Building relativistic mean field models for finite nuclei and neutron stars

Background: Theoretical approaches based on density functional theory provide the only tractable method to incorporate the wide range of densities and isospin asymmetries required to describe finite nuclei, infinite nuclear matter, and neutron stars.

Purpose: A relativistic energy density functional (EDF) is developed to address the complexity of such diverse nuclear systems. Moreover, a statistical perspective is adopted to describe the information content of various physical observables.

Methods: We implement the model optimization by minimizing a suitably constructed ${\chi }^2$ objective function using various properties of finite nuclei and neutron stars. The minimization is then supplemented by a covariance analysis that includes both uncertainty estimates and correlation coefficients.

Results: A new model, “FSUGold2,” is created that can well reproduce the ground-state properties of finite nuclei, their monopole response, and that accounts for the maximum neutron-star mass observed up to date. In particular, the model predicts both a stiff symmetry energy and a soft equation of state for symmetric nuclear matter, suggesting a fairly large neutron-skin thickness in ${}^{208}\mathrm{Pb}$ and a moderate value of the nuclear incompressibility.

Conclusions: We conclude that without any meaningful constraint on the isovector sector, relativistic EDFs will continue to predict significantly large neutron skins. However, the calibration scheme adopted here is flexible enough to create models with different assumptions on various observables. Such a scheme—properly supplemented by a covariance analysis—provides a powerful tool to identify the critical measurements required to place meaningful constraints on theoretical models.

Figure 1

Constrained giant monopole energies for ${}^{90}\mathrm{Zr},{}^{116}\mathrm{Sn},{}^{144}\mathrm{Sm}$, and ${}^{208}\mathrm{Pb}$. Experimental data were obtained from experiments carried out at TAMU [62] and RCNP [63, 64, 65, 66, 67]. Theoretical predictions are presented for NL3 [8], FSUGold [10], and FSUGold2 supplemented with theoretical errors. The red solid line represents a best fit to the FSUGold2 predictions of the form $E_{\mathrm{fit}}=72.8A^{-0.31}$ MeV.

Figure 2

Mass-vs-radius relation predicted by the three models considered in the text: NL3 [8], FSUGold [10], and FSUGold2. Also shown are recent observational constraints on neutron-star masses [18, 19] and radii [28, 29, 30, 31]. The FSUGold2 results are supplemented with two sets of theoretical errors: one (red) in which the maximum neutron-star mass was included in the calibration of the functional and the other (gray) estimated also using FSUGold2, but with the impact of the maximum neutron-star mass being removed from the curvature matrix, as explained in the text.

Figure 3

Amplitude decomposition of the eigenvectors of the curvature matrix corresponding to the two largest [(a) and (b)] and the two smallest [(c) and (d)] eigenvalues, with the largest eigenvalue normalized to one. The two different colors (blue and red) indicate that the amplitudes contribute with opposite signs.

Figure 4

(a) Binding energy per nucleon of symmetric nuclear matter and (b) symmetry energy as a function of density in units of nuclear-matter saturation density ${\rho }_{{}_0}=0.148{\mathrm{fm}}^{-3}$. Predictions are included from the three models discussed in the text: NL3 [8], FSUGold [10], and FSUGold2 supplemented with theoretical errors.

Figure 5

Correlation coefficients (in absolute value) depicted in graphical form for a representative set of observables. The set includes four GMR energies (for ${}^{90}\mathrm{Zr},{}^{116}\mathrm{Sn},{}^{144}\mathrm{Sm}$, and ${}^{208}\mathrm{Pb})$, two neutron radii (for ${}^{48}\mathrm{Ca}$ and ${}^{208}\mathrm{Pb})$, several bulk properties of nuclear matter $({\varepsilon }_{{}_0},{\rho }_{{}_0},M^{*},K,J$, and $L)$, and two neutron-star observables (the maximum mass $M_{\max }$ and the radius of a $1.4M_{\odot }$ neutron star $R_{1.4})$.

Figure 6

Correlation coefficients (absolute values) between Lagrangian model parameters depicted in graphical form.

Figure 7

Correlation coefficients (absolute values) between Lagrangian model parameters and a representative set of physical observables. The set of observables are the same as those considered in Fig. 5.