# Constraints on Phase Transitions in Neutron Star Matter

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Equation of State of Neutron Star Matter

#### 2.1. Observational Constraints

#### 2.1.1. Neutron Star Masses and Radii

#### 2.1.2. Binary Neutron Star Mergers and Tidal Deformabilities

#### 2.2. Inference of Sound Velocity and the EoS in Neutron Stars

#### 2.3. Selected Neutron Star Properties

## 3. Constraints on Phase Transitions in Neutron Stars

#### 3.1. Evidence against a Very Low Squared Sound Speed in Neutron Stars

#### 3.2. Evidence against a Strong First-Order Phase Transition in the Cores of Neutron Stars

#### 3.3. Intermediate Summary

## 4. Phenomenology and Models

#### 4.1. Reminder of Low-Energy Nucleon Structure and a Two-Scale Scenario

#### 4.2. Quark–Hadron Continuity and Crossover

#### 4.3. Restoration of Chiral Symmetry in Dense Matter: From First-Order Phase Transition to Crossover

#### 4.4. Dense Baryonic Matter: A Fermi Liquid Picture

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Squared speed of sound [31] as a function of the energy density inferred from the empirical dataset listed in Section 2.1. Median (solid curve) and highest posterior density credible bands at levels of 68% (gray band) and 95% (dashed curves) are displayed. Intervals are indicated for the 68% ranges of central energy densities in the cores of 1.4 and 2.3 ${M}_{\odot}$ neutron stars.

**Figure 2.**Equation of state $P\left(\epsilon \right)$ [31] deduced from the inferred sound velocity based on the empirical dataset listed in Section 2.1. The median, 68%, and 95% posterior credible bands are displayed, as in Figure 1. The APR EoS [41] (dotted line) is shown for comparison. Also shown for orientation is the baryon density scale, $\rho /{\rho}_{0}$ (in units of the equilibrium density of nuclear matter ${\rho}_{0}=0.16$ fm${}^{-3}$), which was computed for the median of $P\left(\epsilon \right)$.

**Figure 3.**Baryon chemical potential $\mu \left(\rho \right)$ in neutron star matter, normalized at $\mu (\rho =0)=939$ MeV. Posterior credible bands [31] at the 68% level (gray band) and 95% level (dashed lines) are shown.

**Figure 4.**Median and credible bands [31] of the trace anomaly measure, $\Delta =\frac{1}{3}-\frac{P}{\epsilon}$, at the 68% level (gray band) and 95% level (dashed lines).

**Figure 5.**Posterior credible bands [31] of the radius R as a function of the neutron star mass M at the 68% level (gray band) and 95% level (dashed line) compared to the analysis of NICER data by Riley et al. for PSR J0030+0451 and PSR J0740+6620 [10,12]. In addition, the mass–radius credible interval at the 68% level of the thermonuclear burster 4U 1702-429 (blue) is displayed [46] (which was not included in the Bayesian analysis).

**Figure 7.**Characteristic behaviors of the squared sound velocity in the presence of a phase transition or a crossover.

**Figure 8.**Illustration of the constraint on the maximum width of a Maxwell-constructed coexistence region for a first-order phase transition within the 68% credible band of $P\left(\epsilon \right)$. The upper (baryon density) scale refers to the median of the $P\left(\epsilon \right)$ distribution, as in Figure 2.

**Figure 9.**Sketch of low- and high-density baryonic matter. Baryons (e.g., nucleons) are viewed as valence quark cores surrounded by clouds of quark–antiquark pairs (e.g., chiral meson clouds). At densities of $\rho \gtrsim 2-3\phantom{\rule{0.166667em}{0ex}}{\rho}_{0}$, the percolation of quark–antiquark pairs over larger distances starts, as indicated. Valence quark cores begin to touch and overlap at baryon densities $\rho \gtrsim 5\phantom{\rule{0.166667em}{0ex}}{\rho}_{0}$.

**Figure 10.**Chiral order parameters in symmetric nuclear matter and neutron matter at a temperature of $T=0$ as a function of baryon density in units of nuclear ground state equilibrium density, ${\rho}_{0}=0.16$ fm${}^{-3}$. Dotted lines: liquid–gas phase transition (L-G) in symmetric nuclear matter. Dashed lines: first-order chiral phase transitions emerging from the mean-field (MF) approximation of a relativistic chiral nucleon–meson (ChNM) field-theoretical model. Solid lines show the results of extended mean-field (EMF) calculations (with the inclusion of fermionic vacuum fluctuations) and functional renormalization group (FRG) calculations based on the same ChNM model. The figures were adapted from Refs. [55,75].

**Figure 11.**The Landau effective mass ${m}_{L}^{*}\left(\rho \right)=\sqrt{{p}_{F}^{2}+{m}^{2}\left(\rho \right)}$ and potential $U\left(\rho \right)$ of quasiparticles representing the median of the posterior distribution of the baryon chemical potential $\mu \left(\rho \right)$ (see Figure 3).

**Figure 12.**The Landau Fermi liquid parameters derived from quasiparticle properties that are based on the median of the data-inferred baryon chemical potential $\mu $ (see Figure 3).

**Table 1.**Bayes factors ${\mathcal{B}}_{{c}_{s,min}^{2}\le 0.1}^{{c}_{s,min}^{2}>0.1}$ comparing EoS samples with competing scenarios: (a) minimum squared speed of sound (following a maximum) with ${c}_{s,min}^{2}>0.1$ versus (b) equations of state with ${c}_{s,min}^{2}\le 0.1$. The minimum speeds of sound are computed up to the maximum neutron star masses, as indicated (taken from [31]).

${\mathit{M}}_{\mathbf{m}\mathbf{a}\mathbf{x}}\mathbf{/}{\mathit{M}}_{\mathbf{\odot}}$ | 1.9 | 2.0 | 2.1 | 2.2 | 2.3 |
---|---|---|---|---|---|

${\mathcal{B}}_{{c}_{s,min}^{2}\le 0.1}^{{c}_{s,min}^{2}>0.1}$ | 500.9 | 229.8 | 15.0 | 3.6 | 2.2 |

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Brandes, L.; Weise, W.
Constraints on Phase Transitions in Neutron Star Matter. *Symmetry* **2024**, *16*, 111.
https://doi.org/10.3390/sym16010111

**AMA Style**

Brandes L, Weise W.
Constraints on Phase Transitions in Neutron Star Matter. *Symmetry*. 2024; 16(1):111.
https://doi.org/10.3390/sym16010111

**Chicago/Turabian Style**

Brandes, Len, and Wolfram Weise.
2024. "Constraints on Phase Transitions in Neutron Star Matter" *Symmetry* 16, no. 1: 111.
https://doi.org/10.3390/sym16010111