a heavy ion-collision (bottom row). While the former is shown for two neutron stars, each $1.35M_{\odot}$, merging into a compact object, the latter is of a non-central Au+Au collision at 2.42 GeV per nucleon. Despite dramatically different spatial and temporal scales, both events show similar densities and temperatures. Figure from HADES:2019auv.
• Figure 3: Schematic overview of information sources of the dense matter EOS. Dark blue lines show candidate EOS up to their maximum mass configurations, while coloured bands mark the density ranges constrained by different inputs. Figure from Koehn:2024set.
• Figure 4: QCD Phase Diagram shown as a function of temperature ($T$) and baryon chemical potential ($\mu_B$). It highlights lattice QCD results, critical point predictions, freeze-out data and freeze-out calculations. The shaded areas show lattice QCD results for the chiral crossover Borsanyi:2020fev (dark blue shaded area), the deconfinement crossover Borsanyi:2024xrx (magenta shaded area), a constant entropy guided $2\sigma$ exclusion for the CEP Borsanyi:2025dyp (light blue shaded area) and the chiral crossover HotQCD:2018pds (red shaded area). The grey shaded area and the grey dot show the liquid-gas phase transition and its critical point calculated with the QvdW-HRG Poberezhnyuk:2019pxs. The critical point predictions are all shown as blue symbols. The CEP predictions are from: lQCD guided truncated Dyson-Schwinger approach with backcoupled mesons Gunkel:2021oya (star), lQCD guided functional Renormalization Group approach Fu:2019hdw (diamond), lQCD guided generalized DSE/fRG approach Gao:2020qsj (cross), finite size scaling of measured proton cumulants Sorensen:2024mry (square), lQCD guided extrapolations along constant entropy density contours Shah:2024img (circle), lQCD guided Bayesian inference in a holographic model Hippert:2023bel (triangle-down), lQCD guided non-perturbative holographic model Cai:2022omk (triangle-left), Chiral-Mean-Field model prediction constrained by combined neutron star $M-R$ and heavy-ion data Steinheimer:2025hsr (triangle-up), lQCD guided Padé resummation using a conformal map to track Lee-Yang edge singularities Basar:2023nkp (triangle-right), lQCD guided and Ising-, Thirring- and Roberge-Weiss consistent multi-point Padé expansion Clarke:2024seq (pentagon) and lQCD guided multi-point pade expansion to locate Lee-Yang singularities of pressure Clarke:2024ugt (thick diamond). Also shown are "freeze-out data" (various thermal model fits to measured yields, ratios or cumulants) as red dots. The data points are from STAR:2017salVovchenko:2015idtVovchenko:2018fmhLysenko:2024hqpAlba:2014ebaAndronic:2017pugBecattini:2016xctSagun:2017eye and fitted to data measured at SIS18, AGS, SPS, RHIC and LHC. The parametrized chemical freeze-out lines are taken from Andronic:2017pug (black dashed line), Lysenko:2024hqp (magenta dashed line) and using $\langle E\rangle /\langle N\rangle=0.94$ GeV Cleymans:1998fq (orange dashed line) and the hadronic transport model based calculations of the chemical freeze-out is from Reichert:2020yhx (green plusses).
• Figure 5: Selected constraints on the pressure $P(n_B)$ of symmetric matter Danielewicz:2002puFuchs:2003pcLynch:2009vcLeFevre:2015pajDrischler:2020yadOliinychenko:2022uvyOmanaKuttan:2022amlMohs:2024gyc (left) and the symmetry energy $S(n_B)$Tsang:2008fdRussotto:2011hqRussotto:2016ucmLynch:2021xkqKortelainen:2010hvKortelainen:2011ftBrown:2013mgaDanielewicz:2013upaPREX:2021umo (right) as functions of baryon density ($n_B/n_0$), extracted from comparisons of experimental data with hadronic transport simulations. Right figure from Sorensen:2023zkk.