Constraining the nuclear symmetry energy and properties of neutron star from GW170817 by Bayesian analysis
aa r X i v : . [ nu c l - t h ] A ug Constraining the nuclear symmetry energy and properties ofneutron star from GW170817 by Bayesian analysis
Yuxi Li, Houyuan Chen,
1, 2
Dehua Wen ∗ , and Jing Zhang School of Physics and Optoelectronics,South China University of Technology, Guangzhou 510641, P.R. China School of Physics and Astronomy,Sun Sat-Sen University, Zhuhai 519082, P.R. China (Dated: August 10, 2020)
Abstract
Based on the distribution of tidal deformabilities and component masses of binary neutron starmerger GW170817, the parametric equation of states (EOS) are employed to probe the nuclearsymmetry energy and the properties of neutron star. To obtain a proper distribution of theparameters of the EOS that is consistent with the observation, Bayesian analysis is used and theconstraints of causality and maximum mass are considered. From this analysis, it is found thatthe symmetry energy at twice the saturation density of nuclear matter can be constrained within E sym (2 ρ ) = 34 . +20 . − . MeV at 90% credible level. Moreover, the constraints on the radii anddimensionless tidal deformabilities of canonical neutron stars are also demonstrated through thisanalysis, and the corresponding constraints are 10.80 km < R . < < Λ . < R . = 12.60 km and ¯Λ . = 500, respectively.With respect to the prior, our result (posterior result) prefers a softer EOS, corresponding to alower expected value of symmetry energy, a smaller radius and a smaller tidal deformability. PACS numbers: 97.60.Jd; 04.40.Dg; 04.30.-w; 95.30.Sf ∗ Corresponding author. [email protected] . INTRODUCTION On August 17, 2017, the Advanced LIGO and Advanced Virgo first observed the mergerof two neutron stars GW170817 [1]. Through continual research based on the data of thisobservation, the understanding of properties of neutron stars (such as the radius, the tidaldeformability, etc.) and the state of dense nuclear matter is improved continually [2–7].Considering the case that the detection of gravitational radiation from the coalescence ofa neutron star binary system is occasional [1, 8], Bayesian inference becomes a popularmethod to analyze the observational data. In fact, the Bayesian analysis is frequently usedto investigate the properties and the state of the compact-star-matter in recent years [9–19].Based on Bayesian analysis with equation of states described by chiral effective fieldtheory, Lim et al. constrained the dimensionless tidal deformability of a 1.4 M ⊙ neutronstar in a range of 136 ≤ Λ . ≤
519 at 95% credible level. Moreover, they found an empiricalrelation between the tidal deformability of a canonical neutron star and the pressure at twicenuclear saturation density, which provides a useful clue to investigate the state of the densenuclear matter [11]. By performing Bayesian analysis with the distance and source locationderived by electromagnetic observations of GW170817 event, De et al. constrained the ˜Λ ( ˜Λis defined as equation 3 in [12]) at 90% credible level as follows: 84 ≤ ˜Λ ≤
642 for uniformcomponent mass prior, 94 ≤ ˜Λ ≤
698 for the distribution of component mass prior deducedfrom radio observations of Galactic binary neutron stars and 89 ≤ ˜Λ ≤
681 for a componentmass prior derived by radio pulsars [12].As we know, the density of the neutron star matter covers a large range of magnitude,from a density far lower than the saturation density at the outer crust to a density closeto 10 times the saturation density at the stellar center. At present, there is relativelysmall discrepancy in the EOS at density near or lower than the saturation density. Butfor the matter in the core with supra-saturation density, the EOS is far from certain. Innuclear theory, there are too many EOS predictions based on various nuclear theories byusing different interactions, and the predicted EOS often diverge at the supra-saturationdensity. In fact, the uncertainty of the symmetry energy at supra-saturation density is themain factor leading to the divergency of the EOS [20]. With the aid of the astronomicalobservations, people find a practice way to narrow the divergency. For example, by usingthe representative stellar radius data of canonical neutron star, Xie and Li [21] inferred thehigh-density nuclear symmetry energy through Bayesian inference by employing an isospin-dependent parametric EOS model for neutron star matter recently. They obtained constrainton the symmetry energy at twice the saturation density of nuclear matter as E sym (2 ρ ) =39 . +12 . − . MeV at 68% credible level.Motivated by the above interesting works, we will investigate the constraint on thenuclear symmetry energy and some of the properties of canonical neutron star throughBayesian inference based on the distribution of tidal deformabilities and component massesof GW170817 in this work.The paper is organized as follows. In the next section, the isospin-dependent parametricEOS for dense neutron-rich nucleonic matter and properties of neutron star are outlined. InSec. 3, through performing Bayesian analysis by correlating the EOS with the GW170817data released by LIGO and VIRGO, the posterior distribution of the parameter space ofthe EOS and the symmetry energy of the super dense nuclear matter are presented. Then,In Sec. 4, we present the constraint on the radii and tidal deformabilities of a canonicalneutron star through the corresponding posterior distribution. A brief summary is given at2he end.
2. ISOSPIN-DEPENDENT PARAMETRIC EOS AND NEUTRON STAR PROP-ERTIES
Here we give a brief outline of the isospin-dependent parametric EOS, where the densenuclear matter is supposed to be composed of neutrons, protons, electrons and muons at β -equilibrium and charge neutral [21, 22].The energy density of dense nuclear matter with isospin asymmetry δ = ( ρ n − ρ p ) /ρ atdensity ρ can be expressed as ǫ ( ρ, δ ) = ρ [ E ( ρ, δ ) + M N ] + ǫ l ( ρ, δ ) , (1)where M N is the average nucleon mass ( M N = 939 MeV), ǫ l ( ρ, δ ) is the lepton energydensity, and E ( ρ, δ ) is the nucleon specific energy. The pressure of dense nuclear matter canbe calculated by P ( ρ, δ ) = ρ dǫ ( ρ, δ ) /ρdρ . (2)The nucleon specific energy E ( ρ, δ ) for neutron-rich nuclear matter can be well approximatedby the empirical parabolic law as [20, 23] E ( ρ, δ ) = E ( ρ ) + E sym ( ρ ) · δ + O ( δ ) , (3)where E ( ρ ) and E sym ( ρ ) are the energy in symmetric nuclear matter and the symmetry en-ergy of asymmetric nuclear matter, respectively. They can be parameterized by the followingequations [22] E ( ρ ) = E ( ρ ) + K ρ − ρ ρ ) + J ρ − ρ ρ ) , (4) E sym ( ρ ) = E sym ( ρ ) + L ( ρ − ρ ρ ) + K sym ρ − ρ ρ ) + J sym ρ − ρ ρ ) , (5)where ρ is the nuclear saturation density. According to the researches near nuclear satura-tion density, the most probable values of parameters in Eqs. (4) and Eqs. (5) are as follows: K = 240 ±
20 MeV, E sym ( ρ ) = 31.7 ± L = 58.7 ± − ≤ J ≤
400 MeV, − ≤ K sym ≤
100 MeV, − ≤ J sym ≤
800 MeV [24–29]. It is shown that theparameters K , E sym ( ρ ) and L have already been constrained in a very narrow range, while J , K sym and J sym have large uncertainties. To simplify the calculation, here we choose themost probable values for K , E sym ( ρ ) and L as K = 240 MeV, E sym ( ρ ) = 31.7 MeV and L = 58.7 MeV. For more details about this EOS please refer to Ref. [22]. Through varyingthe parameters J , K sym and J sym within their allowed ranges, we can generate sufficientlylarge number of EOS to perform the Bayesian analysis. Compared with the multisegmentpolytropic EOS, the parametric EOS model builds a more convenient way to extract thesymmetry energy of the asymmetric nuclear matter from the astronomical observations. Inthis work, the core matter of neutron star is described by the parametric EOS model, whilethe inner crust and the outer crust of neutron star are described by the NV EOS model [30]and BPS EOS model [31], respectively. We choose resolution for EOS tables as Ref. [32].The structure of neutron star is governed by Tolman-Oppenheimer-Volkoff (TOV) equa-tions [33, 34] 3 Pdr = − G [ m ( r ) + 4 πr P ( r ) /c ][ ǫ ( r ) + P ( r )] c r [ r − Gm ( r ) /c ] , (6) dmdr = 4 πr ǫ ( r ) /c , (7)where ǫ ( r ) and P ( r ) are the energy density and pressure at radius r , m ( r ) denotes the massenclosed within radius r , G is the gravitational constant and c is the speed of light. For agiven EOS, the TOV equations can be numerically integrated from the origin ( r = 0) to thesurface ( r = R ), where the pressure vanishs, to obtain the M - R relation of neutron star.The tidal deformability describes how a neutron star deforms under an external gravita-tional field produced by its companion star. It can be given by [9, 35–40]Λ = 23 k ( c G RM ) , (8)where the k denotes the second tidal Love number which has to be solved together withthe TOV equations [36]. M ( M ) R (km) FIG. 1: The prior distribution of the M - R relation, where the color from red to blue indicates theprobability density from high to low. The black dash line denotes the 90% credible interval. From terrestrial experiments, there are rudimentary constraints on K sym , J sym , and J [41]. One of our prior assumptions is independent and uniform in their parameter space. Inthis work, we use Monte Carlo random sampling method to generate two million EOS. Thenwe screen the generated EOS by causality and by supporting the recently observed heavieststellar mass 2.14 M ⊙ of neutron star J0740+6620 [42]. The remaining EOS are about 1.6million. 4 Prior P D F E sym (MeV) FIG. 2: The prior distribution of E sym (2 ρ ), where the solid vertical lines represent the 90% credibleinterval for E sym (2 ρ ). The PDF is short for probability density function. Calculating the TOV equations by inputting the remaining EOS, we obtain the priordistribution of the M - R relation, as shown in Fig. 1. It is shown that most of the radii ofneutron stars with lower mass (1 . − . M ⊙ ) are concentrated in a relatively narrow rangeof 12 . − . K sym , J sym and J has a relatively weak effect on theradius of neutron star with lower mass. It is worth noting that the prior distribution reflectsthe general features of the parametric EOS, but not including the impact from GW170817.According to Eqs. (5), we can obtain the prior probability density of E sym (2 ρ ), as shownin Fig. 2. Within 90% credible level, the prior probability of E sym (2 ρ ) is constrained in arange of E sym (2 ρ ) = 54 . +4 . − . MeV.Normally, we call a neutron star with mass of 1.4 M ⊙ as canonical neutron star as mostobserved neutron stars have stellar masses near 1.4 M ⊙ [43]. There have been massiveresearches on the properties of canonical neutron star in recent years [44–48], especiallyafter the GW170817 event [2, 5, 7, 11, 49–54]. Here we will also focus on the propertiesof canonical neutron stars. In Fig. 3, we show the prior distribution of R . (a) and Λ . (b), where R . and Λ . denote the radius and the dimensionless tidal deformability of acanonical neutron star, respectively. It is shown that in the prior distribution and within90% credible level, the radius is distributed in a range of 11.30 km ≤ R . ≤ ≤ Λ . ≤ R . is 12.9 km and the most probable value of Λ . is620, respectively. The relatively large value of the expected E sym (2 ρ ), radius and tidaldeformability indicate that the prior results, which are consistent with the former studies5
10 11 12 13 140.000.030.060.090.12
Prior P D F R (km) (a) Prior P D F (b) FIG. 3: The prior distribution of R . (a) and Λ . (b), where R . and Λ . denote the radii andthe dimensionless tidal deformabilities of canonical neutron stars (with mass of 1.4 M ⊙ ), and thesolid vertical lines represent the 90% credible interval for R . and Λ . , respectively. [47, 55–64], prefer a stiffer EOS for a neutron star.6 . CONSTRAINT ON THE PARAMETER SPACE OF EOS AND THE SYMME-TRY ENERGY AT TWICE SATURATION DENSITY -400 -300 -200 -100 0 1000200400600800 J sy m ( M e V ) K sym (MeV) J = -100 MeV (a) -400 -300 -200 -100 0 100-2000200400600800 J sy m ( M e V ) K sym (MeV) J = 100 MeV (b) -400 -300 -200 -100 0 100-2000200400600800 J sy m ( M e V ) K sym (MeV) J = 300 MeV (c) -400 -300 -200 -100 0 100-2000200400600800 J sy m ( M e V ) K sym (MeV) J = 400 MeV (d) FIG. 4: The posterior probability distribution in parameter space, where the color from red to blueindicates the probability density from high to low. The white areas are the forbidden parameterspaces, where the maximum mass M max =2.14 M ⊙ can not be supported. The black dash linedenotes the 90% credible interval. Based on the distribution of stellar masses and tidal deformabilities of the binary neutronstar in GW170817 [7], we define the likelihood function of each set of parameters of EOSand perform Bayesian inference to analyze the impact on the parameter space of EOS andthe symmetry energy. For the details of the Bayesian inference approach adopted in thiswork please see the Appendix.For the convenience to show the posterior probability of the EOS parameters, we plot theposterior probability of K sym and J sym in a two-dimensional diagram by fixing the parameter J , as shown in Fig. 4, where four values of J : (a) −
100 MeV, (b) 100 MeV, (c) 300 Mev and(d) 400 MeV are adopted. From Fig. 4, we can see that the higher probability density areais located in the left area. Moreover, it is also shown that for a given J , lower values of J sym and K sym are preferred to support the observed data of GW170817. From the definitionsof the parameters K sym , J sym and J in Eqs. (4) and Eqs. (5) and according to Eq. (3), itis easy to understand that higher values of the set of parameters correspond with a stifferEOS. Therefore, the posterior probability of the parameters of EOS as shown in Fig. 47
400 -300 -200 -100 0 1000.0100.0150.0200.0250.0300.035
Posterior P D F K sym (MeV) (a) -200 0 200 400 600 8000.0000.0070.0140.0210.0280.035 P D F J sym (MeV) Posterior (b) -300 -200 -100 0 100 200 300 4000.0000.0070.0140.0210.0280.035 Posterior P D F J (MeV) (c) FIG. 5: The posterior distribution of K sym , J sym and J , where the solid vertical lines representthe 90% credible interval for K sym , J sym and J , respectively. indicates that the observation of GW170817 prefers a relatively softer EOS. In addition, thewhite areas at the left bottom in Fig. 4 are forbidden areas, where the related EOS can notsupport the maximum mass of M max =2.14 M ⊙ . The black dash line denotes the boundaryof 90% credible interval.In addition, according to posterior probability distribution of parameter space , we canget a roughly linear constraint of the parameter space (from the boundary of 90% credibleinterval). That is, the upper boundary (located in the right area) can be approximatelyexpressed as 8 . K sym + 1 . J sym + J ≤ .
25 MeV , (9)and the lower boundary (located in the left area) can be approximately expressed as1 . K sym + 1 . J sym + J ≥ − .
00 MeV , (10)where K sym , J sym , and J are the values of corresponding parameters in unit of MeV. Thetwo constraints can reduce the parameter space to about 50% of the original parameterspace.The posterior distributions of K sym , J sym and J are presented in Fig. 5. Within 90%credible level, it is shown that the parameters K sym , J sym and J are constrained in − . P D F E sym (MeV) Posterior
FIG. 6: The posterior distribution of E sym (2 ρ ), where the solid vertical lines represent the 90%credible interval for E sym (2 ρ ). MeV < K sym < . − . < J sym < . − . < J < . ρ ) of nuclear matter [45]. Thus theknowledge of the EOS, especially the symmetry energy around the twice saturation densityis very important in understanding the radius of neutron star. Up to now, a lot of researchworks have been done to constrain the symmetry energy at twice the saturation density[21, 44, 65–78]. For example, through employing three sets of observational related radiidata and three sets of imaginary radii data of canonical neutron star to perform Bayesiananalysis, Xie and Li [21] inferred the nuclear symmetry energy E sym ( ρ ) by the parametricEOS and constrained the symmetry energy at twice the saturation density as E sym (2 ρ ) =39 . +12 . − . MeV at 68% credible level. Based on the oscillation modes of canonical neutronstars, Wen et al. [44] predicted that the symmetry energy at twice the saturation density E sym (2 ρ ) should be in a range of 54 . +6 . − . MeV if the frequency of f-mode takes a value of f = 1 . +0 . − . kHz, while E sym (2 ρ ) should be in a range of 43 . +6 . − . MeV if the frequency of f-mode takes a value of f = 1 . +0 . − . kHz. By comprehensively combining the observationalconstraints on the radius, maximum mass, tidal deformability and causality condition ofneutron stars, Zhang and Li [65] deduced that the symmetry energy at twice the saturationdensity should be in a range of E sym (2 ρ ) = 46.9 ± et al. [66] derived the symmetry energy at twice the saturation density9 sym (2 ρ ) in a range of [39 . +7 . − . , 54 . +3 . − . ] MeV, respectively. To sum up, current studiesconstrain the symmetry energy at twice the saturation density in a range of [30, 60] MeV.By employing the posterior distribution of the parameter space and Eqs. (5), we obtainthe posterior probability density of E sym (2 ρ ), as shown in Fig. 6. It is shown that thesymmetry energy at twice the saturation density of nuclear matter is constrained in a rangeof E sym (2 ρ ) = 34 . +20 . − . MeV at 90% credible level. Obviously, there is a big differenceof the maximum probability point between the prior and posterior distribution. The latterone prefers a relatively low symmetry energy. Moreover, both of the prior and posteriordistribution of E sym (2 ρ ) are consistent with the conclusions of above literatures.
4. THE CONSTRAINT ON THE RADII AND TIDAL DEFORMABILITIESTHROUGH THE GW170817 M ( M ) R (km) FIG. 7: The posterior distribution of M - R relation, where the color from red to blue indicates theprobability density from high to low. The black dash line denotes 90% credible interval. For comparison with prior distribution of the M - R relation, here we present the corre-sponding posterior distribution in Fig. 7. We use Monte Carlo random sampling method (bythe probability density of posterior parameter space) to generate two million EOS, and then M - R relation, R . , and Λ . are calculated statistically. Of course, according to the priorassumptions, it can be known that the EOS generated by posterior parameter space haveensured constraints mentioned in section III. The distribution of posterior M - R relation isshown in Fig. 7. It can be seen from Fig. 7 that, comparing with the prior M - R relationdistribution, the posterior M - R relation distribution spreads to the left, especially at highstellar mass. So the posterior M - R relation prefers a softer EOS.10
10 11 12 13 140.000.020.040.060.080.10
Posterior P D F R (km) (a) Posterior P D F (b) FIG. 8: The posterior distribution of radii R . (a) and dimensionless tidal deformabilities Λ . (b) of canonical neutron stars, where the solid vertical lines represent the 90% credible interval for R . and Λ . , respectively. The posterior distribution of the radii R . and the dimensionless tidal deformabilities Λ . of canonical neutron stars are presented in Fig. 8 (a) and (b), respectively. Unlike the prior11istribution (see Fig. 3), there are two peaks in the posterior distribution of R . and Λ . in Fig. 8. We notice that in Fig. 4, each subgraph has two higher probability density areasof the posterior probability in the EOS parameter space and one of them has a relativelylower probability density. Considering a certain relevance between the tidal deformabilityand the radius, it is easy to understand that the two peaks in Fig. 8 are consistent with thetwo higher probability density areas in Fig. 4.According to our calculations, at 90% credible level in the posterior distribution, theradius of a canonical neutron star is distributed in a range of R . = 12 . +0 . − . km and itsdimensionless tidal deformability is distributed in a range of Λ . = 500 +186 − . Comparingwith the prior distribution of radius ( R . = 12 . +0 . − . km) and the tidal deformability (Λ . =620 +103 − ), we can find that the most probable value of the radius and the tidal deformabilityof the posterior distribution is smaller than that of the prior distribution, which means thatthe posterior distribution prefers a relatively softer EOS.
5. SUMMARY
The detection of the gravitational waves of the binary neutron star merger eventGW170817 provides us important information, such as the distribution of the tidal de-formabilities and the stellar masses of the binary neutron star, to further investigate theproperties and the state of matter of neutron stars. In this work, we investigate the radiusand tidal deformability of canonical neutron star and the symmetry energy of the superdense matter through the Bayesian analysis based on the distribution of component massesand tidal deformabilities of binary neutron star merger GW170817 released by LIGO andVIRGO. To perform the Bayesian analysis, one need to generate a huge number of EOS.Normally, the polytropic EOS model is adopted to generate the EOS. Normally, the poly-tropic EOS model is adopted to generate the EOS. Here we adopt the isospin-dependentparametric EOS model as this kind of model that can provide a more convenient way toextract the symmetry energy of the asymmetric nuclear matter from the astronomical ob-servations. In this work, two million isospin-dependent parametric EOS are generated bythe Monte Carlo random sampling method, and the generated EOS are further screenedby the recently observed heaviest stellar mass 2.14 M ⊙ of J0740+6620 and the causality toperform the Bayesian analysis. From this analysis, we find that the parameter space of EOScan be reduced to about 50% of the original parameter space at 90% credible level. In theposterior distribution, the symmetry energy at twice the saturation density of nuclear mat-ter can be constrained within E sym (2 ρ ) = 34 . +20 . − . MeV at 90% credible level, the radiusis distributed in a range of R . = 12 . +0 . − . km and the dimensionless tidal deformability isdistributed in a range of Λ . = 500 +186 − at 90% credible level. Comparing with the priordistribution of E sym (2 ρ ) ( E sym (2 ρ ) = 54 . +4 . − . MeV), radii ( R . = 12 . +0 . − . km) and thetidal deformabilities (Λ . = 620 +103 − ), one can see that the posterior distribution prefers asofter EOS.
6. ACKNOWLEDGEMENTS
We thank Bao-An Li for helpful discussions. This work is supported by NSFC (GrantsNo. 11975101 and No. 11722546), Guangdong Natural Science Foundation (Grant No.2020A151501820) and the talent program of South China University of Technology (Grant12o. K5180470). This project has made use of NASA’s Astrophysics Data System.
Appendix A: Bayesian Inference Approach
Bayesian statistics give the posterior probability as P ( −→ θ | D ) = P ( D |−→ θ ) P ( −→ θ ) R P ( D |−→ θ ) P ( −→ θ ) d −→ θ , (A1)where P ( −→ θ | D ) is the posterior probability for the model −→ θ (isospin-dependent parametricEOS) given the data set D which is the distribution of tidal deformabilities and stellarmasses in GW170817, P ( D |−→ θ ) is the likelihood function for a given neutron star model −→ θ to correctly infer the data D; and P ( −→ θ ) is the prior probability of the model −→ θ beforebeing correlated with the data D. The denominator in Eq. A1 is the normalization constant,which is a constant for each parametric EOS.Here the probability distributions of tidal deformabilities and masses (gravitational wavemodel PhenomPNRT and low spin prior, χ ≤ P ( D |−→ θ ) = Z dM · P Λ · P M , (A2)For the convenience of understanding the construction of the likelihood function, we discussit in more details as follows. For a given parametric EOS, we can get the properties (mass,tidal deformability) of a series of neutron stars by using our codes. The P Λ and P M ofeach neutron star are derived by combining the tidal deformability and mass of a specifiedneutron star with the probability distribution of tidal deformabilities and component massesin GW170817. Therefore, for a given parametric EOS, the value of the non-normalizedlikelihood function can be determined by integrating the Eqs.(A2).As we know, one of the tightly constrained physical quantities in GW170817 is the chirpmass, which is defined as M c = ( m m ) / ( m + m ) / , (A3)where the m and m are the component masses of the merged binary neutron stars. Thechirp mass in GW170817 is constrained to M c = 1 . +0 . − . M ⊙ at 90% credible level, whichis independent of the waveform model [1]. Here we employ correlated prior for the masswith the specified chirp mass at the most probable value M c = 1 . M ⊙ and completelyuncorrelated prior for the deformabilities of binary neutron star in GW170817. Then weuse the fixed chirp mass value to set m for each given m . From this way, we can get theposterior probability of each sample (each parametric EOS). [1] The LIGO Scientific Collaboration and the Virgo Collaboration (B.P. Abbott et al .), Phys.Rev. Lett. , 161101 (2017).[2] The LIGO Scientific Collaboration and the Virgo Collaboration (B.P. Abbott et al .), Phys.Rev. Lett. , 161101 (2018).
3] B. Margalit, B.D. Metzger, Astrophys. J. , L19 (2017).[4] A. Bauswein, O. Just, H.T. Janka, N. Stergioulas, Astrophys. J. , L34 (2017).[5] The LIGO Scientific Collaboration and the Virgo Collaboration (B.P. Abbott et al .), Phys.Rev. Lett. , 061104 (2019).[6] G. Baym et al ., Rep. Prog. Phys. , 056902 (2018).[7] The LIGO Scientific Collaboration and the Virgo Collaboration (B.P. Abbott, et al .), Phys.Rev. X. , 011001 (2019).[8] B.P. Abbott, et al ., arXiv:2001.01761v1.[9] Z. Carson, A.W. Steiner, K. Yagi, Phys. Rev. , 043010 (2019).[10] W. Kastaun, F. Ohme, Phys. Rev. D. , 103023 (2019).[11] Y. Lim, J.W. Holt, Phys. Rev. Lett. , 062701 (2018).[12] S. De, D. Finstad, J.M. Lattimer, D.A. Brown, E. Berger, C.M. Biwer, Phys. Rev. Lett. ,091102 (2018).[13] T. Carreau, F. Gulminelli, J. Margueron, Eur. Phys. J. A , 188 (2019).[14] Y. Lim, J.W. Holt, Eur. Phys. J. A , 209 (2019).[15] Y. Lim, J.W. Holt, R.J. Stahulak, Phys. Rev. C. , 035802 (2019).[16] T. Carreau, F. Gulminelli, J. Margueron, Phys. Rev. C. , 055803 (2019).[17] F. Hernandez et al ., Phys. Rev. D. , 103009 (2019).[18] M. Fasano, T. Abdelsalhin, A. Maselli, V. Ferrari, Phys. Rev. Lett. , 141101 (2019).[19] T.E. Riley et al ., Astrophys. J. , L21 (2019).[20] B.A. Li, L.W. Chen, C.M. Ko, Phys. Rep. , 113 (2008).[21] W.J. Xie, B.A. Li, Astrophys. J. , 2 (2019).[22] N.B. Zhang, B.A. Li, J. Xu, Astrophys. J. , 90 (2018).[23] I. Bombaci, U. Lombardo, Phys. Rev. C. , 1892 (1991).[24] S. Shlomo, V.M. Kolomietz, G. Colo, Eur. Phys. J. A , 23 (2006).[25] J. Piekarewicz, J. Phys. G. , 064038 (2010).[26] B.A. Li, X. Han, Phys. Lett. B. , 276 (2013).[27] N.B. Zhang, B.J. Cai, B.A. Li, W.G. Newton, J. Xu, Nucl. Sci. Tech. , 181 (2017).[28] M. Oertel, M. Hempel, T. Klahn, S. Typel, Rev. Mod. Phys. , 015007 (2017).[29] B.A. Li, Nuclear Physics News. , 7 (2017).[30] J.W. Negele, D. Vautherin, Nucl. Phys. , A298 (1973).[31] G. Baym, C.J. Pethickm, P. Sutherland, Astrophys. J. , 299 (1971).[32] H.Y. Chen, D.H. Wen, N. Zhang, Chin. Phys. C. , 054108 (2019).[33] R.C. Tolman, Phys. Rev. , 364 (1939).[34] J.R. Oppenheimer, G.M. Volkoff, Phys. Rev. , 374 (1939).[35] ´E.´E. Flanagan, T. Hinderer, phys. Rev. D. , 021502 (2008).[36] T. Damour, A. Nagar, Phys. Rev. D. , 084035 (2009).[37] T. Damour, A. Nagar, Phys. Rev. D. , 084016 (2010).[38] T. Hinderer, Astrophys. J. , 1216 (2008).[39] T. Hinderer, B.D. Lackey, R.N. Lang, J.S. Read, Phys. Rev. D. , 123016 (2010).[40] K. Yagi, N. Yunes, Phys. Rev. D. , 023009 (2013).[41] I. Tews, J.M. Lattimer, A. Ohnishi, E.E. Kolomeitsev, Astrophys. J. , 105 (2017).[42] H.T. Cromartie, E. Fonseca, S.M. Ransom et al ., Nature, , 0880 (2019).[43] https://stellarcollapse.org/nsmasses[44] D.H. Wen, B.A. Li, H.Y. Chen, N.B. Zhang, Phys. Rev. C. , 045806 (2019).[45] J.M. Lattimer, M. Prakash, Astroph. J. , 426 (2001).
46] D.H. Wen, W.G. Newton, B.A. Li, Phys. Rev. C. , 025801 (2012).[47] J.M. Lattimer, A.W. Steiner, Eur. Phys. J. A , 40 (2014).[48] R.R. Jiang, D.H. Wen, H.Y. Chen, Phys. Rev. D. , 123010 (2019).[49] T.Q. Zhao, J.M. Lattimer, Phys. Rev. D. , 063020 (2018).[50] C. Raithel, F. ¨Ozel, D. Psaltis, Astrophys. J. , L23 (2018).[51] F.J. Fattoyev, J. Piekarewicz, C.J. Horowitz, Phys. Rev. Lett. , 172702 (2018).[52] E. Annala, T. Gorda, A. Kurkela, A. Vuorinen, Phys. Rev. Lett. , 172703 (2018).[53] A. Bauswein, O. Just, H. Janka, N. Stergioulas, Astrophys. J. , L34 (2017).[54] I. Tews, J. Margueron, S. Reddy, Phys. Rev. C. , 045804 (2018).[55] F. ¨Ozel, Nature, , 1115 (2006).[56] F. ¨Ozel, T. G¨uver, D. Psaltis, Astrophys. J. , 1775 (2009).[57] T. G¨uver, F. ¨Ozel, A. Cabrera-Lavers, P. Wroblewski, Astrophys. J. , 964 (2010).[58] F. ¨Ozel, D. Psaltis, T. G¨uver, G. Baym, C. Heinke, S. Guillot, Astrophys. J. , 28 (2016).[59] V.F. Suleimanov, J. Poutanen, M. Revnivtsev, K. Werner, Astrophys. J. , 122 (2011).[60] B.A. Li, A.W. Steiner, Phys. Lett. B. , 436 (2006).[61] A.W. Steiner, J.M. Lattimer, E.F. Brown, Astrophys. J. , L5 (2013).[62] S. Guillot, M. Servillat, N.A. Webb, R.E. Rutledge, Astrophys. J. , 7 (2013).[63] S. Guillot, R.E. Rutledge, Astrophys. J. , L3 (2014).[64] S. Bogdanov, C.O. Heinke, F. ¨Ozel, T. G¨uver, Astrophys. J. , 184 (2016).[65] N.B. Zhang, B.A. Li, Eur. Phys. J. A , 39 (2019).[66] Y. Zhou, L.W. Chen, Z. Zhang, Phys. Rev. D. , 121301 (2019).[67] X. Roca-Maza, M. Centelles, X. Vinas, M. Warda, Phys. Rev. Lett. , 252501 (2011).[68] M. Dutra, O. Louren, J.S.S. Martins, A. Delfino, J.R. Stone, P.D. Stevenson, Phys. Rev. C. , 035201 (2012).[69] M. Dutra, O. Louren, S.S. Avancini, B.V. Carlson, A. Delfino, D.P. Menezes, C. Providencia,S. Typel, J.R. Stone, Phys. Rev. C. , 055203 (2014).[70] I. Vidana, C. Providencia, A. Polls, A. Rios, Phys. Rev. C. , 045806 (2009).[71] Z.H. Li, H.J. Schulze, Phys. Rev. C. , 028801 (2008).[72] F. Sammarruca, Int. J. Mod. Phys. E. , 1259 (2010).[73] A.Akmal, V.R. Pandharipande, D.G. Ravenhall, Phys. Rev. C. , 1804 (1998).[74] B. Friedman, V.R. Pandharipande, Nucl. Phys. A. , 502 (1981).[75] R.B. Wiringa, V. Fiks, A. Fabrocini, Phys. Rev. C. , 1010 (1988).[76] F. Sammarruca, Phys. Rev. C. , 064312 (2014).[77] X.T. He, F.J. Fattoyev, B.A. Li, W.G. Newton, Phys. Rev. C. , 015810 (2015).[78] T. Kl¨ahn, D. Blaschke, S. Typel, E.N.E. van Dalen, A. Faessler, C. Fuchs, T. Gaitanos, H.Grigorian, A. Ho, E.E. Kolomeitsev, M.C. Miller, G. R¨opke, J. Tr¨umper, D.N. Voskresensky,F. Weber, H.H. Wolter, Phys. Rev. C. , 035802 (2006)., 035802 (2006).