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Register a new userAuxiliary Function Approach for Determining Symmetry Energy at Supra-saturation Densities
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Nuclear symmetry energy $E_{rm{sym}}(rho)$ at density $rho$ is normally expanded or simply parameterized as a function of $chi=(rho-rho_0)/3rho_0$ in the form of $E_{rm{sym}}(rho)approx S+Lchi+2^{-1}K_{rm{sym}}chi^2+6^{-1}J_{rm{sym}}chi^3+cdots$ using its magnitude $S$, slope $L $, curvature $K_{rm{sym}}$ and skewness $J_{rm{sym}}$ at the saturation density $rho_0$ of nuclear matter. Much progress has been made in recent years in constraining especially the $S$ and $L$ parameters using various terrestrial experiments and astrophysical observations. However, such kind of expansions/parameterizations do not converge at supra-saturation densities where $chi$ is not small enough, hindering an accurate determination of high-density $E_{rm{sym}}(rho)$ even if its characteristic parameters at $rho_0$ are all well determined by experiments/observations. By expanding the $E_{rm{sym}}(rho)$ in terms of a properly chosen auxiliary function $Pi_{rm{sym}}(chi,Theta_{rm{sym}})$ with a parameter $Theta_{rm{sym}}$ fixed accurately by an experimental $E_{rm{sym}}(rho_{rm{r}})$ value at a reference density $rho_{rm{r}}$, we show that the shortcomings of the $chi$-expansion can be completely removed or significantly reduced in determining the high-density behavior of $E_{rm{sym}}(rho)$. In particular, using two significantly different auxiliary functions, we show that the new approach effectively incorporates higher $chi$-order contributions and converges to the same $E_{rm{sym}}(rho)$ much faster than the conventional $chi$-expansion at densities $lesssim3rho_0$. Several quantitative demonstrations using Monte Carlo simulations are given.
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Within an isospin- and momentum-dependent hadronic transport model it is shown that the recent FOPI data on the $pi^-/pi^+$ ratio in central heavy-ion collisions at SIS/GSI energies (Willy Reisdorf {it et al.}, NPA {bf 781}, 459 (2007)) provide circumstantial evidence suggesting a rather soft nuclear symmetry energy esym at $rhogeq 2rho_0$ compared to the Akmal-Pandharipande-Ravenhall prediction. Some astrophysical implications and the need for further experimental confirmations are discussed.
Motivated by the historical detection of gravitational waves from GW170817, the neutron star and the neutron drop, i.e., a certain number of neutrons confined in an external field, are systematically investigated by ab initio calculations as well as the nonrelativistic and relativistic state-of-art density functional theories. Strong correlations are found among the neutron star tidal deformability, the neutron star radius, the root-mean-square radii of neutron drops, and the symmetry energies of nuclear matter at supra-saturation densities. From these correlations and the upper limit on the tidal deformability extracted from GW170817, the neutron star radii, the neutron drop radii, and the symmetry energy at twice saturation density are respectively constrained as $R_{1.4M_{odot}}leqslant 12.94$ km, $R_{rm nd} leqslant 2.36$ fm, and $E_{mathrm{sym}}(2rho_0) leqslant 53.2$ MeV.
The elliptic-flow ratio of neutrons with respect to protons in reactions of neutron rich heavy-ions systems at intermediate energies has been proposed as an observable sensitive to the strength of the symmetry term in the nuclear Equation Of State (EOS) at supra-saturation densities. The recent results obtained from the existing FOPI/LAND data for $^{197}$Au+$^{197}$Au collisions at 400 MeV/nucleon in comparison with the UrQMD model allowed a first estimate of the symmetry term of the EOS but suffer from a considerable statistical uncertainty. In order to obtain an improved data set for Au+Au collisions and to extend the study to other systems, a new experiment was carried out at the GSI laboratory by the ASY-EOS collaboration in May 2011.
By directly inverting several neutron star observables in the three-dimensional parameter space for the Equation of State of super-dense neutron-rich nuclear matter, we show that the lower radius limit for PSR J0740+6620 of mass $2.08pm 0.07~M_{odot}$ from Neutron Star Interior Composition Explorer (NICER)s very recent observation sets a much tighter lower boundary than previously known for nuclear symmetry energy in the density range of $(1.0sim 3.0)$ times the saturation density $rho_0$ of nuclear matter. The super-soft symmetry energy leading to the formation of proton polarons in this density region of neutron stars is clearly disfavoured by the first radius measurement for the most massive neutron star observed reliably so far.
We examine the correlations of neutron star radii with the nuclear matter incompressibility, symmetry energy, and their slopes, which are the key parameters of the equation of state (EoS) of asymmetric nuclear matter. The neutron star radii and the EoS parameters are evaluated using a representative set of 24 Skyrme-type effective forces and 18 relativistic mean field models, and two microscopic calculations, all describing 2$M_odot$ neutron stars. Unified EoSs for the inner-crust-core region have been built for all the phenomenological models, both relativistic and non-relativistic. Our investigation shows the existence of a strong correlation of the neutron star radii with the linear combination of the slopes of the nuclear matter incompressibility and the symmetry energy coefficients at the saturation density. Such correlations are found to be almost independent of the neutron star mass in the range $0.6text{-}1.8M_{odot}$. This correlation can be linked to the empirical relation existing between the star radius and the pressure at a nucleonic density between one and two times saturation density, and the dependence of the pressure on the nuclear matter incompressibility, its slope and the symmetry energy slope. The slopes of the nuclear matter incompressibility and the symmetry energy coefficients as estimated from the finite nuclei data yield the radius of a $1.4M_{odot}$ neutron star in the range $11.09text{-}12.86$ km.
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