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 _ { c } = 0.11 fm ^ { -3 } extracted from nuclear mass and the neutron skin thickness of Sn isotopes .
Our results indicate that the symmetry energy can not 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 .