Equation of state for dense nucleonic matter from metamodeling. I. Foundational aspects

Metamodeling for the nucleonic equation of state (EOS), inspired from a Taylor expansion around the saturation density of symmetric nuclear matter, is proposed and parameterized in terms of the empirical parameters. The present knowledge of nuclear empirical parameters is first reviewed in order to estimate their average values and associated uncertainties, and thus defining the parameter space of the metamodeling. They are divided into isoscalar and isovector types, and ordered according to their power in the density expansion. The goodness of the metamodeling is analyzed against the predictions of the original models. In addition, since no correlation among the empirical parameters is assumed a priori, all arbitrary density dependences can be explored, which might not be accessible in existing functionals. Spurious correlations due to the assumed functional form are also removed. This meta-EOS allows direct relations between the uncertainties on the empirical parameters and the density dependence of the nuclear equation of state and its derivatives, and the mapping between the two can be done with standard Bayesian techniques. A sensitivity analysis shows that the more influential empirical parameters are the isovector parameters $L_{\mathrm{sym}}$ and $K_{\mathrm{sym}}$, and that laboratory constraints at supersaturation densities are essential to reduce the present uncertainties. The present metamodeling for the EOS for nuclear matter is proposed for further applications in neutron stars and supernova matter.

Figure 1

Correlation between the empirical parameters $K_{\mathrm{sat}}$ and $Q_{\mathrm{sat}}$ for different kind of nuclear interactions: Skyrme, Gogny, RMF, and relativistic Hartree-Fock (RHF). Points from EFT approach are also plotted. The points are obtained from Tables 9–12, except for the Gogny model, which is extracted from Ref. [33], and the colored bands come from fits of the data including their dispersion considering 67% of the best models.

Figure 2

Comparison of the energy per nucleon in symmetric matter between Skyrme SLy5 and ELFa metamodeling, where the empirical parameters of Skyrme SLy5 has been used as a function of the order $N$ (lines with different colors). The crosses show the reference value given by SLy5.

Figure 3

Same as Fig. 2 for ELFb metamodeling.

Figure 4

Coefficients $a_N^{\mathrm{is}}/{(-3)}^{N+1}$ (left) and $a_N^{\mathrm{iv}}/{(-3)}^{N+1}$ (right) as a function of the order $N$ for various relativistic (bottom panels) and nonrelativistic (top panels) nuclear interactions.

Figure 5

Same as Fig. 2 for ELFc metamodeling.

Figure 6

Comparison of the ELFc (dashed lines) and ELFd (solid lines) metamodels against the reference models SLy5 (left panel) and PK1 (right panel). Only the orders $N=3$ and 4 are plotted.

Figure 7

Comparison of the pressure (top panels) and sound velocity (bottom panels) for the metamodels ELFc (dashed lines) and ELFd (solid lines) against the reference models SLy5 (left panels) and PK1 (right panels). Only the orders $N=3$ and 4 are plotted.

Figure 8

Equations of state at $1\sigma , 2\sigma$, and $3\sigma$ obtained from the first calculation of neutron matter with chiral EFT at next-to-next-to-next leading order ($\mathrm{N}{}^3\mathrm{LO}$) (points with error bars) and given in Ref. [11]. See text for more details.

Figure 9

Analysis of the probability distribution $p(E_{\mathrm{NM}},L_{\mathrm{sym}},K_{\mathrm{NM}})$ deduced from the best fits to neutron matter calculations shown in Fig. 8. The centroids and standard deviations for the parameters $E_{\mathrm{NM}}, L_{\mathrm{sym}}$, and $K_{\mathrm{NM}}$ are given in the figure. See text for more details.

Figure 10

Effect of varying the value of $L_{\mathrm{sym}}$ and $K_{\mathrm{sym}}$ on the symmetry energy around saturation density. Note that we vary the parameters by $\pm 2\sigma$ to enhance the effects. Top panels: varying $K_{\mathrm{sym}}$ at fixed $L_{\mathrm{sym}}$. Bottom panels: varying $L_{\mathrm{sym}}$ at fixed $K_{\mathrm{sym}}$.

Figure 11

Effect of varying the value of $E_{\mathrm{sat}}$ (top panels) and $E_{\mathrm{sym}}$ (bottom panels) around the mean value given in Tab. 7 considering $1\sigma$ deviation. Here $1\sigma$ is also taken from Tab. 7. From left to right: energy per nucleon, symmetry energy and pressure in SNM ($\delta =0$), ANM ($\delta =0.5$), and PNM ($\delta =1$).

Figure 12

Same as Fig. 11 for $n_{\mathrm{sat}}$ (top panel) and $L_{\mathrm{sym}}$ (bottom panel).

Figure 13

Same as Fig. 11 for $K_{\mathrm{sat}}$ (top panel) and $K_{\mathrm{sym}}$ (bottom panel).

Figure 14

Same as Fig. 11 for $Q_{\mathrm{sat}}$ (top panel) and $Q_{\mathrm{sym}}$ (bottom panel).

Figure 15

Same as Fig. 11 for $Z_{\mathrm{sat}}$ (top panel) and $Z_{\mathrm{sym}}$ (bottom panel).

Figure 16

Same as Fig. 11 for $m_{\mathrm{sat}}^{*}$ (top panel) and $\mathrm{\Delta }{m}_{\mathrm{sat}}^{*}$ (bottom panel).

Figure 17

Effect of varying the value of all empirical parameters in the isoscalar (top panels) and isovector (bottom panels) channels. From left to right: energy per nucleon, symmetry energy, and pressure in SNM ($\delta =0$), ANM ($\delta =0.5$), and PNM ($\delta =1$).

Figure 18

Same as Fig. 17, varying the value of the empirical parameters all together.