Decoding the Density Dependence of the Nuclear Symmetry Energy

The large imbalance in the neutron and proton densities in very neutron rich systems increases the nuclear symmetry energy so that it governs many aspects of neutron stars and their mergers. Extracting the density dependence of the symmetry energy therefore constitutes an important scientific objective. Many analyses have been limited to extracting values for the symmetry energy, $S_0$, and its ``derivative'', $L$, at saturation density $\rho_0 \approx 2.6 \times 10^{14}~\mathrm{g/cm^3}$ $\approx 0.16~\mathrm{nucleons/fm^{3}}$, resulting in constraints that appear contradictory. We show that most experimental observables actually probe the symmetry energy at densities far from $\rho_0$, making the extracted values of $S_0$ or $L$ imprecise. By focusing on the densities these observables actually probe, we obtain a detailed picture of the density dependence of the symmetry energy from $0.25\rho_0$ to $1.5\rho_0$. From this experimentally derived density functional, we extract $L_{01}=53.1\pm6.1 MeV$ at $\rho \approx 0.10~\mathrm{fm^{-3}}$, a neutron skin thickness for $^{208}Pb$ of $R_{np} =$ $0.23\pm0.04$ fm, a symmetry pressure at saturation density of $P_0=3.2\pm1.2 MeV/fm^3$ and suggests a radius for a 1.4 solar mass neutron star of $13.1\pm0.6$ km.