# Bayesian Exploration of Phenomenological EoS of Neutron/Hybrid Stars with Recent Observations

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## Abstract

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## 1. Introduction

## 2. The Structure of Neutron Stars

## 3. The Equation of State

#### Piecewise Polytropic Representation

## 4. Markov Chain Monte Carlo and Bayesian Inference

## 5. Bayesian Power Regression Model with Heteroscedastic Errors

## 6. Conclusions and Perspectives

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Piecewise model representation of the equations of state with the polytropic Equation (7). The black continuous line represents the SLy4 EoS region, the orange dashed line the EoS for first politrope and the red dotted line the EoS for the last politrope. The vertical lines represent the transition points ${\rho}_{1}$ and ${\rho}_{2}$ of each piece of the EoS.

**Figure 2.**MCMC chain (

**a**,

**d**) and respective posterior distributions (

**b**,

**e**) for both ${\Gamma}_{1}$ (upper panels) and ${\Gamma}_{2}$ (lower panels) obtained with ${\rho}_{1}={\rho}_{0}$ and ${\rho}_{2}=2{\rho}_{0}$ with $T=5\times {10}^{4}$ iterations. The deviation of the posterior average values along the chain is small ${\sigma}_{P\left(\Gamma \right)}/T\phantom{\rule{0.166667em}{0ex}}\approx 0.02$, indicating that a small portion of the posterior distribution is due to sampling error. This can be visualized in the moving average of the MCMC chains. Autocorrelation functions are shown in (

**c**,

**f**).

**Figure 3.**Gelmen–Rubin diagnostics for the case shown in Figure 2 for both ${\Gamma}_{1}$ (

**left**) and ${\Gamma}_{2}$ (

**right**). The Gelmen–Rubin coefficient $\approx 1$, the black dashed line, shows the numerical convergence of the MCMC algorithm.

**Figure 4.**On the left side: Mass–radius relationship for the MD1 parametrization from Table 1. The blue continuous line at $2.0\phantom{\rule{4pt}{0ex}}{M}_{\odot}$ corresponds to the two massive pulsars J0348+0432 and J1614-2230. The filled green region represents the pulsar J0740+6620 and the filled dashed salmon region is the pulsar J2215+5135. The red line is the low mass compact object in the binary system GW190414. The dark dots with errors bars are the NICER estimations of PSR J0030+0451. The purple curves in the left panel are the mass–radius relationships for the EoS generated by the MCMC algorithm. In the upper right corner, in purple, we have the MD1 EoS generated by the algorithm. In the middle right panel, we have the sound speed, and, in the lower panel, the masses for different central densities. The two vertical lines represent the transition regions, and the dashed-dotted horizontal lines in the middle right panel are the luminal and conformal velocities. Dark lines represent the SLy4 EoS.

**Figure 5.**Same as Figure 4: in the upper panel, the blue one, the transition regions are ${\rho}_{1}={\rho}_{0}$ and ${\rho}_{2}=3{\rho}_{0}$, and, in the lower, the green one, ${\rho}_{1}={\rho}_{0}$ and ${\rho}_{2}=5{\rho}_{0}$.

**Figure 6.**Analysis considering ${\rho}_{2}$ as a free parameter together with ${\Gamma}_{1}$ and ${\Gamma}_{2}$. Cyan and red curves show the first transition at ${\rho}_{1}=0.5{\rho}_{0}$ and ${\rho}_{1}=2{\rho}_{0}$, respectively. The intermittent behavior of the series represents “unstable” solutions of the minimization problem, more observational data points are needed here to reduce the variability of the parameters.

**Figure 7.**Representative example of heteroscedastic (

**a**) and homoscedastic (

**b**) residuals. Notice how the variance of the residuals changes with the value of x for the first, while it remains constant for the second.

**Figure 8.**Bayesian Power Regression model heteroscedastic errors. Black solid lines are the 65 EoS from the LIGO Lalsuite [92] data set, while the yellow ones are posterior samples generated by the BPR-HE model.

**Table 1.**Summary of the hyperparameters and averaged adjusted polytrope indices for piecewise EoS models. We employed 49 thousand EoS for each case. ${\rho}_{0}=2.8\times {10}^{14}\phantom{\rule{4pt}{0ex}}\mathrm{g}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{-3};{\mathrm{n}}_{0}=0.17\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{-3}$ is the nuclear saturation density.

Label | ${\mathit{\rho}}_{1}$ | ${\mathit{\rho}}_{2}$ | $\langle {\mathbf{\Gamma}}_{1}\rangle $ | $\langle {\mathbf{\Gamma}}_{2}\rangle $ | Color |
---|---|---|---|---|---|

MD1 | ${\rho}_{0}$ | $2{\rho}_{0}$ | 3.6 | 2.3 | purple |

MD2 | ${\rho}_{0}$ | $3{\rho}_{0}$ | 3.2 | 2.2 | blue |

MD3 | ${\rho}_{0}$ | $5{\rho}_{0}$ | 3.1 | 4.7 | green |

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## Share and Cite

**MDPI and ACS Style**

Chimanski, E.V.; Lobato, R.V.; Goncalves, A.R.; Bertulani, C.A.
Bayesian Exploration of Phenomenological EoS of Neutron/Hybrid Stars with Recent Observations. *Particles* **2023**, *6*, 198-216.
https://doi.org/10.3390/particles6010011

**AMA Style**

Chimanski EV, Lobato RV, Goncalves AR, Bertulani CA.
Bayesian Exploration of Phenomenological EoS of Neutron/Hybrid Stars with Recent Observations. *Particles*. 2023; 6(1):198-216.
https://doi.org/10.3390/particles6010011

**Chicago/Turabian Style**

Chimanski, Emanuel V., Ronaldo V. Lobato, Andre R. Goncalves, and Carlos A. Bertulani.
2023. "Bayesian Exploration of Phenomenological EoS of Neutron/Hybrid Stars with Recent Observations" *Particles* 6, no. 1: 198-216.
https://doi.org/10.3390/particles6010011