# A Modern View of the Equation of State in Nuclear and Neutron Star Matter

^{*}

## Abstract

**:**

**Background:**We analyze several constraints on the nuclear equation of state (EOS) currently available from neutron star (NS) observations and laboratory experiments and study the existence of possible correlations among properties of nuclear matter at saturation density with NS observables.

**Methods:**We use a set of different models that include several phenomenological EOSs based on Skyrme and relativistic mean field models as well as microscopic calculations based on different many-body approaches, i.e., the (Dirac–)Brueckner–Hartree–Fock theories, Quantum Monte Carlo techniques, and the variational method.

**Results:**We find that almost all the models considered are compatible with the laboratory constraints of the nuclear matter properties as well as with the largest NS mass observed up to now, $2.{14}_{-0.09}^{+0.10}\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ for the object PSR J0740+6620, and with the upper limit of the maximum mass of about 2.3–2.5$\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ deduced from the analysis of the GW170817 NS merger event.

**Conclusion:**Our study shows that whereas no correlation exists between the tidal deformability and the value of the nuclear symmetry energy at saturation for any value of the NS mass, very weak correlations seem to exist with the derivative of the nuclear symmetry energy and with the nuclear incompressibility.

## 1. Introduction

## 2. Equations of State

## 3. Bulk Properties of Nuclear Matter

## 4. EOS for Betastable Matter

## 5. Constraints on the EOS from Terrestrial Laboratories and Astrophysical Observations

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The relation between the symmetry energy at saturation density ${S}_{0}$ and its slope L. The full symbols represent the predictions of microscopic approaches (black circles), Skyrme EOSs (green triangles), NLWM models (red squares) and DDM approaches (blue diamonds), see Table 1 for the numerical values. The shaded areas represent experimental bands, see text for details.

**Figure 2.**The symmetry energy vs. the baryon density for all the discussed EOSs. The green, blue and grey bands represent experimental data from HICs, whereas the orange contour represents the IAS calculations. See text for details.

**Figure 3.**Pressure vs. baryon density for the symmetric case (

**left**panels), and the beta-stable case (

**right**panels). The upper (lower) panels display results for microscopic (phenomenological) EOSs. Constraints derived from HIC data are displayed in the left panels as orange (KaoS experiment) and grey (flow data) bands. Limits deduced by the GW170817 event are labelled by blue bands in the right panels. See text for details.

**Figure 4.**Mass-radius relations predicted by the different EOSs listed in Table 1. The observed masses of the millisecond pulsar PSR J0740 + 6620 [12] and of J0384-0432 [11] are also shown, as well as constraints inferred from the analysis of the GW170817 event and observations reported by the NICER mission [18,19]. See text for details.

Model | EOS | ${\mathit{\rho}}_{0}$[fm${}^{-3}$] | $-{\mathit{E}}_{0}$ [MeV] | ${\mathit{S}}_{0}$ [MeV] | L [MeV] | ${\mathit{K}}_{0}$ [MeV] | ${\mathit{M}}_{\mathbf{max}}$[${\mathit{M}}_{\odot}$] | ${\mathbf{\Lambda}}_{1.2}$ | ${\mathbf{\Lambda}}_{1.4}$ | ${\mathbf{\Lambda}}_{1.6}$ |
---|---|---|---|---|---|---|---|---|---|---|

Micro. | BOB | 0.170 | 15.4 | 33.6 | 70 | 238 | 2.50 | 1366 | 570 | 252 |

V18 | 0.178 | 13.9 | 32.3 | 67 | 207 | 2.36 | 1082 | 442 | 188 | |

N93 | 0.185 | 16.1 | 36.5 | 77 | 229 | 2.25 | 1234 | 473 | 190 | |

UIX | 0.171 | 14.9 | 33.5 | 61 | 171 | 1.96 | 848 | 309 | 112 | |

APR | 0.159 | 15.9 | 33.4 | 51 | 233 | 2.20 | 720 | 274 | 110 | |

DBHF(A) | 0.181 | 16.2 | 34.4 | 69 | 218 | 2.31 | 1635 | 681 | 295 | |

DBHF(B) | 0.186 | 16.2 | 32.8 | 67 | 272 | - | 830 | 327 | 133 | |

FSS2CC | 0.157 | 16.3 | 31.8 | 52 | 219 | 1.94 | 814 | 295 | 106 | |

FSS2GC | 0.170 | 15.6 | 31.0 | 51 | 185 | 2.08 | 697 | 262 | 101 | |

AFDMC | 0.160 | 16.0 | 31.3 | 60 | 239 | 2.21 | 822 | 293 | 109 | |

Skyrme | Gs | 0.158 | 15.6 | 31.2 | 94 | 239 | 2.13 | 1769 | 659 | 253 |

Rs | 0.158 | 15.1 | 30.8 | 86 | 248 | 2.12 | 1652 | 618 | 238 | |

SLy4 | 0.160 | 16.0 | 31.8 | 45 | 232 | 2.05 | 756 | 287 | 111 | |

SV | 0.155 | 16.0 | 33.0 | 97 | 305 | 2.43 | 2224 | 914 | 393 | |

SkI4 | 0.158 | 16.2 | 33.7 | 106 | 245 | 2.17 | 1203 | 474 | 194 | |

SkMP | 0.158 | 15.6 | 34.3 | 82 | 244 | 2.11 | 1295 | 487 | 188 | |

SkO | 0.157 | 15.8 | 29.7 | 70 | 230 | 2.01 | 1252 | 451 | 164 | |

BSk22 | 0.158 | 16.1 | 32.0 | 69 | 246 | 2.26 | 1553 | 632 | 268 | |

BSk24 | 0.158 | 16.1 | 30.0 | 46 | 246 | 2.28 | 1260 | 523 | 227 | |

BSk26 | 0.159 | 16.1 | 30.0 | 38 | 241 | 2.17 | 830 | 327 | 133 | |

NLWM | SFHO | 0.157 | 16.2 | 32.8 | 53 | 244 | 2.06 | 862 | 334 | 132 |

GM1 | 0.153 | 16.3 | 32.5 | 94 | 300 | 2.36 | 2223 | 913 | 393 | |

GM3 | 0.153 | 16.4 | 32.5 | 90 | 241 | 2.02 | 1688 | 617 | 228 | |

DDM | DDME1 | 0.152 | 16.2 | 33.1 | 55 | 245 | 2.47 | 1765 | 773 | 355 |

DDME2 | 0.152 | 16.1 | 32.3 | 51 | 251 | 2.51 | 1834 | 806 | 374 | |

TW99 | 0.153 | 16.2 | 32.8 | 55 | 240 | 2.08 | 1041 | 404 | 162 | |

Exp. | ∼ 0.14–0.17 | ∼ 15–17 | 28.5–34.9 | 30–87 | 220–260 | $>2.{14}_{-0.09}^{+0.10}$ | 70–580 | |||

Ref. | [71] | [71] | [6,72] | [73,74] | [6,72] | [12] | [2] |

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Burgio, G.F.; Schulze, H.-J.; Vidaña, I.; Wei, J.-B.
A Modern View of the Equation of State in Nuclear and Neutron Star Matter. *Symmetry* **2021**, *13*, 400.
https://doi.org/10.3390/sym13030400

**AMA Style**

Burgio GF, Schulze H-J, Vidaña I, Wei J-B.
A Modern View of the Equation of State in Nuclear and Neutron Star Matter. *Symmetry*. 2021; 13(3):400.
https://doi.org/10.3390/sym13030400

**Chicago/Turabian Style**

Burgio, G. Fiorella, Hans-Josef Schulze, Isaac Vidaña, and Jin-Biao Wei.
2021. "A Modern View of the Equation of State in Nuclear and Neutron Star Matter" *Symmetry* 13, no. 3: 400.
https://doi.org/10.3390/sym13030400