Nuclear incompressibility and fourth moment of the nuclear density in Skyrme functionals

Recent experimental advances could soon allow the accurate extraction of not only the root-mean-square radius but also the fourth radial moment of the nuclear electric charge density distribution. The fourth radial moment of the nuclear density distribution, $R_4\equiv\sqrt[4]{\left<r^4\right>}$, provides a sensitive probe of the nuclear surface thickness, as it is more susceptible to the large-$r$ distributions than the root-mean-square radius ($R_2$). In this work, we examine the utility of $R_4$ for constraining the nuclear equation of state (EoS) at subsaturation densities, specifically for the proton distribution and within the framework of Skyrme energy density functionals. Using a statistical analysis based on predictions from one hundred Skyrme functional models, we demonstrate strong correlations between the energy per particle curvature $K(\rho)$ at $\rho = 0.08 \text{ fm}^{-3}$ and $R_4$ (or the ratio $R_{4/2}=R_4/R_2$) in representative nuclei such as $\text{}^{48}\text{Ca}$ and $\text{}^{208}\text{Pb}$. We establish that $R_{4/2}$, being sensitive to the density tail, serves as an efficient proxy for sub-saturation $K(\rho)$ within the tested Skyrme functional space. Knowledge of $R_{4/2}$ within 0.5\% precision or better, for example in $^{48}$Ca or $^{208}$Pb, could constrain the curvature of the energy per particle of symmetric matter at $0.08$ fm$^{-3}$ within 20 MeV or less.