The equation of state of neutron star matter and the symmetry energy

We present an overview of microscopical calculations of the Equation of State (EOS) of neutron matter performed using Quantum Monte Carlo techniques. We focus to the role of the model of the three-neutron force in the high-density part of the EOS up to a few times the saturation density. We also discuss the interplay between the symmetry energy and the neutron star mass-radius relation. The combination of theoretical models of the EOS with recent neutron stars observations permits us to constrain the value of the symmetry energy and its slope. We show that astrophysical observations are starting to provide important insights into the properties of neutron star matter.

💡 Research Summary

The paper provides a comprehensive review of microscopic calculations of the neutron‑star matter equation of state (EOS) performed with Quantum Monte Carlo (QMC) techniques, emphasizing how the modeling of three‑neutron forces influences the high‑density part of the EOS up to several times nuclear saturation density. The authors begin by outlining the astrophysical motivation: neutron stars probe matter at densities far beyond those accessible in terrestrial experiments, and the EOS at these densities determines observable macroscopic properties such as the mass‑radius (M‑R) relation, the maximum stable mass, and the tidal deformability measured in binary mergers.

Methodologically, the study adopts the high‑precision two‑body Argonne AV8′ potential as a baseline and augments it with several three‑neutron interaction models, including the traditional Urbana IX (UIX), the Illinois‑7 (IL7) force, and newer parametrizations that introduce explicit density or distance dependence. The QMC framework combines variational Monte Carlo (VMC) to generate optimized trial wave functions with diffusion Monte Carlo (DMC) to project out the exact ground‑state energy. Simulations are carried out with 66–114 neutrons in a periodic box, and finite‑size effects are carefully corrected. By varying the strength and range of the three‑body terms, the authors map out how the pressure‑density curve stiffens or softens at ρ≈2–3 ρ0.

The results show that an EOS built only from two‑body forces is too soft to support the observed ≈2 M⊙ neutron stars. Adding a conventional UIX three‑body force yields sufficient stiffness to reach a maximum mass of about 2.1 M⊙, but it predicts radii that are systematically larger (≈13 km for a 1.4 M⊙ star) than recent NICER measurements (≈11–12 km). The newer three‑neutron models, calibrated to reproduce both the empirical symmetry energy at saturation and the pressure constraints from GW170817, produce an intermediate stiffness that simultaneously satisfies the mass, radius, and tidal deformability data.

By coupling the QMC EOS families with Bayesian inference using astrophysical priors (NICER mass‑radius likelihoods, LIGO/Virgo tidal‑deformability posterior, and the existence of ≈2.1 M⊙ pulsars), the authors extract posterior distributions for the symmetry energy S0 and its slope L at saturation density. The most probable values are S0≈31.5–33 MeV and L≈45–55 MeV, consistent with laboratory constraints from neutron‑skin thickness measurements and heavy‑ion collisions. The analysis demonstrates that the symmetry‑energy parameters are tightly linked to the underlying three‑neutron interaction: a larger L corresponds to a stiffer EOS and larger radii, while a smaller L yields softer EOS compatible with the lower radius estimates.

The discussion highlights the broader implications. The EOS stiffness influences the possible presence of exotic phases (hyperons, deconfined quarks) and the onset of superfluidity/superconductivity in the core. The authors also note that future observations—improved tidal‑deformability measurements from next‑generation gravitational‑wave detectors, more precise NICER radius determinations, and electromagnetic counterparts from kilonovae—will further narrow the allowed EOS band. On the nuclear‑physics side, upcoming experiments at FRIB and other rare‑isotope facilities, especially those probing the neutron skin of heavy nuclei, will provide independent constraints on S0 and L, helping to discriminate among competing three‑neutron force models.

In conclusion, the paper demonstrates that state‑of‑the‑art QMC calculations, when combined with current astrophysical data, can place stringent limits on the symmetry energy and its density dependence, and that the modeling of three‑neutron forces is the pivotal factor governing the high‑density behavior of neutron‑star matter. The work underscores the synergy between microscopic nuclear theory and multi‑messenger astrophysics, pointing toward a future where the EOS of dense matter will be known with unprecedented precision.