Consistent relativistic mean field models constrained by GW170817
Odilon Lourenço Mariana Dutra César H. Lenzi César V. Flores Débora P. Menezes


 






 

Depto de Física - CFM - Universidade Federal de Santa Catarina, 
Florianópolis - SC - CP. 476 - CEP 88.040 - 900 - Brazil





We have obtained the Love number and corresponding tidal deformabilites (Λ) associated with 
the relativistic mean-field parametrizations shown to be consistent (CRMF) with the nuclear matter, 
pure neutron matter, symmetry energy and its derivatives [Dutra et al., Phys. Rev. C 90, 
055203 (2014)]. Our results show that CRMF models present very good agreement with the recent data 
from binary neutron star merger event GW170817. They also confirm the strong correlation 
between Λ_1.4 and the radius of canonical stars (R_1.4). When a recently GW170817 
constraint on Λ_1.4 and the corresponding radius R_1.4 is used, the majority of the 
models tested are shown to satisfy it.


21.30.Fe, 21.65.Cd, 26.60.Kp, 24.10.Jv

Consistent relativistic mean field models constrained by GW170817
    Débora P. Menezes
    December 30, 2023
=================================================================





§ INTRODUCTION


The discovery of the first binary pulsar PSR1913+16 by Russel Hulse and Joseph
Taylor in 1974 <cit.> with its very stable and precise pulse period, and the 
observation in 1978  that its orbit period was declining with time <cit.>,
opened a clear possibility  for the detection of gravitational
waves (GW). A probable explanation for the change in the period was the
loss of energy by the binary system in the form of GW. The detection
of these waves was expected since then until in
2015 the first signal was clearly seen (GW150914) and shown to be produced by
two colliding black holes <cit.>.  Finally, in 2017, LIGO and
Virgo made the first detection of GW170817 produced by colliding neutron
stars <cit.> and the event was observed also as light in the optical, UV, IR,
X-ray and γ-ray emissions <cit.>, what was then
called a multi-messenger observation. 

When one of the neutron stars in a binary system gets close to its companion just before
merging, a mass quadrupole develops as a response
to the tidal field induced by the companion. This is known as tidal
deformability <cit.> and can be used to constrain neutron star
macroscopic properties <cit.>, which
in turn, are obtained from appropriate equations of state (EOS). A nice and
simple review on the basic ingredients necessary to construct an EOS
is given in <cit.>.

If one searchs the literature for an equation of state, hundreds of
models are found. Not too long ago, 263 relativistic mean-field (RMF) 
parametrizations were analyzed in <cit.> and confronted
with different sets of constraints, all of them 
related to symmetric nuclear matter, pure neutron matter, symmetry energy and its 
derivatives. The different sets differ from one another in the choice of validity ranges of certain 
quantities and in the level of restriction.  Only a small number of parametrizations of 
these models (35) were shown to satisfy adequately the chosen
constraints. This fact reinforces the idea that the proliferation of
models and the production of new parameter sets with a limited range 
of application should not be encouraged.

In <cit.>, the relativistic models were divided into 7 families, namely, 
linear finite range models (Walecka-type models <cit.>, type 1), nonlinear 
σ models (Boguta-Bodmer models <cit.>, type 2), nonlinear σ and 
ω models with a self-quartic interaction in the ω field (type 3),
nonlinear σ and ω terms and cross terms involving these fields (type 4), 
density dependent models <cit.> with couplings adjusted to nuclear properties 
(type 5), nonlinear point coupling models <cit.> (type 6) and models with 
δ mesons (type 7). Thirty of the approved models are of type 4, two are of 
type 5, one of type 6, and two of type 7 being both density dependent.

Later on, 34 RMF models that were shown to satisfy several nuclear matter constraints 
in <cit.>, namely, the same previous 35 models excluding the point-coupling 
one as discussed in <cit.>, were confronted with astrophysical constraints 
<cit.>. The more important of these constraints are the
neutron stars with maximum mass in the range 
of  <cit.>, 
but the direct Urca process and the sound velocity also give hints on
the star cooling mechanism and its internal matter distribution.  
From the 34 analyzed models with nucleonic matter included, only 15
can sustain massive stars and none if hyperons are 
included, a result that accounts for the famous hyperon puzzle. Once hyperons are included 
in the calculations, the situation becomes more complicated because the EOS must be soft 
at subsaturation densities and hard at higher densities to predict massive stars, but 
hyperons soften the EOS. A possibility that reconciles the measurements of massive stars
with canonical stars with small radii present in the same family (another recently 
imposed constraint), is either the inclusion of strange mesons or of a new degree of 
freedom (not necessarily  known) in the calculations <cit.>. 

The measurements and analyses of data from this specific
gravitational wave established limits both on the dimensionless tidal
deformability of the binary system Λ̃ and on the
tidal deformability of the canonical  star Λ_1.4 as
being ⩽ 800 for the low spin priors upper boundary <cit.> and 
contributed to the exclusion of very stiff EOS that would give rise to values larger than 
800. A lower limit was estimated  as Λ̃> 400 <cit.>. Recently, LIGO 
and Virgo collaboration updated the Λ_1.4 values to be constrained to the range 
of 70⩽Λ_1.4⩽580 <cit.>. Moreover, the chirp mass, which 
relates the masses of both NS in the binary system  was observed to be  M=1.188 
M_⊙ <cit.>.
The above mentioned boundaries combined with the chirp mass can be used to calculate
the bounds on the tidal deformability of the individual neutron stars
in the binary system <cit.>.

Since the detection of GW170817, several studies were dedicated to
look for correlations and sensitivity of important nuclear bulk
properties, i.e., the symmetry energy, its slope, compressibility and values of the
tidal deformability for the canonical 1.4M_⊙ and other
slightly less and slightly more massive stars. In <cit.>
the authors analyzed 4 Skyrme-type models and 1 obtained from a
density functional theory; in <cit.> 18 relativistic and 24
nonrelativistic models were analyzed; in <cit.> many Skyrme
type models were investigated and in <cit.> 67 RMF models
were considered.  In the 3 last works, different correlations were
found between, for instance, Λ and R,
Λ_1.4 and R_1.4, Λ_1.4 and M_max or
Λ_1.4 and Λ̃.

In <cit.>, a parametrized EOS was built at high-densities and one Skyrme EOS at
low densities and was confronted with GW170817 tidal deformability
information to obtain NS radii. The suggested values for the two
neutron stars in the binary system lie in the range
 and . If a further
    restriction is imposed to account for EOS that support massive
    stars, both radii are constrained to the range of 11.9 ± 1.4 km. All models
analyzed in <cit.> with a maximum mass of (1.97 ± 0.04)M_⊙ or larger 
bear radii within the proposed range. In
<cit.>, the authors established the upper limit on the
canonical stars radii as R_1.4⩽ 12.9 km. In <cit.>,
only 7 models are excluded by this constraint.

Also, in a recent paper <cit.>, the authors show that an
infinite number of combinations of EOS with large slopes and small
compressibilities or small slopes and large compressibilities can lead
to the same Λ_1.4 and R_1.4, pointing to the need of
more observables so that the density dependence of the symmetry energy
be completely determined.

In all above mentioned papers, the models used to test constraints and
to look for correlations were randomly chosen. However, in
the present work we follow a more direct line of work by choosing
models we have already tested previously based on exactly the same constraints, 
avoiding models either with flamboyant degrees of
freedom or that have been forcefully corrected with extra mixed meson interactions
and return to the more conventional 34 RMF parametrizations that were
shown to satisfy the nuclear matter constraints
in <cit.>  to confront them with tidal deformabilities
 inferred from GW170817. 



§ RESULTS AND DISCUSSION


In a binary neutron star system, the tidal deformability is the
measurement of the perturbation generated by  the quadrupole moment in
one star as a response to the external field created by its 
companion. From the mathematical point of view, the dimensionless
tidal deformability, in terms of the Love number k_2, is given by

    Λ= 2k_2/3C^5,

where C=m/R is the compactness of the neutron star of mass m. The Love number k_2 is 
calculated by the following expression,  

    k_2 =8C^5/5(1-2C)^2[2+C(y_R-1)-y_R]×
       ×{2C [6-3y_R+3C(5y_R-8)]
    
       +4C^3[13-11y_R+C(3y_R-2) + 2C^2(1+y_R)]
    
       +3(1-2C^2)[2-y_R+2C(y_R-1)] ln(1-2C)}^-1,

where y_R≡ y(r) is found from the solution of

    r dy(r)/dr + y(r)^2 + y(r) F(r) + r^2Q(r)=0,

with

    F(r) = r - 4π r^3[ ϵ(r) - p(r) ]/r - 2M(r)

and

    Q(r)   =4π r[5ϵ(r) + 9p(r) + ϵ(r)+p(r)/∂ 
    p(r)/∂ϵ(r)-6/4π r^2]/r - 2M(r)
       - 4[ M(r)+4π r^3 p(r)/r^2(1-2M(r)/r)]^2.

The set of Eqs. (1)-(5) can also be found in Refs. <cit.>, 
for instance, in which earlier new developments were performed by using a large number of EOS's used 
to calculate k_2 and Λ.

Actually, Eq. (<ref>) must be solved together with the well known TOV 
equations <cit.>, 
in which ϵ and p are the energy density and pressure, respectively, given as 
input. In our case, these quantities are given by the CRMF parametrizations with protons, 
neutrons, electrons and muons with the charge neutrality and β-equilibrium 
conditions together with the Baym-Pethick-Sutherland (BPS) equation 
of state <cit.> in the low density regime, namely, 
0.1581× 10^-10 fm^-3⩽ρ⩽ 0.008 907 fm^-3. The initial 
condition for Eq. (<ref>) is y(0) = 2 (related to the Love number order) and M(r) 
is the neutron star mass enclosed within the radius r. At the surface of the star, in 
which r=R, one has M(R)=m. For detailed discussions on such calculations, we address 
the reader to Refs. <cit.>, for instance.

 The compactness of one recently measured  isolated neutron star <cit.> is equal 
 to 0.105 ± 0.002. Notice that the GW170817 constrains NS in a binary system and 
hence, more measurements are necessary before this value is used as a constraint. 
Nevertheless, in Table <ref>, we show the compactness for the cases of 
m=m_ (C_), m=1.4M_⊙ (C_1.4), and 3 more 
cases obtained from the limits of possible masses in the binary system. The mass-radius 
diagrams used to calculate these C values for the CRMF parametrizations are found in 
Ref. <cit.>.


Models belonging to the same families present very similar compactness both for the 
maximum mass star and for the canonical one. In the low limit mass case, both stars of the binary 
system present practically the same mass, m_1≈ m_2≈ 1.37M_⊙. This is the 
reason they present the same compactness. However, this is  no longer true in the other 
cases, when one of the star is always more compact than its companion.

In Fig. <ref> we display the Love number k_2 for the CRMF parametrizations, obtained through 
the definition given in Eq. (<ref>) with y_R calculated from the solution of Eq. (<ref>) 
coupled to the TOV equations.

The pattern exhibited by k_2 is similar to that found in calculations involving other relativistic hadronic model, as 
one can verify in Ref. <cit.>, for instance. It is important to point out that k_2 is very sensitive to the description of the crust of the star.
Once k_2 is calculated, it is possible to analyze the 
dimensionless tidal deformability by using the definition presented in  Eq. (<ref>).

In Fig. <ref> we display a diagram of the dimensionless tidal deformabilities of 
each star in the binary system. Λ_1 is associated to the neutron star with mass 
m_1 
which corresponds to the integration of every EOS in the range 1.37⩽ 
m/M_⊙⩽ 1.60 obtained  from GW170817. On the other hand, the mass m_2 of the 
companion star is determined by solving the chirp mass ℳ = (m_1 
m_2)^3/5/(m_1 + m_2 )^1/5, whose value is 1.188M_⊙, as determined in 
Ref. <cit.>. 

One can see that all the investigated models lie in between the confidence
lines, what corroborates the fact that the previously constrained models to satisfy 
nuclear bulk properties are reliable to investigate neutron stars in binary systems, although many 
of them do not describe massive stars, as explained in the introduction of this letter. 

As already shown in Refs. <cit.>, we have also found a strong correlation 
between the tidal deformability of the canonical star and its radius both in linear and log scale 
(not shown), namely, Λ_1.4=2.65×10^-5R_1.4^6.58, as can be seen in 
Fig. <ref>.

Since the second Love number k_2 depends on R, for a given EOS, through a nontrivial differential equation coupled 
to the TOV one, Λ_1.4 as a function of R_1.4 is not simply given by Λ_1.4∝ R_1.4^5 as 
Eq. (<ref>) suggests.

When we analyze the constraint for Λ_1.4 in the range 
, as proposed in <cit.>, with 
corresponding R_1.4 values, depicted in Fig. <ref> by the shaded square, we 
observe that 24 parametrizations (out of 34) are in accordance with this proposition. 
They are: BSR15, BSR16, BSR17, BSR18, BSR19, BSR20, , , 
FSUGold, , FSUGZ06,  G2^*, , Z271s2, Z271s3, Z271s4, Z271s5, 
Z271s6, Z271v4, Z271v5, Z271v6, , TW99, and DD-MEδ. It is interesting to 
notice that not all models capable of describing massive stars in the range 
 <cit.> 
discussed in <cit.> lie inside the box with values obtained from GW170817, 
and not all the 24 models inside the gray area can describe massive stars. Only 5 models 
satisfy both constraints, namely, G2^*, , DD-F, TW99, and DD-MEδ. 

Although the current range for Λ_1.4 is not very restrictive, we remind the reader that 
its values were still more imprecise, as one can verify in Ref. <cit.> in which the range 
was computed as Λ_1.4⩽ 800. Furthermore, there is a huge number of hadronic 
parameterizations coming from relativistic and non-relativistic models, around 500 if we take into 
account only those from RMF and Skyrme models. Thus, it is important to find which particular set 
of parameterizations among these huge number is able to describe simultaneously different nuclear 
environments. In that sense, a constraint coming from the analysis of the recent GW170817, even being not 
so restrictive (70⩽Λ_1.4⩽ 580) can be useful for this purpose.




§ FINAL REMARKS


In the present work, we have revisited 34 relativistic mean-field parametrizations 
shown to be consistent (CRMF) with the  nuclear matter, pure neutron matter, symmetry 
energy and its derivatives in <cit.> and used them to compute the Love number 
and corresponding tidal deformabilities. We have checked that all analyzed models lie in 
between the confidence lines in the plot Λ_2 versus Λ_1. They also confirm 
previously obtained correlation between the tidal deformability and the radius of 
canonical stars. Once we use the GW170817 constraints on the tidal deformabilities to 
identify the corresponding neutron star radii range, as proposed in  <cit.>, 24 
parametrizations are shown to satisfy them.  As far as the compactness, an important 
ingredient in the calculation of the Love numbers, is investigated, we have seen that, 
generally,  one of the star is always more compact than its companion, except in the low 
limit mass case m_1=1.37, when both stars in the binary system present the same 
compactness. 

It is also worth pointing out that only 5 parametrizations of the CRMF models, namely, 
G2^*, , , TW99, and , can simultaneously describe 
massive stars in the range  <cit.>, as shown in <cit.>, and 
constraints from GW170817. 

To further constrain the existing EOS or confirm the results obtained so far, we look 
forward to the next detections of gravitational waves.




§ ACKNOWLEDGMENTS
 
This work is a part of the project INCT-FNA Proc. No. 464898/2014-5,
was partially supported by CNPq (Brazil) under grants 301155.2017-8
(DPM) and 310242/2017-7 (OL), by Capes-PNPD program (CVF), and by Fundação de 
Amparo à Pesquisa do Estado de São Paulo (FAPESP) under the thematic project 
2013/26258-4 (OL,MD,CL).


99

achar1 R. A. Hulse and J. H. Taylor, Astrophys. J. 191 L59 (1974); 195 
L51 (1975).

achar2 J. H. Taylor, L. A. Fowler and P. M. McCulloch, Nature 277 437 
(1979).

achar3  B. P. Abbott et al. (LIGO Scientific Collaboration and
Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016); 116, 241103 
(2016); Phys. Rev. X 6, 041015 (2016).

PRL119 B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo
Collaboration), Phys. Rev. Lett. 119, 161101 (2017).


AJL848 P. S. Cowperthwaite et al. (Ligo Scientific Collaboration and Virgo
  Collaboration), Astrophys. Jour. Lett. 848: L12 (2017).

Damour86 T. Damour, M. Soffel and C.-M. Xu, Phys. Rev. D  45, 1017 (1992).

Flanagan87 E. E. Flanagan and T. Hinderer,  Phys. Rev. D  77, 021502 (2008).

Piekarewicz2018 F. J. Fattoyev, J. Piekarewicz and C. J. Horowitz,  Phys. Rev. Lett.  120, 172702 (2018).

Providencia2018 Tuhin Malik et al.,  Phys. Rev. C  98, 035804 (2018).

Hadrons2018  J. Piekarewicz, Proceedings of the XIV International Workshop on Hadron Physics, Florianópolis, 18-23 March 2018 pages 12-37, arXiv:1805.04780.  

PRC90-055203 M. Dutra, O. Lourenço, S. S. Avancini, B. V. Carlson, A. 
Delfino, D. P. Menezes, C. Providência, S. Typel, and J. R. Stone, Phys. Rev. C 90, 055203 (2014).

walecka J. D. Walecka, Ann. Phys. 83, 491 (1974).

boguta  J. Boguta and A. R. Bodmer, Nucl. Phys. A 292, 413 (1977).
  
NPA656-331 S. Typel and H. H. Wolter, Nucl. Phys. A 656, 331 (1999).

PRC46-1757 B. A. Nikolaus, T. Hoch, and D. G. Madland, Phys. Rev. C 46, 
1757 (1992); O. Lourenço, M. Dutra, A. Delfino, and R. L. P. G. Amaral, Int. J. Mod. 
Phys. E, 16, 3037 (2007).

PRC93-025806 Mariana Dutra, Odilon Lourenço and Débora Peres Menezes, Phys. Rev. C 93, 025806 (2016); 94, 049901(E) (2016).

nature467-2010 P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts, and J. W. T. Hessels, Nature 467, 1081 (2010).

science340-2013 J. Antoniadis, P. C. C. Freire, N. Wex et al., Science 340, 448 (2013).

Luiz2018 L. Lopes and D. Menezes, Journal of Cosmology and Astroparticle Physics  5, 038 (2018).
  
lower D. Radice, A. Perego, F. Zappa, and S. Bernuzzi, The Astrophysical Journal Letters  852, L29 (2018).  

PRL121 B. P. Abbott et al. (The LIGO Scientific Collaboration and the Virgo Collaboration), Phys. Rev. Lett.  121, 161101 (2018). 

constanca Tuhin Malik, N. Alam, M. Fortin, C. Providência, B. K. Agrawal, T. K. Jha, Bharat Kumar and S. K. Patra, Phys. Rev. C  98, 035804 (2018).
  
 1805.00219 Young-Min Kim, Yeunhwan Lim, Kyujin Kwak, Chang Ho Hyun and Chang-Hwan Lee, arXiv 1805.00219, to appear in Phys. Rev. C.

tsang M.B. Tsang, C.Y. Tsang, P. Danielewicz, and W.G. Lynch, arXiv: 1811.04888.

1809.07108 Rana Nandi, Prasanta Char and Subrata Pal, arXiv: 1809.07108.

bao-an Nai-Bo Zhang and Bao-An Li, J. Phys. G: Nuclear and Particle Physics  46, 014002 (2019).

new S. Postnikov, M. Prakash, and J. M. Lattimer, Phys. Rev. D 82, 024016 (2010).

tanija1 T. Hinderer, Astrophys. J. 677, 1216 (2008).

tanija2 T. Hinderer, B. D. Lackey, R. N. Lang, and J. S. Read, Phys. Rev. D 81, 123016 (2010).

binnington1 T. Binnington and E. Poisson, Phys. Rev. D 80, 084018 (2009).

damour1 T. Damour and A. Nagar, Phys. Rev. D 81, 084016 (2010).

tov R. C. Tolman, Phys. Rev. 55, 364 (1939); J. R. Oppenheimer and G. M. 
Volkoff, Phys. Rev. 55, 374 (1939).

bps N. K.  Glendenning, Compact Stars, 2nd ed. (Springer, New York, 2000); G. Baym, 
C. Pethick, and P. Sutherland, Astrophys. J. 170, 299 (1971).

bharat1 Bharat Kumar, S. K. Biswal, and S. K. Patra, Phys. Rev. C 95, 015801, (2017)

comp V. Hambaryan, V. Suleimanov, F. Haberl, A. D. Schwope, R. Neuhauser, M. 
Hohle, and K. Werner, Astronomy & Astrophysics 601, A108 (2017).




