Based on an extended Skyrme interaction that includes the terms in relative momenta up to sixth order, corresponding to the so-called Skyrme pseudopotential up to next-to-next-to-next-to leading order (N3LO), we derive the expressions of Hamiltonian density and single nucleon potential under general non-equilibrium conditions which can be applied in transport model simulations of heavy-ion collisions induced by neutron-rich nuclei. While the conventional Skyrme interactions, which include the terms in relative momenta up to second order, predict an incorrect behavior as a function of energy for nucleon optical potential in nuclear matter, the present extended N3LO Skyrme interaction can give a nice description for the empirical nucleon optical potential. We also construct three interaction sets with different high-density behaviors of the symmetry energy, by fitting both the empirical nucleon optical potential up to energy of 1 GeV and the empirical properties of isospin asymmetric nuclear matter. These extended N3LO Skyrme interactions will be useful in transport model simulations of heavy-ion collisions induced by neutron-rich nuclei at intermediate and high energies, and they can also be useful in nuclear structure studies within the mean-field model.

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While the equation of state (EOS) of symmetric nuclear matter (SNM) at suprasaturation densities has been relatively well constrained from heavy-ion collisions, the EOS of high-density neutron-rich matter is still largely uncertain due to the poorly known high-density behavior of the symmetry energy. Using the constraints on the EOS of SNM at suprasaturation densities from heavy-ion collisions together with the data of finite nuclei and the existence of 2M⊙ neutron stars from electromagnetic (EM) observations, we show that the high-density symmetry energy cannot be too soft, which leads to lower bounds on dimensionless tidal deformability of Λ1.4 ≥ 193 and radius of R1.4 ≥ 11.1 km for 1.4M⊙ neutron star. Furthermore, we find that the recent constraint of Λ1.4 ≤ 580 from the gravitational wave signal GW170817 detected from the binary neutron star merger by the LIGO and Virgo Collaborations rules out too stiff high-density symmetry energy, leading to an upper limit of R1.4 ≤ 13.3 km. All these terrestrial nuclear experiments and astrophysical observations based on strong, EM and gravitational measurements together put stringent constraints on the high-density symmetry energy and the EOS of SNM, pure neutron matter and neutron star matter.

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