Bayesian inference of neutron-star observables based on effective nuclear interactions

Based on the Skyrme-Hartree-Fock model (SHF) as well as its extension [the Korea-IBS-Daegu-SKKU (KIDS) model] and the relativistic mean-field (RMF) model, we have studied the constraints on the parameters of the nuclear matter equation of state (EOS) from adopted astrophysical observables using a Bayesian approach. While the masses and radii of neutron stars generally favor a stiff isoscalar EOS and a moderately soft nuclear symmetry energy, model dependence on the constraints is observed and mostly originates from the incorporation of higher-order EOS parameters and differences between relativistic and nonrelativistic models. At twice saturation density, the value of the symmetry energy is constrained to be ${48}_{-11}^{+15}\mathrm{MeV}$ in the standard SHF model, ${48}_{-15}^{+8}\mathrm{MeV}$ in the KIDS model, and ${48}_{-6}^{+5}\mathrm{MeV}$ in the RMF model, around their maximum a posteriori values within $68\%$ confidence intervals. Our study helps to obtain a robust constraint on nuclear matter EOS, and, meanwhile, to understand the model dependence of the results.

Figure 1

Dependence of the core-crust transition density ${\rho }_t$ (first row), the radius $R_{1.4}$ (second row) and the tidal deformability ${\mathrm{\Lambda }}_{1.4}$ (third row) of a canonical neutron star, the maximum mass of the neutron star $M_{\max }/M_{\odot }$ (fourth row), and the radius $R_{2.08}$ (fifth row) of a neutron star with mass $M=2.08M_{\odot }$ individually on $K_0$, $E_{\mathrm{sym}}^0$, $L$, $m_s^{★}/m$, and $m_v^{★}/m$ within their prior ranges, with the values of other parameters fixed at their default values as in Table 1, based on the standard SHF model. Results from different values of EOS coefficients $\gamma$ for the inner crust are compared, together with those without considering the crust (${\rho }_t=0$).

Figure 2

Similar to Fig. 1 but for the KIDS model showing dependence of observables individually on $K_0$, $Q_0$, $E_{\mathrm{sym}}^0$, $L$, $K_{\mathrm{sym}}$, $Q_{\mathrm{sym}}$, $m_s^{★}/m$, and $m_v^{★}/m$ within their prior ranges.

Figure 3

Similar to Fig. 1 but for the RMF model showing dependence of observables individually on $K_0$, $Q_0$, $E_{\mathrm{sym}}^0$, $L$, $m_s^{★}/m$, and $m_v^{★}/m$ within their prior ranges.

Figure 4

Posterior correlated PDFs in the ($L$, $K_0$) and ($L$, $E_{\mathrm{sym}}^0$) planes in the standard SHF model (top row), in the ($Q_0$, $K_0$), ($L$, $K_0$), ($L$, $E_{\mathrm{sym}}^0$), and ($L$, $K_{\mathrm{sym}}$) planes in the KIDS model (middle row), and in the ($Q_0$, $K_0$), ($L$, $K_0$), and ($L$, $E_{\mathrm{sym}}^0$) planes in the RMF model (bottom row) from the constraints of astrophysical observables.

Figure 5

Comparison of the posterior PDFs of $K_0$, $Q_0$, $E_{\mathrm{sym}}^0$, $L$, $K_{\mathrm{sym}}$, $Q_{\mathrm{sym}}$, $m_s^{★}$, and $m_v^{★}$ in the standard SHF model (top row), the KIDS model (middle row), and the RMF model (bottom row) from the constraints of astrophysical observables with different crust EOSs. Results from different crust EOSs with different coefficients $\gamma$ are compared, and the prior PDF of each individual quantity is also displayed.

Figure 6

Probability distributions of $E_{\mathrm{SNM}}\left(\rho \right)$ and $E_{\mathrm{sym}}\left(\rho \right)$ from parameter ranges in Table 1 (prior) and under the constraints of astrophysical observables (posterior) in the standard SHF model (top), the KIDS model (middle), and the RMF model (bottom).

Figure 7

Posterior and prior PDFs of $E_{\mathrm{sym}}\left(\rho \right)$ at $\rho =1.5{\rho }_0$ (a) and $2{\rho }_0$ (b) in the standard SHF model, the KIDS model, and the RMF model.

Figure 8

Illustration of the limited parameter space in the $K_0\text{−}{Q}_0$ plane (a), the $m_s^{★}/m\text{−}{Q}_0$ plane (b), and the $E_{\mathrm{sym}}^0\text{−}L$ plane (c) with other parameters set as their default values in Table 1 in the RMF model.