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We cover here the present state-of-the-art in constraining the nuclear equation-of-state (EoS) and the symmetry energy using heavy-ion collisions (HIC), from sub- to supra-saturation densities, from Fermi to (ultra-) relativistic beam energies. We also discuss how HIC constraints on the EoS contribute to the knowledge of thermodynamical properties of neutron star matter. Necessary improvements and challenges are outlined, in particular in the perspective, for HICs, of staying competitive with future astrophysical multimessenger observations.

The ASY-EOS experiment at GSI laboratory measured the direct and elliptic flow of neutrons and light charged particles in the reaction 197Au+197 Au at 400 A MeV incident energy. The ratio of elliptic flow of neutrons with respect to that of the light charged particles was used as main experimental observable to probe the density dependence of the symmetry energy term of the nuclear equation of state. Results, obtained by comparison of the experimental data with the UrQMD model predictions, strongly support a moderately soft to linear density dependence of the symmetry energy at suprasaturation densities below 2ρ0.

A QMD transport model that employs a modified momentum dependent interaction (MDI2) potential, supplemented by a phase-space coalescence model fitted to experimental multiplicities of free nucleons and light clusters, is used to study the density dependence of the symmetry energy above the saturation point by a comparison with experimental elliptic flow ratios measured by the FOPI-LAND and ASYEOS collaborations for 197Au+197Au collisions at 400 MeV/nucleon impact energy. Comparing theoretical predictions with experimental data for neutron-to-proton and neutron-to-charged particles elliptic flow ratios the following constraint is extracted for the slope L and curvature Ksym of symmetry energy at saturation: L=59±24(exp)±16(th)±10(sys) MeV and Ksym=0±370(exp)±220(th)±150(sys) MeV. Theoretical errors are the result of poorer known model ingredients. Systematical uncertainties are generated by the inability of the transport model to reproduce experimental light-cluster-to-proton multiplicity ratios. A more accurate value for L, free of systematical theoretical uncertainties, can be extracted from the neutron-to-proton elliptic flow ratio alone: L=63±18(exp)±14(th) MeV.

An upgraded version of the isospin dependent T¨ubingen QMD transport model, which allows the conservation of the total energy, is presented. This is achieved by including in the energy-balance equations of the density, isospin asymmetry and momentum dependent inmedium baryon potential energies. It leads to an effective modification of particle production thresholds with respect to the vacuum ones. Compatible constraints for the symmetry energy stiffness from π−/π+ multiplicity ratio and elliptic flow experimental data of Au+Au collisions at 400 MeV/nucleon can be extracted in this case. However, an important dependence of the π−/π+ observable on the strength of the isovector part of the Δ(1232) isobar potential is also demonstrated. The present lack of information on this quantity prevents a precise extraction of the value for the symmetry energy stiffness employing the mentioned observable alone.

The elliptic-flow ratio of neutrons with respect to protons or light complex particles in reactions of heavy ions at pre-relativistic energies has been proposed as an observable sensitive to the strength of the symmetry term of the nuclear equation of state at supra-saturation densities. In the ASY-EOS experiment at the GSI laboratory, flows of neutrons and light charged particles were measured for 197Au+197Au collisions at 400 MeV/nucleon. Flow results obtained for the Au+Au system, in comparison with predictions of the UrQMD transport model, confirm the moderately soft to linear density dependence of the symmetry energy deduced from the earlier FOPI-LAND data.

The elliptic-flow ratio of neutrons with respect to protons or light complex particles in reactions of heavy ions at pre-relativistic energies has been proposed as an observable sensitive to the strength of the symmetry term of the nuclear equation of state at supra-saturation densities. In the ASY-EOS experiment at the GSI laboratory, flows of neutrons and light charged particles were measured for 197Au+197Au, 96Zr+96Zr and 96Ru+96Ru collisions at 400 MeV/nucleon with the Large Area Neutron Detector LAND as part of a setup with several additional detection systems used for the event characterization. Flow results obtained for the Au+Au system, in comparison with predictions of the UrQMD transport model, confirm the moderately soft to linear density dependence of the symmetry energy deduced from the earlier FOPI-LAND data.

The elliptic-flow ratio of neutrons with respect to protons or light complex particles in reactions of heavy-ions at pre-relativistic energies has been proposed as an observable sensitive to the strength of the symmetry term of the nuclear equation of state at supra-saturation densities. The results obtained from the existing FOPI/LAND data for 197Au+197Au collisions at 400 MeV/nucleon in comparison with the UrQMD model simulations favoured a moderately soft symmetry term, but suffer from a considerable statistical uncertainty. These results have been confirmed by an independent analysis based on the Tübingen QMD simulations. In order to obtain an improved data set for Au+Au collisions and to extend the study to other systems, a new experiment was carried out at the GSI laboratory by the ASY-EOS collaboration. The present status of the data analysis is reported