# Neutron Stars and Gravitational Waves: The Key Role of Nuclear Equation of State

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## Abstract

**:**

## 1. Introduction

## 2. Cold Neutron Stars

#### 2.1. The Momentum-Dependent Interaction Nuclear Model

#### 2.2. Speed of Sound Formalism

#### 2.3. Construction of the EoS

#### 2.4. Structure Equations

#### 2.4.1. Nonrotating Neutron Stars

#### 2.4.2. Rotating Neutron Stars

## 3. Hot Neutron Stars

#### 3.1. Thermodynamical Description of Hot Neutron Star Matter

#### 3.2. Bulk Thermodynamic Quantities

#### 3.3. Lepton’s Contribution

#### 3.4. Isothermal Configuration

#### 3.5. Isentropic Configuration and Neutrino Trapping

#### 3.6. Construction of the Hot EoSs

#### 3.7. Rapidly Rotating Hot Neutron Stars

## 4. Results and Discussion

#### 4.1. Speed of Sound and Tidal Deformability

- The overall thickness decreases as the transition density ${n}_{\mathrm{tr}}$ reaches higher values. This behavior can be explained by the variation of the radius $M\left(R\right)$ presented in the M-R diagram (see Figure 2).
- The thickness of each shaded region decreases as the ${n}_{\mathrm{tr}}$ reaches higher values.

#### 4.2. GW190814: A Postulation of the Most Massive Neutron Star

#### 4.3. The Case of a Very Massive Neutron Star

#### 4.3.1. Isolated Non-Rotating Neutron Star

#### 4.3.2. A Very Massive Neutron Star in a Binary Neutron Stars System

#### 4.4. Finite Temperature Effects on Rapidly Rotating Neutron Stars

#### 4.4.1. Sequences of Constant Baryon Mass

#### 4.4.2. Moment of Inertia, Kerr Parameter, and Ratio $T/W$

- compactness parameter: ${\beta}_{\mathrm{rem}}^{\mathrm{iso}}\le 0.19$ and ${\beta}_{\mathrm{rem}}^{\mathrm{ise}}\le 0.27$,
- Kerr parameter: ${\mathcal{K}}_{\mathrm{rem}}^{\mathrm{iso}}\le 0.42$ and ${\mathcal{K}}_{\mathrm{rem}}^{\mathrm{ise}}\le 0.68$,
- ratio $T/W$: ${(T/W)}_{\mathrm{rem}}^{\mathrm{iso}}\le 0.05$ and ${(T/W)}_{\mathrm{rem}}^{\mathrm{ise}}\le 0.127$,

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EoS | Equation of state |

MDI | Momentum dependent interaction |

APR | Akmal, Pandharipande, and Ravenhall |

SNM | Symmetric nuclear matter |

ANM | Asymmetric nuclear matter |

PNM | Pure neutron matter |

NM | Nuclear model |

TOV | Tolman-Oppenheimer-Volkoff |

N.R. | Non-rotating configuration |

M.R. | Maximally-rotating configuration |

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**Figure 1.**(

**a**,

**c**) Dependence of the pressure on the rest mass density and (

**b**,

**d**) dependence of the square sound speed in units of light speed on the transition density. (

**a**,

**b**) The speed of sound is fixed at the two boundary cases, ${({v}_{s}/c)}^{2}=1/3$ and ${({v}_{s}/c)}^{2}=1$, and the value p takes the arguments $[1.5,2,3,4,5]$. (

**c**,

**d**) The value p takes the arguments $[1.5,2]$, and the speed of sound is parametrized in the range ${({v}_{s}/c)}^{2}=[1/3,1]$ (the lower values of the speed of sound correspond to the darker colored curves). In all figures, the vertical dotted lines indicate the transition cases, while the shaded regions note the credibility interval derived from Reference [24].

**Figure 2.**Mass vs. radius for an isolated neutron star and for the two cases of speed of sound bounds. The blue (green) lines correspond to the upper (lower) bound. The black diagonal shaded region corresponds to NICER’s observation (data taken from Reference [121]), while the purple upper (orange lower) shaded region corresponds to the higher (smaller) component of GW170817 event (data taken from Reference [24]). The solid (dashed) contour lines describe the 90% (50%) confidence interval.

**Figure 3.**The effective tidal deformability $\tilde{\Lambda}$ as a function of the binary mass ratio q for the event (

**a**) GW170817 and (

**b**) GW190425. The measured upper limits for $\tilde{\Lambda}$ are also indicated, with the grey shaded region corresponding to the excluded area. The green (blue) curves correspond to the ${({v}_{s}/c)}^{2}=1/3$ (${({v}_{s}/c)}^{2}=1$) case.

**Figure 4.**$\tilde{\Lambda}$ as a function of the transition density ${n}_{\mathrm{tr}}$ (in units of saturation density ${n}_{s}$) at the maximum mass configuration for the two speed of sound bounds ${v}_{s}=c/\sqrt{3}$ and ${v}_{s}=c$ and for the events (

**a**) GW170817 and (

**b**) GW190425. The measured upper limits for $\tilde{\Lambda}$ [14,98] as well as the corresponding lower values of transition density are also indicated for both events. The green (blue) arrow marks the accepted region of transition density for the ${v}_{s}=c/\sqrt{3}$ (${v}_{s}=c$) case. The green lower (blue upper) curved shaded region corresponds to the ${v}_{s}=c/\sqrt{3}$ (${v}_{s}=c$) limit. The yellow shaded area indicates the region between the two cases of bounds of the speed of sound.

**Figure 5.**The effective tidal deformability $\tilde{\Lambda}$ as a function of the maximum mass for the two speed of sound bounds ${v}_{s}=c/\sqrt{3}$ and ${v}_{s}=c$ and for the events (

**a**) GW170817 and (

**b**) GW190425. The measured upper limits for $\tilde{\Lambda}$ (black dashed lines with arrows; see References [14,98]); the corresponding maximum mass shaded regions, for the ${v}_{s}=c/\sqrt{3}$ (left green) case, for the ${v}_{s}=c$ case (right blue), and for the middle cases (yellow); and the current observed maximum neutron star mass $M=2.{14}_{-0.09}^{+0.10}{M}_{\odot}$ (purple shaded vertical area; see Reference [17]) are also displayed. The green left (blue right) arrow marks the accepted region of maximum mass ${M}_{\mathrm{max}}$ for ${v}_{s}=c/\sqrt{3}$ (${v}_{s}=c$) case.

**Figure 6.**The effective tidal deformability $\tilde{\Lambda}$ as a function of ${R}_{1.4}$ for both events and all the bounds of the speed of sound. The dashed and dash-dotted horizontal black lines correspond to the upper limit on $\tilde{\Lambda}$ for the GW190425 and GW170817 events, respectively, taken from References [14,98]. The grey shaded regions correspond to the excluded areas. The horizontal arrows indicate the allowed area for ${R}_{1.4}$ in each case. The purple dotted curve demonstrates the proposed expression by References [133,134].

**Figure 7.**Dependence of the gravitational mass on the Kerr parameter. The lower solid line represents Equation (63) with ${M}_{\mathrm{TOV}}=2.08{M}_{\odot}$, while the upper solid line represents Equation (63) with ${M}_{\mathrm{TOV}}=2.3{M}_{\odot}$. The dashed line corresponds to the MDI-APR EoS. In addition, the horizontal shaded region notes the mass of the second component of GW190814 event, and the vertical wide shaded region marks the Kerr parameter, $\mathcal{K}=[0.49,0.68]$, according to Reference [139]. Furthermore, the narrow vertical shaded region indicates the Kerr parameter, ${\mathcal{K}}_{\mathrm{max}}=[0.67,0.69]$, extracted from Reference [63] by assuming that the low mass component was rotating at its mass-shedding limit. The cross, the plus sign, and the diamond show the maximum mass configuration at the mass-shedding limit.

**Figure 8.**Dependence of the maximum gravitational mass on the central energy/baryon density both at nonrotating and rapidly rotating with the Kepler frequency configurations. Circles and squares correspond to 23 hadronic EoSs [63] at the nonrotating (N.R.) and maximally-rotating (M.R.) cases, respectively, and stars and triangles correspond to data of Cook et al. [140] and Salgado et al. [141], respectively. Furthermore, diamonds and plus signs note the nonrotating configuration, while crosses and polygons show the maximally-rotating one, in the cases of the two limiting values of the sound speed. The horizontal dashed lines mark the current observed neutron star mass limits ($2.01{M}_{\odot}$ [16], $2.14{M}_{\odot}$ [17], and $2.27{M}_{\odot}$ [18]). Equation (66) is noted with the dashed-dotted line, while for comparison, the Tolman VII analytical solution [63] is added with the solid line. The horizontal shaded region notes the mass range of the second component of GW190814 event.

**Figure 9.**(

**a**) Mass vs. radius for an isolated nonrotating neutron star, for each transition density ${n}_{\mathrm{tr}}$ and all speed of sound cases. The darker curves’ color corresponds to the lower values of speed of sound. The blue horizontal line and region indicate the mass estimation of the massive compact object of Reference [19]. The dashed-dotted and dotted curves correspond to the MDI-APR and APR EoS, respectively. (

**b**) The maximum mass ${M}_{max}$ of a nonrotating neutron star as a relation to the bounds of the speed of sound ${({v}_{s}/c)}^{2}$ for each transition density ${n}_{\mathrm{tr}}$ (in units of saturation density ${n}_{s}$). The purple vertical shaded region corresponds to the ${n}_{\mathrm{tr}}=1.5{n}_{s}$ case, while the green one corresponds to the ${n}_{\mathrm{tr}}=2{n}_{s}$ case. The purple (green) vertical line indicates the corresponding value of the speed of sound for a massive object with $M=2.59{M}_{\odot}$.

**Figure 10.**Tidal parameters (

**a**) ${k}_{2}$ and (

**b**) $\lambda $ as a function of an neutron star’s mass. The blue vertical line and shaded region indicate the estimation of the recently observed massive compact object of Reference [19]. The solid (dashed) curves correspond to the ${n}_{\mathrm{tr}}=1.5{n}_{s}$ (${n}_{\mathrm{tr}}=2{n}_{s}$) case. The lower values of the speed of sound correspond to the darker-colored curves.

**Figure 11.**The effective tidal deformability $\tilde{\Lambda}$ as a function of (

**a**) the chirp mass ${\mathcal{M}}_{c}$ and (

**b**) binary mass ratio q, in the case of a very massive neutron star component, identical to Reference [19]. The darker colored curves correspond to lower values of speed of sound. The black dashed vertical line shows (

**a**) the corresponding chirp mass ${\mathcal{M}}_{c}$ and (

**b**) mass ratio q, of a binary neutron star system with ${m}_{1}=2.59{M}_{\odot}$ and ${m}_{2}=1.4{M}_{\odot}$, respectively.

**Figure 12.**The effective tidal deformability $\tilde{\Lambda}$ as a function of the radius ${R}_{1.4}$ of an ${m}_{2}=1.4{M}_{\odot}$ neutron star. The heavier component of the system was taken to be ${m}_{1}=2.59{M}_{\odot}$. The darker colors correspond to lower values of speed of sound bounds. The grey lines show the expression of Equation (68). The black dotted vertical line indicates the proposed upper limit of Reference [132].

**Figure 13.**Dependence of the Kepler frequency on (

**a**) the temperature and (

**b**) the central baryon density for baryon masses in the range $[1.6,2.2]{M}_{\odot}$. (

**a**) Solid lines represent the fits originated from Equation (69). (

**b**) The solid line represents Equation (70), while open circles note the high-temperature region ($T\ge 30\mathrm{MeV}$).

**Figure 14.**Dependence of the dimensionless moment of inertia on the compactness parameter at the mass-shedding limit in the case of (

**a**) isothermal and (

**b**) isentropic profiles. Dashed lines note the hot configurations, while the solid line notes the cold configuration.

**Figure 15.**Dependence of the Kerr parameter on the gravitational mass at the mass-shedding limit in the case of (

**a**) isothermal and (

**b**) isentropic profiles. Dashed lines note the hot configurations, while the solid line notes the cold configuration. The horizontal dotted line corresponds to the Kerr bound for astrophysical Kerr black holes, ${\mathcal{K}}_{\mathrm{B}.\mathrm{H}.}=0.998$ [144], while the shaded region corresponds to the neutron star limits from Reference [64]. (

**c**) Dependence of the Kerr parameter on the temperature for gravitational masses in the range $[1.4,2.3]{M}_{\odot}$ and in the case of isothermal profile.

**Figure 16.**Dependence of the angular velocity on the ratio of rotational kinetic to gravitational binding energy at the mass-shedding limit in the case of (

**a**) isothermal and (

**b**) isentropic profiles. Dashed lines note the hot configurations, while the solid line notes the cold configuration. Markers correspond to the ${M}_{\mathrm{gr}}=1.4{M}_{\odot}$ configuration. The vertical dotted line notes the critical value, $T/W=0.08$, for gravitational radiation instabilities.

$\mathbf{Bounds}$ | $\mathbf{GW}170817$ | $\mathbf{GW}190425$ | ||||||
---|---|---|---|---|---|---|---|---|

${\mathit{c}}_{\mathbf{1}}$ | ${\mathit{c}}_{\mathbf{2}}$ | ${\mathit{c}}_{\mathbf{3}}$ | ${\mathit{c}}_{\mathbf{4}}$ | ${\mathit{c}}_{\mathbf{1}}$ | ${\mathit{c}}_{\mathbf{2}}$ | ${\mathit{c}}_{\mathbf{3}}$ | ${\mathit{c}}_{\mathbf{4}}$ | |

c | $500.835$ | $0.258$ | $53.457$ | $0.873$ | $47.821$ | $0.055$ | $10.651$ | $1.068$ |

$c/\sqrt{3}$ | $503.115$ | $0.325$ | $38.991$ | $1.493$ | $43.195$ | $0.069$ | $5.024$ | $1.950$ |

**Table 2.**Coefficients of Equation (62) for the two bounds of the speed of sound.

Event | Bounds | ${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ |
---|---|---|---|

$\mathrm{GW}170817$ | c | $0.12357\times {10}^{-4}$ | $6.967$ |

$c/\sqrt{3}$ | $0.12179\times {10}^{-4}$ | $6.967$ | |

$\mathrm{GW}190425$ | c | $0.870\times {10}^{-6}$ | $7.605$ |

$c/\sqrt{3}$ | $0.088\times {10}^{-6}$ | $8.422$ |

${\mathit{n}}_{\mathbf{tr}}$ | ${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | ${({\mathit{v}}_{\mathit{s}}/\mathit{c})}_{\mathbf{min}}^{2}$ | ${({\mathit{v}}_{\mathit{s}}/\mathit{c})}^{2}$ | ${({\mathit{v}}_{\mathit{s}}/\mathit{c})}_{\mathbf{max}}^{2}$ |
---|---|---|---|---|---|---|---|

$1.5{n}_{s}$ | $-1.6033\times {10}^{3}$ | $-7.56\times {10}^{-4}$ | $-1.64\times {10}^{-1}$ | $1.6068\times {10}^{3}$ | 0.448 | 0.485 | 0.52 |

$2{n}_{s}$ | 5.5754 | 0.2742 | −0.6912 | −1.9280 | 0.597 | 0.659 | 0.72 |

${({\mathit{v}}_{\mathit{s}}/\mathit{c})}^{2}$ | ${\mathit{c}}_{5}$ (${\mathbf{km}}^{-1}$) | ${\mathit{c}}_{6}$ | $\tilde{\mathit{\Lambda}}$ |
---|---|---|---|

$0.8$ | $4.1897\times {10}^{-9}$ | $9.3518$ | $109.536$ |

$0.9$ | $5.3213\times {10}^{-9}$ | $9.2652$ | $111.416$ |

1 | $6.1109\times {10}^{-9}$ | $9.2159$ | $112.729$ |

**Table 5.**Coefficients ${\alpha}_{i}$ with i = 0–3 for the empirical relation (69) and baryon masses in the range [1.6–2.2] ${M}_{\odot}$.

Coefficients | Baryon Mass | |||
---|---|---|---|---|

$\mathbf{1}.\mathbf{6}{\mathbf{M}}_{\odot}$ | $\mathbf{1}.\mathbf{8}{\mathbf{M}}_{\odot}$ | $\mathbf{2}.\mathbf{0}{\mathbf{M}}_{\odot}$ | $\mathbf{2}.\mathbf{2}{\mathbf{M}}_{\odot}$ | |

${a}_{0}(\times {10}^{2})$ | 4.259 | 5.284 | 6.414 | 7.863 |

${a}_{1}(\times {10}^{-3})$ | −4.787 | −3.202 | −2.099 | −1.443 |

${a}_{2}(\times {10}^{2})$ | 5.401 | 4.929 | 4.363 | 3.530 |

${a}_{3}(\times {10}^{-1})$ | −1.468 | −1.443 | −1.424 | −1.636 |

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Koliogiannis, P.S.; Kanakis-Pegios, A.; Moustakidis, C.C.
Neutron Stars and Gravitational Waves: The Key Role of Nuclear Equation of State. *Foundations* **2021**, *1*, 217-255.
https://doi.org/10.3390/foundations1020017

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Koliogiannis PS, Kanakis-Pegios A, Moustakidis CC.
Neutron Stars and Gravitational Waves: The Key Role of Nuclear Equation of State. *Foundations*. 2021; 1(2):217-255.
https://doi.org/10.3390/foundations1020017

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Koliogiannis, Polychronis S., Alkiviadis Kanakis-Pegios, and Charalampos C. Moustakidis.
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https://doi.org/10.3390/foundations1020017