Symmetry energy systematics and its high density behavior

We explore the systematics of the density dependence of nuclear matter symmetry energy in the ambit of microscopic calculations with various energy density functionals, and find that the symmetry energy from subsaturation density to supra-saturation density can be well determined by three characteristic parameters of the symmetry energy at saturation density $\rho_0$, i.e., the magnitude $E_{\text{sym}}({\rho_0 })$, the density slope $L$ and the density curvature $K_{\text{sym}}$. This finding opens a new window to constrain the supra-saturation density behavior of the symmetry energy from its (sub-)saturation density behavior. In particular, we obtain $L=46.7 \pm 12.8$ MeV and $K_{\text{sym}}=-166.9 \pm 168.3$ MeV as well as $E_{\text{sym}}({2\rho _{0}}) \approx 40.2 \pm 12.8$ MeV and $L({2\rho _{0}}) \approx 8.9 \pm 108.7$ MeV based on the present knowledge of $E_{\text{sym}}({\rho_{0}}) = 32.5 \pm 0.5$ MeV, $E_{\text{sym}}({\rho_c}) = 26.65 \pm 0.2$ MeV and $L({\rho_c}) = 46.0 \pm 4.5$ MeV at $\rho_{\rm{c}}= 0.11$ fm$^{-3}$ extracted from nuclear mass and the neutron skin thickness of Sn isotopes. Our results indicate that the symmetry energy cannot be stiffer than a linear density dependence.In addition, we also discuss the quark matter symmetry energy since the deconfined quarks could be the right degree of freedom in dense matter at high baryon densities.