Systematic analysis of the impacts of symmetry energy parameters on neutron star properties

The impacts of various symmetry energy parameters on the properties of neutron stars (NSs) have been recently investigated, and the outcomes are at variance, as summarized in Table III of Phys. Rev. D 106, 063005 (2022). We have systematically analyzed the correlations of slope and curvature parameters of symmetry energy at the saturation density (${\rho }_0=0.16{\text{fm}}^{-3}$) with the tidal deformability and stellar radius of nonspinning neutron stars in the mass range of $1.2–1.6M_{\odot }$ using a large set of minimally constrained equations of state (EoSs). The EoSs at low densities correspond to the nucleonic matter and are constrained by empirical ranges of a few low-order nuclear matter parameters from the finite nuclei data and the pure neutron matter EoS from chiral effective field theory. The EoSs at high densities ($\rho >1.5–2{\rho }_0$) are obtained by a parametric form for the speed of sound that satisfies the causality condition. Several factors affecting the correlations between the NS properties and the individual symmetry energy parameters usually considered in the literature are explored. These correlations are quite sensitive to the choice of the distributions of symmetry energy parameters and their interdependence. But, variations of NS properties with the pressure of $\beta$-equilibrated matter at twice the saturation density remain quite robust which may be due to the fact that the pressure depends on the combination of multiple nuclear matter parameters that describe the symmetric nuclear matter as well as the density dependence of the symmetry energy. Our results are practically insensitive to the behavior of EoS at high densities.

Figure 1

Corner plot for the nuclear matter parameters (in MeV). The one-dimensional marginalized posterior distributions (salmon) and the prior distributions (green) lines are displayed along the diagonal plots. The vertical dashed lines indicate $68\%$ ($1\sigma$) confidence interval. The confidence ellipses for two-dimensional posterior distributions are plotted with $1\sigma , 2\sigma$, and $3\sigma$ confidence intervals along the off-diagonal plots.

Figure 2

The energy per neutron $[E_n\left(\rho \right)=E(\rho ,1)]$ for pure neutron matter as a function of neutron density. The colored bands correspond to $6\times {\mathrm{N}}^3\mathrm{LO}$ (light green), $2\times {\mathrm{N}}^3\mathrm{LO}$ (light red), and 90% confidence interval for the EoSs (light blue) obtained in our calculation (see text for details).

Figure 3

The pressure for $\beta$-equilibrated charge neutral matter as a function of nucleon density. The colored bands correspond to 90% confidence intervals for the EoSs with different ranges of the square of the speed of sound at the center of NS with maximum mass $(c_{s,\max }^2$) (see text for details).

Figure 4

The mass-radius relationship (left panel) obtained for the EoSs as shown in Fig. 3. The outer and inner gray shaded regions indicate the 90% (solid) and 50% (dashed) confidence intervals from the LIGO-Virgo analysis for BNS components of GW170817 event [7, 84, 85]. The rectangular regions enclosed by blue and black dashed lines indicate the constraints from the millisecond pulsar PSR $\text{J0030}+0451$ from NICER x-ray observation [21, 22], and PSR $\text{J0740}+6620$ orange dashed line [24]. The right panel displays tidal deformability versus NS mass. The orange shaded region is the observation for $\mathrm{\Lambda }$ with 90% posterior interval from the LIGO/Virgo Collaboration (GW170817 event) [84]. The black line correspond to observational bounds on ${\mathrm{\Lambda }}_{1.4}={190}_{-120}^{+390}$ [7]. For the comparison, we have shown violet and gold lines corresponding to ${\mathrm{\Lambda }}_{1.8}=70–270$, and ${\mathrm{\Lambda }}_{2.0}=30–150$ obtained from a few well-known theoretical models [18, 86].

Figure 5

Corner plot for the central density (${\rho }_c$ in ${\rho }_0$) and corresponding pressure ($P_c$ in MeV ${\mathrm{fm}}^{-3}$) for the neutron star with canonical and maximum mass, radius ($R_{1.4}$ in km), tidal deformability (${\mathrm{\Lambda }}_{1.4}$), and maximum mass ($M_{\max }$ in $M_{\odot }$) of NS. The confidence ellipses for two-dimensional posterior distributions are plotted with $1\sigma$(solid line) and $2\sigma$ (dashed line) confidence intervals along the off-diagonal plots. The vertical dashed lines indicate 68% confidence intervals.

Figure 6

The tidal deformability (${\mathrm{\Lambda }}_{1.4}$) as a function of slope parameter ($L_0$), curvature parameter ($K_{\mathrm{sym},0}$) and the pressure of $\beta$-equilibrated matter $\left[P\left(2{\rho }_0\right)\right]$.

Figure 7

Variations of tidal deformability (${\mathrm{\Lambda }}_{1.4}$) with slope parameter ($L_0$) for three different distributions of the nuclear matter parameters which are with uniform uncorrelated (U-Unc), Gaussian uncorrelated (G-Unc), and Gaussian correlated posterior distributions (G-Cor) as discussed in the text in details. The results are obtained by considering the EoSs associated with maximum NS mass $M_{\max }⩾2.1M_{\odot }$. The circle (orange) symbols represent the results obtained by varying all parameters, whereas the (blue) star symbols represent those obtained by fixing $K_0=240\mathrm{MeV}$ and $J_0=32$ MeV.

Figure 8

The variation of the tidal deformability (${\mathrm{\Lambda }}_{1.4}$) with curvature parameter ($K_{\mathrm{sym},0}$) for three different nuclear matter parameter distributions with uniform uncorrelated (U-Unc), Gaussian uncorrelated (G-Unc), and posterior distributions (G-Cor) are discussed in details in the text. The results are shown for those EoSs which are associated with a maximum mass of NS $⩾2.1M_{\odot }$. The symbols in circles (orange) represent the results obtained by varying all parameters, whereas stars (blue) symbols represent the results obtained when $K_0=240\mathrm{MeV}$ and $J_0=32\mathrm{MeV}$ are fixed.

Figure 9

A variation in tidal deformability (${\mathrm{\Lambda }}_{1.4}$) with the pressure of $\beta$-equilibrated matter at $2{\rho }_0$ based on three different nuclear matter parameter distributions, namely uniform uncorrelated (U-Unc), Gaussian uncorrelated (G-Unc), and posterior distributions (G-Cor), are discussed in detail in the text. The results are shown for those EoSs which are associated with a maximum mass of NS $⩾2.1M_{\odot }$. As shown by the circles (orange) symbols, results obtained with all parameters varied, while those obtained with $K_0=240\mathrm{MeV}$ and $J_0=32\mathrm{MeV}$ fixed are represented by the stars (blue) symbols.