Constraining neutron-star matter with microscopic and macroscopic collisions

Abstract

Interpreting high-energy, astrophysical phenomena, such as supernova explosions or neutron-star collisions, requires a robust understanding of matter at supranuclear densities. However, our knowledge about dense matter explored in the cores of neutron stars remains limited. Fortunately, dense matter is not probed only in astrophysical observations, but also in terrestrial heavy-ion collision experiments. Here we use Bayesian inference to combine data from astrophysical multi-messenger observations of neutron stars^1-9 and from heavy-ion collisions of gold nuclei at relativistic energies^10,11 with microscopic nuclear theory calculations^12-17 to improve our understanding of dense matter. We find that the inclusion of heavy-ion collision data indicates an increase in the pressure in dense matter relative to previous analyses, shifting neutron-star radii towards larger values, consistent with recent observations by the Neutron Star Interior Composition Explorer mission^5-8,¹⁸. Our findings show that constraints from heavy-ion collision experiments show a remarkable consistency with multi-messenger observations and provide complementary information on nuclear matter at intermediate densities. This work combines nuclear theory, nuclear experiment and astrophysical observations, and shows how joint analyses can shed light on the properties of neutron-rich supranuclear matter over the density range probed in neutron stars.

Fig. 1. Constraints on the EOS of neutron-star matter.

a–d, Evolution of the pressure as a function of baryon number density for the EOS prior (a, grey), when including only data from multi-messenger neutron-star observations (b, green), when including only HIC data (c, orange), and when combining both (d, blue). The shading corresponds to the 95% and 68% credible intervals (lightest to darkest). The impact of the HIC experimental constraint (HIC data, purple lines at 95% and 68%) on the EOS is shown in c. In b–d, the 95% prior bound is shown for comparison (grey dashed lines). e, f, Posterior distributions for the pressure at 1.5n_sat (e) and 2.5n_sat (f) at different stages of our analysis, with the combined Astro + HIC region shaded in light blue.

Fig. 2. Constraints on the mass and radius of neutron stars.

a–d, The 95% and 68% credible ranges for the neutron-star radius across various masses (up to the 95% upper bound on the maximum allowed mass, as only few EOSs support mass beyond that, which would result in an unrepresentative credible range) for the prior (a, grey), when including only multi-messenger constraints (b, green), when including only HIC experiment data (c, orange) and for the joint constraint (d, blue). The prior 95% contour is shown in b–d for comparison. e, f, Posterior distributions for the radii of $1.4M_{\odot }$ (e) and $2M_{\odot }$ (f) stars at different stages of our analysis, with the combined Astro + HIC region shaded in light blue.

Extended Data Fig. 1. Constraints on γ_asy versus symmetry energy S₀ from two Quantum Molecular Dynamics models.

We show the exponent γ_asy of the density dependence of the potential part of the symmetry energy, see Eq. (7), as deduced from the analysis of ASY-EOS experimental data using the UrQMD model used in this work (red points) and new simulations from the IQMD model (blue points). The red line indicates the mean value for γ_asy along the linear interpolation for the chosen range of S₀. Overall, the models are in good agreement with each other and the results suggest that a linear interpolation is reasonable.

Extended Data Fig. 2. Comparison between different sensitivity curves.

We show three sensitivity-to-density curves for different observables and incident energies. In particular, the neutron-over-charged-particle (n/ch, used here) and the neutron-over-proton (n/p) sensitivity curves for 400 MeV/nucleon incident energy from Russotto et al. are compared with the density curve reported by Le Fèvre et al. for the sensitivity of the elliptic flow of protons in Au+Au collisions at 1 GeV/nucleon.

Extended Data Fig. 3. Constraint on the neutron-star mass and radius with successive astrophysics information.

In each panel (except for panel A), EOSs within (outside of) 95% credible interval are shown as blue (grey) lines. Lower panels indicate the probability distribution function (PDF) for the radius of a $1.4M_{\odot }$ neutron star, with the 95% confidence range indicated by dashed lines, in panels (B)-(F) the prior from panel (A) is shown in grey. (A) The EOS prior set constrained by chiral EFT calculations up to 1.5n_sat and $M_{\max }\ge 1.9M_{\odot }$. (B) The EOS set restricted by incorporating information from mass measurements of PSR J0348+0432, PSR J1614-2230, and the maximum-mass constraints obtained from GW170817/AT2017gfo. The 95% confidence interval of the maximum mass posterior probability distribution is shown by the purple band. (C) The EOS set further restricted by the NICER mass-radius measurement of PSR J0030+0451 (purple contours at 68% and 95% confidence) and PSR J0740+6620 (orange contours at 68% and 95% confidence). Note that the latter shows the average of the results obtained by Miller et al. and Riley et al.. (D) Further restrictions on the EOS set from a reanalysis of the GW170817 using Bayesian inference. Contours at 68% and 95% confidence show the mass-radius measurements of the primary (red) and secondary (orange) neutron stars. (E) We use the chirp mass, mass ratio, and the EOSs as Bayesian prior for our analysis of AT2017gfo. (F) Further restrictions by analysing GW190425. Again, contours at 68% and 95% confidence show the mass-radius measurements of the primary (red) and secondary (orange) neutron stars. (G) The radius constraint at each step of this analysis with 95% confidence ranges. The radius constraint after including HIC experimental data is also shown.

Extended Data Fig. 4. Constraints for pure neutron matter.

Energy per particle E/N of neutron matter as a function of density n for various many-body calculations using chiral EFT interactions from Hebeler et al., Tews et al., Lynn et al. (used here), Drischler et al. PRL and GP-B, and low-density quantum Monte Carlo results from Gezerlis and Carlson. Overall, the results from these calculations are in good agreement with each other. We also show the energy per particle of a unitary Fermi gas of neutrons, which has been proposed as a lower bound for the energy of neutron matter. Finally, we compare the theoretical results with the constraint from the ASY-EOS and FOPI experiments (red), which is used as a constraint for neutron matter in the main work.

Extended Data Fig. 5. Comparison of the pressure of symmetric nuclear matter for experiment and theory.

The pressure band from the FOPI experiment at the 1σ level (red) for the incompressibility is consistent with the chiral EFT constraint from Drischler et al.^, at N²LO (light blue) and N³LO (dark blue). The experimental uncertainty band is smaller than the theoretical one because the empirical saturation point used for extracting the experimental results has smaller uncertainties compared to theoretical estimates from chiral EFT. Between 2-3n_sat, we additionally constrain the FOPI results with the constraint from Danielewicz et al. (green), which has no statistical interpretation. This excludes the highest values for the incompressibility K from the FOPI distribution and also influences symmetric matter at smaller densities, which depends on the range of incompressibility K. However, both constraints are in very good agreement with each other and the impact of the additional Danielewicz et al. constraint is small in our analysis.