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Bayesian Analysis for Extracting Properties of the Nuclear Equation of State from Observational Data Including Tidal Deformability from GW170817^{ †}

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## Abstract

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## 1. Introduction

## 2. Hybrid EoS

## 3. Neutron Star Configurations

#### 3.1. Mass and Radius

#### 3.2. Tidal Deformability

## 4. Bayesian Inference for the EoS Models

#### 4.1. Vector of Parameters

#### 4.2. Likelihood of a Model for the ${\Lambda}_{1}$–${\Lambda}_{2}$ Constraint from GW170817

#### 4.3. Likelihood of a Model for the Mass Constraint

#### 4.4. Posterior Distribution

## 5. Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Glendenning, N.K.; Kettner, C. Nonidentical neutron star twins. Astron. Astrophys.
**2000**, 353, L9. [Google Scholar] - Gerlach, U.H. Equation of State at Supranuclear Densities and the Existence of a Third Family of Superdense Stars. Phys. Rev.
**1968**, 172, 1325–1330. [Google Scholar] [CrossRef] - Alford, M.G.; Han, S.; Prakash, M. Generic conditions for stable hybrid stars. Phys. Rev. D
**2013**, 88, 083013. [Google Scholar] [CrossRef] - Alvarez-Castillo, D.E.; Blaschke, D. Proving the CEP with compact stars? In Proceedings of the 17th Conference of Young Scientists and Specialists, Dubna, Russia, 8–12 April 2013; pp. 22–26. Available online: http://xxx.lanl.gov/abs/1304.7758 (accessed on 30 April 2013).
- Benic, S.; Blaschke, D.; Alvarez-Castillo, D.E.; Fischer, T.; Typel, S. A new quark-hadron hybrid equation of state for astrophysics-I. High-mass twin compact stars. Astron. Astrophys.
**2015**, 577, A40. [Google Scholar] [CrossRef] - Blaschke, D.; Alvarez-Castillo, D.E. High-mass twins & resolution of the reconfinement, masquerade and hyperon puzzles of compact star interiors. AIP Conf. Proc.
**2016**, 1701, 020013. [Google Scholar] [Green Version] - Alvarez-Castillo, D.E.; Blaschke, D.B. High-mass twin stars with a multipolytrope equation of state. Phys. Rev. C
**2017**, 96, 045809. [Google Scholar] [CrossRef] - Paschalidis, V.; Yagi, K.; Alvarez-Castillo, D.; Blaschke, D.B.; Sedrakian, A. Implications from GW170817 and I-Love-Q relations for relativistic hybrid stars. Phys. Rev. D
**2018**, 97, 084038. [Google Scholar] [CrossRef] [Green Version] - Seidov, Z.F. The Stability of a Star with a Phase Change in General Relativity Theory. Soviet Astronomy
**1971**, 15, 347. [Google Scholar] - Blaschke, D.; Alvarez-Castillo, D.E.; Benic, S. Mass-radius constraints for compact stars and a critical endpoint. In Proceedings of the 8th International Workshop on Critical Point and Onset of De confinement, Napa, CA, USA, 11–15 March 2013. [Google Scholar]
- Tolman, R.C. Static solutions of Einstein’s field equations for spheres of fluid. Phys. Rev.
**1939**, 55, 364–373. [Google Scholar] [CrossRef] - Oppenheimer, J.R.; Volkoff, G.M. On Massive neutron cores. Phys. Rev.
**1939**, 55, 374–381. [Google Scholar] [CrossRef] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; LIGO Scientific Collaboration and Virgo Collaboration; et al. GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral. Phys. Rev. Lett.
**2017**, 119, 161101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Annala, E.; Gorda, T.; Kurkela, A.; Vuorinen, A. Gravitational-wave constraints on the neutron-star-matter Equation of State. Phys. Rev. Lett.
**2018**, 120, 172703. [Google Scholar] [CrossRef] [PubMed] - Bauswein, A.; Just, O.; Janka, H.T.; Stergioulas, N. Neutron-star radius constraints from GW170817 and future detections. Astrophys. J.
**2017**, 850, L34. [Google Scholar] [CrossRef] - Rezzolla, L.; Most, E.R.; Weih, L.R. Using gravitational-wave observations and quasi-universal relations to constrain the maximum mass of neutron stars. Astrophys. J.
**2018**, 852. [Google Scholar] [CrossRef] - De, S.; Finstad, D.; Lattimer, J.M.; Brown, D.A.; Berger, E.; Biwer, C.M. Tidal Deformabilities and Radii of Neutron Stars from the Observation of GW170817. Phys. Rev. Lett.
**2018**, 121, 091102. [Google Scholar] [CrossRef] [PubMed] - Steiner, A.W.; Lattimer, J.M.; Brown, E.F. The Equation of State from Observed Masses and Radii of Neutron Stars. Astrophys. J.
**2010**, 722, 33–54. [Google Scholar] [CrossRef] - Raithel, C.A.; Özel, F.; Psaltis, D. From Neutron Star Observables to the Equation of State. II. Bayesian Inference of Equation of State Pressures. Astrophys. J.
**2017**, 844, 156. [Google Scholar] [CrossRef] [Green Version] - Salmi, T.; Nättilä, J.; Poutanen, J. Bayesian parameter constraints for neutron star masses and radii using X-ray timing observations of accretion-powered millisecond pulsars. Astron. Astrophys.
**2018**, 618, A161. [Google Scholar] [CrossRef] - Margueron, J.; Hoffmann Casali, R.; Gulminelli, F. Equation of state for dense nucleonic matter from metamodeling. II. Predictions for neutron star properties. Phys. Rev. C
**2018**, 97, 025806. [Google Scholar] [CrossRef] [Green Version] - Blaschke, D.; Alvarez-Castillo, D.E.; Klähn, T. Universal Symmetry Energy Contribution to the Neutron Star Equation of State. 2016. Available online: http://xxx.lanl.gov/abs/1604.08575 (accessed on 29 April 2016).
- Alvarez-Castillo, D.E.; Blaschke, D.B.; Grunfeld, A.G.; Pagura, V.P. Third Family of Compact Stars within A Nonlocal Chiral Quark Model Equation of State. 2018. Available online: http://xxx.lanl.gov/abs/1805.04105v3 (accessed on 1 January 2019).
- Most, E.R.; Weih, L.R.; Rezzolla, L.; Schaffner-Bielich, J. New constraints on radii and tidal deformabilities of neutron stars from GW170817. Phys. Rev. Letters
**2018**, 120, 261103. [Google Scholar] [CrossRef] - Christian, J.E.; Zacchi, A.; Schaffner-Bielich, J. Signals in the tidal deformability for phase transitions in compact stars with constraints from GW170817. Phys. Rev. D
**2019**, 99, 023009. [Google Scholar] [CrossRef] - Montana, G.; Tolos, L.; Hanauske, M.; Rezzolla, L. Constraining Twin Stars with GW170817. 2018. Available online: http://xxx.lanl.gov/abs/1811.10929 (accessed on 28 November 2018).
- Sieniawska, M.; Turczanski, W.; Bejger, M.; Zdunik, J.L. Tidal Deformability and Other Global Parameters of Compact Stars with Phase Transitions. 2018. Available online: http://xxx.lanl.gov/abs/1807.11581 (accessed on 27 July 2018).
- Alvarez-Castillo, D.; Ayriyan, A.; Benic, S.; Blaschke, D.; Grigorian, H.; Typel, S. New class of hybrid EoS and Bayesian M-R data analysis. Eur. Phys. J. A
**2016**, 52, 69. [Google Scholar] [CrossRef] - Kolomeitsev, E.E.; Voskresensky, D.N. Relativistic mean-field models with effective hadron masses and coupling constants, and rho- condensation. Nucl. Phys. A
**2005**, 759, 373–413. [Google Scholar] [CrossRef] - Röpke, G.; Blaschke, D.; Schulz, H. Pauli Quenching Effects in a Simple String Model of Quark / Nuclear Matter. Phys. Rev. D
**1986**, 34, 3499. [Google Scholar] [CrossRef] - Kaltenborn, M.A.R.; Bastian, N.U.F.; Blaschke, D.B. Quark-nuclear hybrid star equation of state with excluded volume effects. Phys. Rev. D
**2017**, 96, 056024. [Google Scholar] [CrossRef] [Green Version] - Ayriyan, A.; Bastian, N.U.; Blaschke, D.; Grigorian, H.; Maslov, K.; Voskresensky, D.N. Robustness of third family solutions for hybrid stars against mixed phase effects. Phys. Rev. C
**2018**, 97, 045802. [Google Scholar] [CrossRef] [Green Version] - Ayriyan, A.; Grigorian, H. Model of the Phase Transition Mimicking the Pasta Phase in Cold and Dense Quark-Hadron Matter. EPJ Web Conf.
**2018**, 173, 03003. [Google Scholar] [CrossRef] [Green Version] - Abgaryan, V.; Alvarez-Castillo, D.; Ayriyan, A.; Blaschke, D.; Grigorian, H. Two Novel Approaches to the Hadron-Quark Mixed Phase in Compact Stars. Universe
**2018**, 4, 94. [Google Scholar] [CrossRef] - Maslov, K.A.; Kolomeitsev, E.E.; Voskresensky, D.N. Relativistic Mean-Field Models with Scaled Hadron Masses and Couplings: Hyperons and Maximum Neutron Star Mass. Nucl. Phys. A
**2016**, 950, 64–109. [Google Scholar] [CrossRef] - Maslov, K.; Yasutake, N.; Ayriyan, A.; Blaschke, D.; Grigorian, H.; Maruyama, T.; Tatsumi, T.; Voskresensky, D.N. Hybrid Equation of State with Pasta Phases and Third Family of Compact Stars I: Pasta Phases and Effective Mixed Phase Model. 2018. Available online: http://xxx.lanl.gov/abs/1812.11889 (accessed on 31 December 2018).
- Hinderer, T.; Lackey, B.D.; Lang, R.N.; Read, J.S. Tidal deformability of neutron stars with realistic equations of state and their gravitational wave signatures in binary inspiral. Phys. Rev. D
**2010**, 81, 123016. [Google Scholar] [CrossRef] - Abbott, B.P.; LIGO Scientific Collaboration and Virgo Collaboration. GW170817: Measurements of neutron star radii and equation of state. Phys. Rev. Lett.
**2018**, 121, 161101. [Google Scholar] [CrossRef] [PubMed] - LIGO Scientific Collaboration and Virgo Collaboration. GW170817: Measurements of Neutron Star Radii and Equation of State. 2018. Available online: https://dcc.ligo.org/LIGO-P1800115/public (accessed on 30 January 2019).
- Antoniadis, J.; Freire, P.C.C.; Wex, N.; Tauris, T.M.; Lynch, R.S.; van Kerkwijk, M.H.; Kramer, M.; Bassa, C.; Dhillon, V.S.; et al. A Massive Pulsar in a Compact Relativistic Binary. Science
**2013**, 340, 1233232. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The set of hybrid EoS obtained by the mixed phase construction for different values of ${\Delta}_{P}$ and $\alpha $ is shown in the

**left**panel, while in the

**right**panel the corresponding set of compact star sequences in the mass–radius diagram is shown. The blue numbers show selected values of critical densities for the onset of the phase transitions obtained by a Maxwell construction.

**Figure 2.**The dependence of the tidal deformability $\Lambda $ on the compact star mass (

**left**panel) and the ${\Lambda}_{1}$–${\Lambda}_{2}$ diagram for selected values of ${\Delta}_{P}$ and $\alpha $ (

**right**panel). Comparison with the 90% confidence line of the LIGO-Virgo Collaboration for the low-spin prior of GW170817 [13] shows that, if the hadronic EoS is as stiff as DD2p40, at least one of the stars in the binary has to be a hybrid star in order to avoid a violation of the ${\Lambda}_{1}$–${\Lambda}_{2}$ constraint. As can be seen from figure 17 of Ref. [23], the merger of two hybrid stars which is admissible when the onset mass for the deconfinement transition is lowered, e.g., by increasing $\alpha $, would lead to the appearance of a new branch in this diagram mimicking the pattern of a merger of two neutron stars with a rather soft nuclear EoS.

**Figure 3.**The PDF reconstructed with ${\Lambda}_{1}$–${\Lambda}_{2}$ data for GW170817 from the LIGO website [39] as a 3D graphics (

**left**panel) and as a contour plot (

**right**panel).

**Figure 4.**

**Left**panel: The posterior distribution of models on the parameter space spanned by ${\Delta}_{P}$ and $\alpha $.

**Right**panel: The compact star sequences in the mass–radius diagram labeled into four probability classes according to the results of the BA for the posterior distribution of the left panel. The grey, brown, orange and black lines show sequences for which the value of the posterior probability exceeds the thresholds of 0.0, 0.02, 0.04 and 0.06, respectively. Due to the restriction to the model and parameter range used in Ref. [32], the sequences with a lower onset mass for the deconfinement transition are not accessed for which an interpretation of GW170817 as a merger of two hybrid stars would be possible.

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**MDPI and ACS Style**

Ayriyan, A.; Alvarez-Castillo, D.; Blaschke, D.; Grigorian, H.
Bayesian Analysis for Extracting Properties of the Nuclear Equation of State from Observational Data Including Tidal Deformability from GW170817. *Universe* **2019**, *5*, 61.
https://doi.org/10.3390/universe5020061

**AMA Style**

Ayriyan A, Alvarez-Castillo D, Blaschke D, Grigorian H.
Bayesian Analysis for Extracting Properties of the Nuclear Equation of State from Observational Data Including Tidal Deformability from GW170817. *Universe*. 2019; 5(2):61.
https://doi.org/10.3390/universe5020061

**Chicago/Turabian Style**

Ayriyan, Alexander, David Alvarez-Castillo, David Blaschke, and Hovik Grigorian.
2019. "Bayesian Analysis for Extracting Properties of the Nuclear Equation of State from Observational Data Including Tidal Deformability from GW170817" *Universe* 5, no. 2: 61.
https://doi.org/10.3390/universe5020061