Toward a unified understanding of the dense-matter equation of state

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 constrain 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 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 collaborative efforts to combine these individual sources of information, focusing on: the Nuclear Physics and Multi-Messenger Astrophysics (NMMA) framework, which relies on Bayesian inference methods; the Modular Unified Solver for the Equation of State (MUSES) Calculation Engine, which integrates EOS priors with HIC data and produces predictions for key neutron star properties; and the Bayesian Analysis of Nuclear Dynamics (BAND) framework, which uses cutting-edge Bayesian methods to produce reliable and trustworthy predictions for nuclear and astrophysical problems. We highlight the scientific advances with respect to the EOS and neutron star properties made possible by each framework and outline the remaining challenges that must be addressed to build a coherent, predictive picture of dense nuclear matter across all relevant regimes. We conclude with a detailed discussion of how these frameworks might be integrated with each other to form a unified workflow for future EOS predictions.