#
Probing the Nuclear Equation of State from the Existence of a ∼2.6 M_{⊙} Neutron Star: The GW190814 Puzzle

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. The MDI-APR Model and the Rapidly Rotating Neutron Star

## 3. Speed of Sound Formalism and Stiffness of Equation of State

## 4. Tidal Deformability

## 5. Results and Discussion

#### 5.1. Slow/Rapid Rotation: Implications to Neutron Star Properties

#### 5.2. Tidal Effects and Speed of Sound: A Very Massive Neutron Star Hypothesis

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

#### 5.2.2. A Very Massive Neutron Star Component

## 6. Concluding Remarks

## 7. Materials and Methods

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EoS | Equation of state |

NS | Neutron star |

QCD | Quantum chromodynamics |

MDI | Momentum dependent interaction |

APR | Akmal, Pandharipande and Ravenhall |

SNM | Symmetric Nuclear Matter |

N.R. | Non-rotating configuration |

M.R. | Maximally-rotating configuration |

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**Figure 1.**(

**a**) Pressure as a function of rest mass density and (

**b**) square speed of sound in units of speed of light as a function of the transition density, where p takes the values $[1.5,2,3,4,5]$, while the speed of sound is parametrized in the two limiting cases, ${({v}_{s}/c)}^{2}=1/3$ and ${({v}_{s}/c)}^{2}=1$. (

**c**) Pressure as a function of rest mass density and (

**d**) square speed of sound in units of speed of light as a function of the transition density, where p takes the values $[1.5,2]$, while the speed of sound is parametrized in the range ${({v}_{s}/c)}^{2}=[1/3,1]$ (As the speed of sound is getting higher values, the curves’ color lightens). The vertical lines display the transition cases, while the shaded regions show the credibility interval extracted from Ref. [5].

**Figure 2.**Gravitational mass as a function of Kerr parameter for the MDI-APR (MDI: momentum dependent interaction, APR: Akmal, Pandharipande and Ravenhall) equation of state (EoS). The solid lines from bottom to top represent the Equation (4) with ${M}_{\mathrm{TOV}}=2.08\phantom{\rule{3.33333pt}{0ex}}{M}_{\odot}$ and ${M}_{\mathrm{TOV}}=2.3\phantom{\rule{3.33333pt}{0ex}}{M}_{\odot}$. The mass range of the second component of GW190814 is noted with the horizontal shaded region, while with the vertical one (

**left**), the possible region of Kerr parameter $\mathcal{K}=[0.49,0.68]$ from Ref. [16] is shown. In addition, the region for the Kerr parameter ${\mathcal{K}}_{\mathrm{max}}=[0.67,0.69]$ from Ref. [35], if the low mass component was rotating at its mass-shedding limit, is presented with the vertical shaded region (

**right**). The markers point the maximum mass configuration at the mass-shedding limit.

**Figure 3.**(

**a**) Gravitational mass and (

**b**) Kerr parameter as a function of transition density at the maximum mass configuration for the two limiting speed of sound bounds. The data at the maximum mass configuration is presented with diamonds for the ${({v}_{s}/c)}^{2}=1/3$ bound and crosses for the ${({v}_{s}/c)}^{2}=1$ bound. The plus marker denotes the lower bound in the speed of sound, ${({v}_{s}/c)}^{2}=0.45$, assuming that the second component was a non-rotating neutron star (NS). The mass range of the second component of GW190814 is noted with the horizontal shaded region. (

**a**) The lighter shaded region marks the allowed range for the transition density at the maximally-rotating (M.R.) configuration, while the darker one, marks the allowed region at the non-rotating (N.R.) configuration. (

**b**) The darker shaded region marks the allowed range for the transition density at the maximally-rotating (M.R.).

**Figure 4.**Gravitational mass as a function of the central energy/baryon density at the maximum mass configuration both at non-rotating and maximally-rotating case. Circles correspond to 23 hadronic EoSs [35] at the non-rotating case (N.R.), squares to the corresponding maximally-rotating (M.R.) one, stars to data of Cook et al. [69], and triangles to data of Salgado et al. [70]. In addition, rhombus and pluses mark the non-rotating configuration at the two limiting values of the sound speed, while crosses and polygons marks the maximally-rotating one. The horizontal dashed lines correspond to the observed NS mass limits ($2.01\phantom{\rule{3.33333pt}{0ex}}{M}_{\odot}$ [41], $2.14\phantom{\rule{3.33333pt}{0ex}}{M}_{\odot}$ [42], and $2.27\phantom{\rule{3.33333pt}{0ex}}{M}_{\odot}$ [43]). Equation (22) is noted with the dashed-dotted line, while for comparison the Tolman VII analytical solution [35] is shown with the solid line. The mass range of the second component of GW190814 is noted with the horizontal shaded region.

**Figure 5.**Mass vs. radius for an isolated non-rotating NS, for each transition density ${n}_{\mathrm{tr}}$ and all speed of sound cases. The higher values of speed of sound correspond to lighter curves’ color. The purple horizontal line and region indicate the mass estimation of the massive compact object of Ref. [1]. The dashdot (dotted) curve corresponds to the MDI-APR (APR) EoS.

**Figure 6.**Dependence of a non-rotating NS’s maximum mass ${M}_{max}$ on the speed of sound values ${({v}_{s}/c)}^{2}$ for each transition density ${n}_{\mathrm{tr}}$ (in units of saturation density ${n}_{s}$). The red 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 red (green) vertical line indicates the corresponding value of the speed of sound for a massive object with $M=2.59\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$.

**Figure 7.**Tidal parameters (

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

**b**) $\lambda $ as a function of a NS’s mass. The purple vertical line and shaded region indicate the estimation of the recently observed massive compact object of Ref. [1]. The solid (dashed) curves correspond to the ${n}_{\mathrm{tr}}=1.5{n}_{s}$ (${n}_{\mathrm{tr}}=2{n}_{s}$) case. As the speed of sound is getting higher values, the curves’ color lightens.

**Figure 8.**The effective tidal deformability $\tilde{\mathsf{\Lambda}}$ as a function of (

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

**b**) binary mass ratio q, in the case of a very massive NS component, identical to Ref. [1]. As the speed of sound bound is getting higher, the color of EoSs lightens. The black dashed vertical line indicates (

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

**b**) mass ratio q, of a binary NS system with ${m}_{1}=2.59\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$ and ${m}_{2}=1.4\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$ respectively.

**Figure 9.**Effective tidal deformability $\tilde{\mathsf{\Lambda}}$ vs. radius ${R}_{1.4}$ of a ${m}_{2}=1.4\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$ NS. The heavier component of the system was taken to be ${m}_{1}=2.59\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$. The lighter colors correspond to higher values of speed of sound bounds. The grey lines indicate the expression of Equation (24). The black dotted vertical line shows the proposed upper limit of Ref. [76].

Speed of Sound Bounds | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\alpha}}_{3}$ | ${\mathit{\alpha}}_{4}$ | ||||
---|---|---|---|---|---|---|---|---|

N.R. | M.R. | N.R. | M.R. | N.R. | M.R. | N.R. | M.R. | |

c | 1.665 | 1.689 | 0.448 | 0.352 | – | 0.683 | – | 1.053 |

$c/\sqrt{3}$ | 1.751 | 2.069 | 0.964 | 0.883 | – | 0.645 | – | 1.348 |

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

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

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

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

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

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

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

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Kanakis-Pegios, A.; Koliogiannis, P.S.; Moustakidis, C.C.
Probing the Nuclear Equation of State from the Existence of a ∼2.6 *M*_{⊙} Neutron Star: The GW190814 Puzzle. *Symmetry* **2021**, *13*, 183.
https://doi.org/10.3390/sym13020183

**AMA Style**

Kanakis-Pegios A, Koliogiannis PS, Moustakidis CC.
Probing the Nuclear Equation of State from the Existence of a ∼2.6 *M*_{⊙} Neutron Star: The GW190814 Puzzle. *Symmetry*. 2021; 13(2):183.
https://doi.org/10.3390/sym13020183

**Chicago/Turabian Style**

Kanakis-Pegios, Alkiviadis, Polychronis S. Koliogiannis, and Charalampos C. Moustakidis.
2021. "Probing the Nuclear Equation of State from the Existence of a ∼2.6 *M*_{⊙} Neutron Star: The GW190814 Puzzle" *Symmetry* 13, no. 2: 183.
https://doi.org/10.3390/sym13020183