Consistent Skyrme parametrizations constrained by GW170817

# Consistent Skyrme parametrizations constrained by GW170817

Odilon Lourenço, Mariana Dutra, César Lenzi, S. K. Biswal, M. Bhuyan, and Débora P. Menezes Departamento de Física, Instituto Tecnológico de Aeronáutica, DCTA, 12228-900, São José dos Campos, SP, Brazil
Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
Depto de Física - CFM - Universidade Federal de Santa Catarina, Florianópolis - SC - CP. 476 - CEP 88.040 - 900 - Brazil
July 20, 2019
###### Abstract

The high-density behavior of the stellar matter composed of nucleons and leptons under -equilibrium and charge neutrality conditions is studied with the Skyrme parametrizations shown to be consistent (CSkP) with the nuclear matter, pure neutron matter, symmetry energy and its derivatives in a set of constraints [Dutra et al., Phys. Rev. C 85, 035201 (2012)]. The predictions of these parametrizations on the tidal deformabilities related to the GW170817 event are also examined. The results points out to a correlation between the Love numbers and tidal deformabilities with the respective radii of the binary neutron stars system (BNSS). We also find that those CSkP supporting massive neutron stars () predict radii of the BNSS in full agreement with recent data from LIGO and Virgo Collaboration (LVC) given by  km. A correlation between dimensionless tidal deformability and radius of the canonical star is found, namely, , with results for the CSkP compatible with the recent range of from LVC. Finally, an analysis of the graph shows that the CSkP compatible with the recent bounds obtained by LVC, namely, GSkI, Ska35s20, MSL0 and NRAPR, can also support massive stars (), and predict a range of for the canonical star radius.

###### pacs:
21.10.-k, 21.10.Gv, 21.65.-f, 21.65.Mn

## I Introduction

Neutron stars are an incredible natural laboratory for the study of nuclear matter at extreme conditions of isospin asymmetry and density (latt04 (); ozel06 (). The properties of nuclear matter at such high densities are mostly governed by the equation(s) of state (EOS), which correlates pressure (), energy density () and other thermodynamical quantities. From the terrestrial experiments, nuclear matter properties are mostly constrained up to saturation density,  fm g/cm tsan12 (); bald16 (); latt16 (); oert17 (). The EOS correlating , and is the sole ingredient to determine the relationship between the mass and radius of a neutron star by using the Tolman-Oppenheimer-Volkoff equations tov39 (); tov39a (). It also plays a vital role in determining other star properties such as the moment of inertia and tidal deformability tanj10 (); phil18 (). The recent observation of the gravitational wave (GW) emission from the first binary neutron stars merger event, GW170817, provided new expectations to constraint the EOS in more efficient ways ligo17 (); ligo18 ().

Since 2015, the observation of the GW emission from the binary compact objects, by LIGO aasi15 () and Virgo acer15 () collaborations, opened a platform to study the GW and related physics in more adequate ways. The GW170817 event, observed on 17 August 2017, has a special importance in nuclear physics since it consists of the emergence of GW from two binary neutron stars. It coincides with the detection of the -ray burst GRB170817 abbo17 (); gold17 () and the components were verified as neutron stars by various electromagnetic spectrum observations abbo17a (); coul17 (); troj17 (); hagg17 (); hall17 (). The measurements of the neutron star mass, spin, radius, and gravitational red shift provide weak constraints on the EOS as these measurements depend on the detailed modeling of the radiation mechanism and are subjected to a lot of systematic errors latt07 (); joce09 (). The GW, however, offers an opportunity to constrain the EOS from the tidal deformability data bhar17 (); tanj10 (); hind08 (); thib09 (); tayl09 (), which establishes a relation between the internal structure of the neutron star and the emitted GW.

In the present context, we use the Skyrme model skyr61 (); bend03 (); ston07 () in order to explore the possible constraint on the EOS by the observation of the GW170817 event. In the work of Ref. dutra12 (), the authors have studied the nuclear matter characteristics of symmetric and asymmetric matter at saturation as well as at high densities by using parametrizations of the Skyrme energy density functional. Following this work, it was observed that only parametrizations, namely, GSkI agrawal2006 (), GSkII agrawal2006 (), KDE0v1 agrawal2005 (), LNS cao2006 (), MSL0 chen2010 (), NRAPR steiner2005 (), Ska25s20 private2 (), Ska35s20 private2 (), SKRA rashdan2000 (), Skxs20 brown2007 (), SQMC650 guichon2006 (), SQMC700 guichon2006 (), SkT1 tondeur1984 (); stone2003 (), SkT2 tondeur1984 (); stone2003 (), SkT3 tondeur1984 (); stone2003 () and SV-sym32 klupfel2009 (), satisfy all constraints from symmetric nuclear matter, pure neutron matter, and a mixture of both related with the symmetry energy and its derivatives dutra12 (). This set was named as Consistent Skyrme Parametrizations (CSkP), which is used in the present manuscript. These parametrizations offer a predictive power starting from sub-saturation density to very high density at very high isospin asymmetry, what has motivated us to analyze the stellar matter behavior for the CSkP, in particular, the tidal deformability related to the GW170817 event. In other words, the tidal deformability of the GW170817 event, using the post-Newtonian model, can provide a suitable constraint to study the predictive capacity of the CSkP in various astrophysical phenomena. We try to correlate the tidal deformability of the canonical neutron star () and the corresponding radius () for the CSkP by addressing a transparent relation between and as a power law. Usually, the proportionality relation , which is based on the definition , with being the neutron star mass, is cited in the literature. It is worth noticing that this proportionality is not exact since the Love number depends on the radius through a complicated second order differential equation. In recent studies, various relations between the and are obtained with different models, like the Skyrme malik18 () and relativistic mean-field fatt18 () ones. Here, we study this correlation with CSkP. The individual radii of the binary neutron stars system components corresponding to the GW170817 event are also discussed.

This manuscript is organized as follows. In Sec. II, we briefly outline the theoretical formalism for the Skyrme model in nuclear and neutron star matter. In Sec. III, we discuss the predictions of CSkP concerning the recent GW170817 event. Special attention is given to the tidal deformability of the neutron stars binary system. We conclude the manuscript with a brief summary in Sec. IV.

## Ii Theoretical Formalism

### ii.1 Infinite nuclear matter

In the following we mention the EOS used in this work related to the Skyrme model at zero temperature. The energy density of infinite nuclear matter, defined in terms of the density and proton fraction, is written as dutra12 ()

 ϵ(ρ,y) =310Mnuc(3π22)2/3ρ5/3H5/3(y) +t08ρ2[2(x0+2)−(2x0+1)H2(y)] +1483∑i=1t3iρσi+2[2(x3i+2)−(2x3i+1)H2(y)] +340(3π22)2/3ρ8/3[aH5/3(y)+bH8/3(y)], (1)

with

 a = t1(x1+2)+t2(x2+2), (2) b = 12[t2(2x2+1)−t1(2x1+1)], (3)

and

 Hl(y) = 2l−1[yl+(1−y)l], (4)

where is the proton fraction, and is the nucleon rest mass. A particular parametrization is defined by a specific set of the following free parameters: , , , , , , , , , , , , , , and .

From Eq. (1), one can construct the pressure of the model as

 p(ρ,y)=ρ2∂(E/ρ)∂ρ=15Mnuc(3π22)2/3ρ5/3H5/3(y) +t08ρ2[2(x0+2)−(2x0+1)H2(y)] +1483∑i=1t3i(σi+1)ρσi+2[2(x3i+2)−(2x3i+1)H2(y)] +18(3π22)2/3ρ8/3[aH5/3(y)+bH8/3(y)], (5)

and also the nucleon chemical potential as

 μq(ρ,y)=∂ϵ∂ρq=12Mnuc(3π22)2/3ρ2/3H5/3(y) +15(3π22)2/3ρ5/3[aH5/3(y)+bH8/3(y)] +t04ρ[2(x0+2)−(2x0+1)H2(y)] +1483∑i=1t3i(σi+2)ρσi+1[2(x3i+2)−(2x3i+1)H2(y)] ±12[1∓(2y−1)]{310Mnuc(3π22)2/3ρ2/3H′5/3(y) −t08ρ(2x0+1)H′2(y)−1483∑i=1t3iρσi+1(2x3i+1)H′2(y) +340(3π22)2/3ρ5/3[aH′5/3(y)+bH′8/3(y)]}, (6)

where for protons and neutrons, respectively. Here one also has that .

### ii.2 Neutron star matter

In order to treat stellar matter, one needs to implement charge neutrality and -equilibrium conditions under the weak processes, , and its inverse process . For densities in which exceeds the muon mass, the reactions , , and energetically favor the emergence of muons. Here, we consider that neutrinos are able to escape the star due to their extremely small cross-sections. By taking these assumptions into account, we can write the total energy density and pressure of the stellar system for the Skyrme model, respectively, as

 E(ρ,ρe,y) =ϵ(ρ,y)+μ4e(ρe)4π2+Mnucρ +1π2∫√μ2μ(ρe)−m2μ0dkk2(k2+m2μ)1/2, (7)

and

 P(ρ,ρe,y) =p(ρ,y)+μ4e(ρe)12π2 +13π2∫√μ2μ(ρe)−m2μ0dkk4(k2+m2μ)1/2, (8)

where, and are given in the Eqs. (1) and (5), respectively. The chemical equilibrium and the charge neutrality conditions are

 μn(ρ,y)−μp(ρ,y)=μe(ρe), (9)

and

 ρp(ρ,y)−ρe=ρμ(ρe), (10)

where and are found from Eq. (6), , , , and , for  MeV and massless electrons. Thus, for each input density , the quantities and are calculated by simultaneously solving conditions (9) and (10).

The properties of a spherically symmetric static neutron star can be studied by taking the energy density and pressure as input for the Tolman-Oppenheimer-Volkoff (TOV) equations, which are given by tov39 (); tov39a (),

 dP(r)dr = −[E(r)+P(r)][M(r)+4πr3P(r)]r2[1−2M(r)r] (11)

and

 dM(r)dr = 4πr2E(r), (12)

where the solution is constrained to the following conditions: (i) at the center, (central pressure), (ii) (central mass), and (iii) (central energy density). Furthermore, at the star surface one has and , with being the neutron star radius. In order to solve the TOV equations in this work, we take and given in Eqs. (7) and (8) as input along with the Baym-Pethick-Sutherland (BPS) equation of state bps () for the low density regime, i. e., for the neutron star crust. In other words, the BPS EOS is included in order to take into account the low density regime, in this case given by  fm fm. The total EOS including hadrons and leptons are coupled to the BPS part from densities greater than  fm.

### ii.3 Tidal deformability

Finally, in order to perform a detailed analysis concerning the prediction of the CSkP on the recent GW170817 event, a very important quantity has to be computed, namely, the tidal deformability. It is one of the observed quantities in the binary neutron stars system ligo17 (); ligo18 (), which plays a major role in constraining hadronic EOS. The induced quadrupole moment in one neutron star of a binary system due to the static external tidal field created by the companion star can be written as tanj10 (); hind08 (),

 Qij=−λEij. (13)

Here, is the tidal deformability parameter, which can be expressed in terms of dimensionless quadrupole tidal Love number as

 λ=23k2R5. (14)

The dimensionless tidal deformability (i.e., the dimensionless version of ) is connected with the compactness parameter through

 Λ=2k23C5. (15)

The tidal Love number is obtained as

 k2=8C55(1−2C)2[2+C(yr−1)−yr]× ×{2C[6−3yr+3C(5yr−8)] +4C3[13−11yr+C(3yr−2)+2C2(1+yr)] +3(1−2C2)[2−yr+2C(yr−1)]ln(1−2C)}−1, (16)

where is found from the solution of

 rdyrdr+y2r+yrF(r)+r2Q(r)=0, (17)

with

 F(r)=r−4πr3[E(r)−P(r)]r−2M(r) (18)

and

 Q(r) =4πr[5E(r)+9P(r)+E(r)+P(r)∂P(r)/∂E(r)−64πr2]r−2M(r) −4[M(r)+4πr3P(r)r2(1−2M(r)/r)]2. (19)

In order to find , Eq. (17) has to be solved as part of a coupled system containing the TOV equations given in Eqs. (11) and (12).

The dimensionless tidal deformabilities of a binary neutron stars system, namely, and , can be combined to yield the weighted average as ligo17 ()

 ~Λ=1613(m1+12m2)m41Λ1+(m2+12m1)m42Λ2(m1+m2)5, (20)

where and are masses of the two stars.

## Iii Results and Discussions

### iii.1 Sound velocity and neutron star matter

As all the CSkP come from a nonrelativistic mean field model, at zero temperature regime, the causal limit may be broken at the high density region, since the sound velocity () increases with density, or equivalently, with energy density. However, for the CSkP we verify that exceeds only at very high energy density values, as we can see in Fig. 1.

From this figure, one can verify that the CSkP obey the causal limit up to a range of  fm. By comparing these results with those obtained for relativistic mean-field (RMF) parametrizations in Fig. 2 of Ref. dutra16 (), a clear difference in behavior is observed. The RMF parametrizations present a saturation for the sound velocity unlike the Skyrme ones, that always increase. Despite this increasing dependence, Fig. 1 shows that it is possible to describe neutron star matter with CSkP within a particular range of energy densities. The mass-radius profiles predicted by the CSkP are obtained next by taking this analysis into account. The results are shown in Fig. 2.

In this figure, horizontal bands in magenta and green colors indicate respectively the observational data of neutron star masses of PSR J1614-2230 nature467-2010 () and PSR J038+0432 science340-2013 () pulsars. We also show the empirical constraints for the mass-radius profile for the cold dense matter inside the neutron star. They were obtained from a Bayesian analysis of type-I x-ray burst observations by Nättliä, et al. in Ref. nat16 () (outer orange and inner red bands), and from a mass-radius coming from six sources, namely, three from transient low-mass x-ray binaries and three from type-I x-ray bursts with photospheric radius, by Steiner et al. in Ref. stein10 () (outer white and inner black bands).

These observations imply that the neutron star mass predicted by any theoretical model should reach the limit of . From the results, we find that the maximum masses obtained for the GSkI, Ska35s20, MSL0, NRAPR, and KDE0v1 parametrizations are consistent with these boundaries nature467-2010 (); science340-2013 (). Furthermore, the radii obtained from these parametrizations (including the crust) for the canonical star of are also inside the bands calculated in Refs. nat16 (); stein10 (). The rest of the CSkP underestimates the observed data regarding the neutron star mass.

In Table 1, we show the maximum neutron star mass and corresponding radius, compactness and central energy density predicted by the CSkP. We also tabulate some properties related to the canonical neutron star. It is worth mentioning that the central energy density of all CSkP are compatible with the causal limit, as one can verify from Fig. 1.

In the recent literature, a lot of effort has been put to constraint the radius of the canonical neutron star, see for instance, Refs. malik18 (); yeun18 (); elia18 (); zhan19 (); caro18 (); tews18 (). In Ref. malik18 (), Tuhin Malik et. al. have discussed this constraint by using Skryme and RMF models and their calculations suggest the range of . By using a set of more realistic models and the neutron skin values as a new constraint, F. J. Fattoyev et. al. have shown the upper limit for as  fatt18 (). In Ref. yeun18 (), Yeunhwan Lim et. al. have used chiral effective field theory and constraints from nuclear experiments to establish the range of . Elias R. Most et. al. have studied the constraint on with a large number of EOS with pure hadronic matter without any kind of phase transition elia18 (). They found the value of inside the range of , with the most likely value of . From the above discussion, we can estimate an specific range for encompassing the previous ones as . Our calculations for from the CSkP show a minimum value of  km (SQMC650 parametrization), while the maximum value is given by  km (Ska35s20 parameter set). Both maximum and minimum values present very good agreement with the composite range. As a consequence, the five CSkP predicting neutron star mass around two solar masses also present compatible with the aforementioned range.

### iii.2 Predictions on the GW170817 event

Here we proceed to give the results provided by the CSkP regarding the binary system, namely, neutron stars of masses and , related to the GW170817 event given in Refs. ligo17 (). In Table 2 we list the binary neutron stars masses and (in units of ), its corresponding radii and (both in km), tidal Love numbers and (dimensionless), tidal deformabilities and (in units of ), and the chirp radius (km).

We present in this table some particular values of chosen from the range of obtained from the analysis of the GW170817 event in Ref. ligo17 (). The mass of the companion star is calculated through the relationship between , and the chirp mass given by

 Mc=(m1m2)3/5(m1+m2)1/5. (21)

In this equation, is fixed at the observed value of according to Ref. ligo17 (). This quantity is also used to compute the chirp radius, defined as chirpradius ()

 Rc=2Mc~Λ1/5, (22)

with given in Eq. (20).

For each CSkP, we give five combinations for the binary stars masses in order to calculate the tidal Love numbers and tidal deformabilities. If we restrict our analysis to those CSkP predicting neutron stars masses around , in agreement with observational data of Refs.nature467-2010 () and science340-2013 (), namely, GSkI, Ska35s20, MSL0, NRAPR, and KDE0v1, one can verify that the calculated radii and lie inside the recent predictions from LIGO and Virgo Collaboration ligo18 (), that found  km for the heavier and lighter star at the 90% credible level. From the table, one can also notice in particular that  () take a range of values of  g.cm.s. It is also worth noting that the values of  (), and  () are strongly correlated with  (. It is verified for all CSkP that Love numbers and tidal deformabilities increase as the respective radii increase, i. e., , , , and , where and are increasing functions of .

In searching for other possible correlations in the context of the neutron star binary system, one can notice from Eq. (15) that is not a good assumption, since the tidal Love number depends on the neutron star radius in a nontrivial way, as seen in Eq. (16). In this context, we try to find a correlation between the radius and tidal deformability for the CSkP for the canonical star, in which . The obtained results for as a function of are shown in Fig. 3, with a similar qualitative behavior in comparison with the study performed in Ref. tsang (), for instance.

From the points shown in the figure, we could establish a fitting curve correlating as a function of , namely, . This correlation presents different numbers in comparison with those found from predictions of EOS constructed by chiral effective field theory at low densities and the perturbative QCD at very high baryon densities using polytropes anna18 (), several energy density functional within RMF models fatt18 (), and both RMF and Skyrme Hartree-Fock energy density functionals malik18 (). In these cited works, the authors found  anna18 (),  fatt18 (), and  malik18 (). However, the values predicted by the CSkP are in full agreement with the very recent data obtained by LIGO and Virgo Collaboration ligo18 () regarding the tidal deformability of the canonical star, given by .

For the sake of completeness, in Fig. 4 we plot the dimensionless tidal deformability of a static neutron star as a function of its mass for the CSkP.

The tidal deformability decreases nonlinearly with the neutron star mass for all parametrizations. At , the resulting values of stand within a range of around for the CSkP, which are within the upper limit of of LIGO + Virgo gravitational detection ligo17 (), and also the recent updated range of  ligo18 (), as mentioned before.

In Fig. 5 we plot the tidal deformabilities and of the binary neutron stars system with component masses of and ().

The diagonal dotted line corresponds to the case in which . The upper and lower dash lines correspond to the 90% and 50% confidence limits respectively, which are obtained from the recent analysis of the GW170817 event ligo18 (). This figure shows that out of 16 CSkP, 4 of them are completely inside the region defined by the upper and lower bounds predicted by the GW170817 data ligo18 (). They are Ska35s20, GSkI, MSL0, and NRAPR. If we look at the various neutron star properties predicted by these parametrizations, we can notice two interesting facts:

• All of them produce neutron stars with maximum mass around . This implies the tidal deformability data of the GW170817 strongly support the maximum mass of the neutron stars close to the observational data of the pulsars PSR J1614-2230 nature467-2010 () and PSR J038+0432 science340-2013 ().

• The radius of the canonical star obtained from these parametrizations follows a trend. Out of them, the minimum value of this quantity is given by the NRAPR parametrization:  km, while the maximum value is found by the Ska35s20 model:  km. All other parametrizations present values in between. In other words, the CSkP which have capacity to reproduce the recent values for the tidal deformability related to the GW170817 event give in the range of . The maximum value of this range is close to the most likely value of given in Ref. elia18 (), namely, .

## Iv Summary and Conclusions

In the present paper we have revisited the Skyrme parametrizations that were shown to satisfy several nuclear matter constraints in Ref. dutra12 (), named as the consistent Skyrme parametrizations (CSkP), and confronted them with astrophysical constraints and predictions on the GW170817 event studied by LIGO and Virgo Collaboration in recent papers ligo17 (); ligo18 (). Concerning the applicability of these nonrelativistic models at the high density regime of the stellar matter, we have shown that causality is not broken at the energy density range of interest, as one can see from Fig. 1, and from the comparison with the central energy density obtained from the CSkP and presented in Table 1. Our calculations also pointed out to a range of according to the predictions of the CSkP. It was also shown that only the GSkI, Ska35s20, MSL0, NRAPR, and KDE0v1 parametrizations are able to produce neutron stars with mass around , value established form observational analysis of PSR J1614-2230 nature467-2010 () and PSR J038+0432 science340-2013 () pulsars.

Concerning the predictions of the CSkP on the GW170817 event, it was shown that the five aforementioned CSkP present radii and , related to the neutron stars components of the binary system, in full agreement with those obtained by LIGO and Virgo Collaboration for the heavier and lighter star at the 90% credible level, namely,  km. The values of and calculated from the CSkP can be seen in Table 2. Also from this table, we could verify that Love numbers and tidal deformabilities are increasing functions of the respective radii, i. e., and are increasing functions of ().

By investigating the results regarding the canonical stars (), our results pointed out to a correlation given by between the dimensionless tidal deformability and the radius. From this correlations, we found that the CSkP present values of completely inside the ranges of  ligo17 (), or even the recent one given by  ligo18 (), as one can see in Figs. 3 and 4.

Finally, we also have calculated the dimensionless tidal deformabilities of the binary neutron stars system, and (see Fig. 5), and found that GSkI, Ska35s20, MSL0, and NRAPR parametrizations are completely inside the region defined by the upper and lower bounds, on the graph, predicted by the recent paper from LIGO and Virgo Collaboration ligo18 (). All of these specific CSkP support massive neutron stars (), and also establish the range of for the canonical star radius.

## Acknowledgments

This work is a part of the project INCT-FNA Proc. No. 464898/2014-5, partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under grants 301155/2017-8 (D.P.M.) and 310242/2017-7 (O.L.), by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under thematic projects 2013/26258-4 (O.L.,M.D.,C.L.), 2014/26195-5, 2017/05660-0 (M.B.), and National key R&D Pogram of China, Grant No. 2018YFA0404402 (S.K.B.).

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