Isovector Effects in Neutron Stars, Radii and the GW170817 Constraint
T. F. Motta, A. M. Kalaitzis, S. Antić, P. A. M. Guichon, J. R. Stone, A. W. Thomas
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Isovector Effects in Neutron Stars, Radii and the GW170817 Constraint
T. F. Motta, A. M. Kalaitzis, S. Anti´c, P. A. M. Guichon, J. R. Stone,
3, 4 and A. W. Thomas CSSM and ARC Centre of Excellence for Particle Physics at the Terascale,Department of Physics, University of Adelaide SA 5005 Australia IRFU-CEA, Universit´e Paris-Saclay, F91191 Gif sur Yvette, France Department of Physics (Astro), University of Oxford, OX1 3RH United Kingdom Department of Physics and Astronomy, University of Tennessee, TN 37996 USA
Submitted to ApJABSTRACTAn isovector-scalar meson is incorporated self-consistently into the quark-meson coupling descriptionof nuclear matter and its most prominent effects on the structure of neutron stars are investigated.The recent measurement of GW170817 is used to constrain the strength of the isovector-scalar channel.With the imminent measurements of the neutron star radii in the NICER mission it is particularlynotable that the inclusion of the isovector-scalar force has a significant impact. Indeed, the effect of thisinteraction on the neutron star radii and masses is larger than the uncertainties introduced by variationsin the parameters of symmetric nuclear matter at saturation, namely the density, binding energy pernucleon and the symmetry energy. In addition, since the analysis of GW170817 has provided constraintson the binary tidal deformability of merging neutron stars, the predictions for this parameter withinthe quark-meson coupling model are explored, as well as the moment of inertia and the quadrupolemoment of slowly rotating neutron stars.
Keywords:
Quark Meson Coupling, GW170817, Neutron Stars, Tidal Deformability, Nuclear Matter INTRODUCTIONAs the repositories of the densest nuclear matter inthe Universe, neutron stars (NS) have long been a focusof nuclear theory studies (Glendenning; Weber (1999);Haensel et al. (2007); Lattimer (2012, 2014); Baym et al.(2018)). The interest in these intriguing objects hasintensified since the LIGO and Virgo observation ofGW170817 (Abbott et al. (2017)), an event identified asalmost certainly the merger of two NS. The GW170817analysis has already demonstrated the potential of grav-itational wave (GW) observations to yield new informa-tion, such as the limits on NS tidal deformability, whichhas not hitherto been accessible.Before the observation by GW170817 but followingthe discovery of particularly heavy NS with massesaround 2 M (cid:12) (Demorest et al. (2010); Antoniadis et al.(2013)), the astrophysics community was already mak-ing great efforts to model nuclear matter equations ofstate (EOS) capable of producing such heavy objects(see for example (Oertel et al. (2017)). This was espe-cially challenging since such heavy stars are expected tobe so dense that, under the constraint of β -equilibrium, they must contain hyperons. The EOS including hy-perons are typically much softer than those containingjust nucleons, leading to maximum masses which arenot compatible with observations. There are several ap-proaches used to solve this issue, such as the inclusionof repulsive three-body forces involving hyperons or theintroduction of a possible phase transition to deconfinedquark matter at densities below the hyperon threshold(for more details see Vida˜na et al. (Vidana (2015)) andreferences therein).Within the quark-meson coupling model (QMC) (Gui-chon (1988); Guichon et al. (1996); Saito et al. (2007);Guichon et al. (2018)) NS with masses of order 1 . M (cid:12) are produced with or without the inclusion of hyperonsin the EOS (Rikovska-Stone et al. (2007); Stone et al.(2010); Whittenbury et al. (2014, 2016)). This remark-able feature, first published three years before the firstheavy NS measurement (Demorest et al. (2010)), is aconsequence of the fact that the self-consistent modi-fication of the internal quark structure of hadrons in-medium naturally leads to repulsive three-body forcesbetween all baryons – see Refs. Guichon and Thomas(2004); Guichon et al. (2018). Furthermore, these forces a r X i v : . [ nu c l - t h ] A p r involve no new parameters but are a direct consequenceof the underlying quark structure, through the so-calledscalar polarizabilities. Here, we extend the model pre-sented in Ref. Rikovska-Stone et al. (2007) by adding theexchange of the isovector-scalar meson a (980), labelled δ , in addition to the original isoscalar-scalar σ , isoscalar-vector ω and isovector-vector ρ mesons and the pion, andinvestigate the effect of this modification on predictionsof NS properties. We note that earlier calculations withthe QMC model (without the Fock terms) have includedsuch an interaction (see Santos et al. (2009)) to studylow density instabilities in asymmetric nuclear matter,while Wang et al. (2011) used the QMC model withthe isovector-scalar channel included in their Hartree-Fock calculation of properties of finite nuclei. This workextends, for the first time, the QMC model with theisovector-scalar exchange and the full Fock terms, topredict properties of high density matter in NS.While NS masses can be determined reasonably ac-curately, the situation regarding their size is much lesssatisfactory. A number of techniques have been usedto put broad limits on their radii, with values typicallyin the range 6 − km (Ozel and Freire (2016)). Acomprehensive recent update can be found in the workof Steiner et al. (Steiner et al. (2018)) and referencestherein. These limits do not constrain the EOS stronglyenough and many different models of dense matter arestill allowed. However, a great deal of anticipation sur-rounds the NICER experiment (Gendreau et al. (2016)),which aims to measure the radii of NS with an accuracyof order 5% (Watts et al. (2016)) In anticipation of re-sults from NICER as well as future observations of NSmergers, with GW170817 yielding its own constraint,our aim is to investigate to what extent the QMC modelwith the isovector-scalar interaction can provide reliablepredictions for the radii of NS, with realistic error esti-mates. As the first positive sign we will show that thecomparison of the QMC model predictions with the re-cent gravitational wave measurement already suggestsan upper bound on the strength of the isovector-scalarcoupling in this model.The isovector-scalar exchange has been extensivelyexplored in relativistic-mean-field models in the Hartreeapproximation ( see, for example, Refs.(Kubis andKutschera (1997); Liu et al. (2002); Menezes and Prov-idencia (2004); Roca-Maza et al. (2011); Singh et al.(2014))). The isovector-isoscalar meson δ has also beenincluded in the chiral mean field model (CMF) Beck-mann et al. (2002). The Dirac-Brueckner-Hartree-Fockmodel (Sammarruca et al. (2012)), including six non-strange bosons with masses below 1GeV, π, η, ρ, ω, σ and δ , was also applied to nuclear matter and NS and the re- sults were compared with the outcome of chiral effectivefield theory. The main conclusions of these investiga-tions, relevant for nuclear matter, are that the effectsof the isovector-scalar channel are almost negligible insymmetric nuclear matter but significant in matter withhigh asymmetry and thus revelant for modelling neutronstars. Predictions for the EoS, the density dependenceof the symmetry energy and its slope, the split of theproton and neutron effective mass and the compositionof asymmetric matter, namely the proton fraction andthe hyperon thresholds have been reported as a func-tion of the isovector-scalar coupling strength, leading toestimates of its value.In section 2 we describe the model used in this work.Fixed and variable parameters of the model are summa-rized in section 3, followed by section 4 containing ourmain results. Conclusions can be found in section 5. THEORETICAL MODELThe QMC model (Guichon (1988); Saito and Thomas(1994); Guichon et al. (1996)) is based on the hypoth-esis that baryons, consisting of bags containing threeconfined, valence quarks, interact amongst themselvesby the exchange of mesons that couple directly to thenon-strange quarks. For vector mesons the mean fieldssimply shift the baryon energies, as in other relativisticmodels. On the other hand, a mean scalar field modi-fies the effective mass of the confined quarks, leading toa change in the valence quark wave functions. This inturn leads to the modification of the scalar field cou-pling to the hadron and hence, in order to find theeffective baryon mass in-medium, one must solve theentire problem self-consistently (Guichon et al. (1996);Rikovska-Stone et al. (2007)). There is immense inter-est in looking for evidence of these changes in baryonstructure (Cloet et al. (2016, 2009)). However, for thepresent purpose, the essential result of the model is thatthe effective baryon mass in-medium no longer has asimple linear dependence on the mean scalar field.Within the QMC model, the internal structure of anucleon, or in general a baryon, is described by the MITbag model (Chodos et al. (1974)), within which the ef-fective in-medium baryon mass with the flavour content N u , N d , N s is given as M (cid:63)B = Ω u N u + Ω d N d + Ω s N s R B − Z R B + ∆ E M + B V B . (1)Here Ω q is the quark’s lowest energy eigenvalue in thebag, taking into account the interaction with the meanscalar field, Z is the so called zero point parameter thatcorrects the energy for gluon fluctuations and centre ofmass effects, ∆ E M is the hyperfine colour interaction(DeGrand et al. (1975)), which is also modified by theapplied scalar field (Guichon et al. (2008)), while B isthe bag constant and V B the volume of the bag.The effective mass can be written as M (cid:63)B = M B − g σB ( σ, δ ) σ − g δB ( σ, δ ) τ · δ , (2)where the functions g σB ( σ, δ ) and g δB ( σ, δ ) are fittedto reproduce Eq. (1) for each baryon, as a function ofthe strength of the isoscalar ( σ ) and isovector ( δ ) scalarfields. Thus, while in practice g σ , g δ , g ω and g ρ (the cou-plings to the nucleon, B = N in free space) are treatedas parameters, these are directly related to the under-lying coupling constants of the mesons to the quarks.These, in turn, allow calculation of the couplings to allof the hyperons with no new parameters.In practice it is sufficient to make an expansion up toterms quadratic in the meson field. So we write M (cid:63) B ( σ, δ ) = M B − g σ σ − t B g δ δ + w σ B σ w δ B δ λ B σδ , (3)where t B and t B are respectively the isospin and theisospin projection of a baryon and w σ,δ B , λ B are weightparameters that we fit to reproduce the result of Eq. (1).The expression for the energy density, including the fullexchange Fock terms, derived in Ref. Rikovska-Stoneet al. (2007), which arise from single pion exchange, andthe meson mean field equations can be found in the Sup-plementary Material. These equations are solved self-consistently, minimizing the energy density subject tothe constraints of charge neutrality, β -equilibrium andbaryon number conservation. INPUT PARAMETERSThe QMC model has two fixed parameters, the bagradius, set to R B = 0.8 fm and the mass of the scalarmeson σ , m σ = 700 MeV, which is not well known exper-imentally. The masses of the pion, ω , ρ and δ and of thebaryon octet were taken from the experiment. The bagconstant, strange quark mass and the colour interactionstrength, α c are fitted within the model to reproducethe bare mass of the free nucleon and the Λ-hyperon, aswell as to fulfil the stability condition ∂ R B M (cid:63)B = 0.The variable parameters, the coupling constants G φ for the mesons, φ = ( σ, ω, ρ, δ ), to the nucleon are de-fined as G φ = g φ /m φ . These are obtained by fitting tothe empirical properties of symmetric nuclear matter,the saturation density, ρ , the binding energy per parti-cle, E , and the symmetry energy, S , at saturation. Thevalues ρ = 0 . − , E = − . S = 30MeVwere chosen as ‘standard’ in this work. In order to probethe effects of a change in nuclear matter parameters on the nuclear matter EOS and, consequently, the NS grav-itational mass and radius, six combinations of the pa-rameters, deviating from the standard values, were con-structed to form additional EOSs. Fits labeled ρ +0 , ρ − deviate by ± . f m − from the standard value of thesaturation density, as detailed in Table 1. We also allowvariations of the symmetry energy and binding energyof ± ρ ± as all results for other parametervariations were found to lie between the results for ρ ± fits.The range within which each parameter is varied de-fines the uncertainty band in ρ , S and E around theStandard EOS. Each set has been used to determine thethree couplings G σ , G ω and G ρ . The coupling G δ waskept constant equal to 3.0 fm during this procedure.However, when examining the isovector effects, G δ wasalso varied about the central value of 3 . (takenfrom the one-boson-exchange potential of Haidenbaueret al. (1992)). The couplings G σ , G ω and G ρ were alsoobtained for two other choices of the δ -nucleon coupling, G δ =0 and 6.0 fm (denoted 2 G δ ) and these are shownin the Supplementary Material. RESULTSBy varying the nuclear matter parameters over thespecified ranges we can define an uncertainty band re-garding the variation in ρ , S and E parameters aroundthe Standard EOS, as given in Fig. 1. In order to cal-culate the M-R curves shown in Figs. 2 and 3, the so-lution of the TOV equations using the QMC model to (MeV/fm³) p ( M e V / f m ³ ) Nucleons OnlyStandard Fit +00
Figure 1.
Illustration of the EOS for matter containing thefull baryon octet for the different sets of parameters chosenhere (see text and Table 1). A single “Nucleons Only” EOS,for which the parameters reproduce the standard fit nuclearmatter parameters, is also shown for comparison. Note thatthe up/down triangles refer directly to an increase/decreaseof the saturation density.
EOS NM parameters QMC couplings ρ S E G ω G ρ G σ G δ L K [fm − ] [MeV] [MeV] [fm ] [fm ] [fm ] [fm ] [MeV] [MeV] Std. Fit .
16 30 − . ρ +0 .
17 30 − . ρ − .
15 30 − . Table 1.
Nuclear matter parameters and coupling values for different EOSs determined by the values of ρ , S and E . Thevalue of K does not change significantly for different values of G δ , while the slope, L , for the cases with G δ = 0 and 6 fm we have, respectively, a decrease and an increase of 10 MeV.
11 12 13 14
R(km) M ( M ) Standard Fit Only
GW17081790% confidenceNo G2 G
Figure 2.
Mass vs Radius diagrams for different values ofthe δ coupling constant, calculated using only the StandardFit parameters for nuclear matter (see Table 1). All curvesinclude the effect of hyperons at sufficiently large density.The box shows the region preferred by the joint LIGO-Virgoanalysis Abbott et al. (2018), while the red band indicatesthe currently preferred mass of PSR J1614223 Arzoumanianet al. (2018).
11 12 13 14
R(km) M ( M ) G Figure 3.
Mass vs Radius diagrams for the variations ofnuclear matter parameters shown in Table 1 for the standard δ coupling strength. The symbols are as defined in Fig. 1. describe the NS core is matched to the low-density EOSthat accounts for the NS crust. In order to describethe crust we have used the EoS developed by Hempel etal. (Hempel et al. (2012); Hempel and Schaffner-Bielich(2010)), which was also recently used by Marques etal. (Marques et al. (2017)) to augment their EoS of aNS core based on the relativistic mean field model withthe density dependent interaction DD2Y, including thefull baryon octet. This choice is similar to that of theQMC model and we regard it as sufficiently realistic forthe purpose of this work. In particular, for stars of massabove 1.4 M (cid:12) the crust occupies only the outer 20% orless of the radial profile of the star.From Fig. 2 we see the fairly dramatic change in radiusfor a typical neutron star as G δ is varied. That this isconsiderably larger than the variation associated withthe choice of nuclear matter parameters may be seen inFig. 3.Next we examine the effect that the introductionof the δ -meson has on NS structure. The gravita-tional mass of the NS, obtained by solving the Tolman-Oppenheimer-Volkoff (TOV) equations (Oppenheimerand Volkoff (1939)) using the standard EoS, is shownin Fig. 2 as a function of the corresponding radius forthree different values of G δ . We observe that the δ cou-pling strength has a sizable effect on the radius, whilechanging the maximum mass by less than 0 . (cid:12) .It is especially interesting to observe that, for the stan-dard choice of nuclear matter parameters (Standard Fitin Table 1), the presence of the δ meson shifts the M-R curve toward larger radii. By centering our searcharound G δ = 3 . , one sees clearly that the valueof this coupling produces transverse shifts of the wholeM-R diagram that exceed the spread arising from varia-tions in the nuclear matter parameters, which is shownin Fig. 3. For NS with gravitational masses around 1 . (cid:12) , the inclusion of the isovector channel results in ra-dius changes of the order of 0 . . . . . . . . . M [M (cid:12) ] Λ No δG δ G δ Figure 4.
Tidal deformability as a function of the mass.The width of the curves illustrates the relatively smalldependence on the choice of nuclear matter parameters.The bar at M =1.4M (cid:12) shows the constraints derived inRef. Abbott et al. (2018). .
06 0 .
08 0 .
10 0 .
12 0 .
14 0 .
16 0 .
18 0 . M/R [M (cid:12) /km] . . . . . . . I / M R I / M R = ( . ± . ) ( + . β + . β ) Standard Fit ρ +0 ρ − Figure 5.
Moment of inertia of the star versus the ratio
M/R , compared with the region preferred in the analysisof Ref. Zhao and Lattimer (2018). per limit on the radius is consistent with other recentwork Raithel et al. (2018); Most et al. (2018); Annalaet al. (2018) and suggests that within the QMC modelthe strength of the isovector-scalar sector is most likelyto be G δ (cid:47) . .Next we demonstrate the effect of the isovector scalarchannel on two macroscopic properties of NS, the mo-ment of inertia, I , and the tidal deformability, Λ. Aswe see in Fig. 4, the variation of this parameter withthe strength of the δ coupling is relatively weak. Fur-thermore, we have verified the universality of the rela-tion between the moment of inertia, the Love numberand the quadrupole moment Hartle (1967); Hartle andThorne (1968), supporting the proposal that this is in-dependent of the EoS Yagi and Yunes (2013). Given that there is also interest in the moment of inertia cal-culated within the model, we show in Fig. 5 the resultsof the calculation using the slow-rotation Hartle-Thorneapproximation Hartle (1967) and an explicit comparisonwith the work carried out in Ref. Zhao and Lattimer(2018). It is clear that the moment of inertia calculatedin the present model is consistent with the constraintregion proposed there.Measurement of the moment of inertia of highly rel-ativistic double pulsar system such as PSR J0737-3039may be within reach after a few years of observation andyield a result with about 10% accuracy Steiner et al.(2013). Given that the masses of both stars in that sys-tem are already accurately determined by observations,a measurement of the moment of inertia of even one neu-tron star would enable accurate estimates of the radiusof the star and the pressure of matter in the vicinityof 1 to 2 times the nuclear saturation density. This,in turn, would provide strong constraints on the equa-tion of state of neutron stars and the physics of theirinteriors. Our calculations show little sensitivity to thevariation in nuclear matter parameters, except for theregion of compactness between 0.15 and 0.16 [M (cid:12) /km]where they differ slightly from the results reported inRef. (Zhao and Lattimer (2018)). In the compactnessregion between 0.10 - 0.15 they are in agreement of thelower bound of the Zhao and Lattimer analysis (Zhaoand Lattimer (2018)). We illustrate the contribution ofthe low density region to the moment of inertia in Fig. 6,showing that, for stars with masses greater than 1M (cid:12) ,the low density region contributes less than 10% of the . . . . . . . M [M (cid:12) ] . . . . . . . . ∆ I / I No δG δ G δ Figure 6.
Fraction of the total contribution to the momentof inertia coming from the crust. The colours indicate thevalue of G δ , while the width of each curve indicates the effectof the variation of nuclear matter parameters. Figure 7.
Binary tidal deformability ˜Λ, with the white areaabove the line M = M being excluded by the constraint80 < ˜Λ <
640 (Steiner et al. (2013); De et al. (2018)). total, going below 5% for masses ≥ . (cid:12) for all fitsand all values of the δ coupling.Using the signal from GW170817, a restriction wasplaced on the binary tidal deformability, ˜Λ, namely thatit should lie inside the window 80 < ˜Λ <
640 (Zhaoand Lattimer (2018)), while the masses of the stars inthe binary system were in the window (1 . < m < . × (1 . < m < . δ meson coupling G δ . Calculations of such quantitieswere performed following the work of Refs. Hartle andThorne (1968); Zhao and Lattimer (2018); Postnikovet al. (2010). Clearly the QMC model results overlapthe constraint region and constrain the NS masses andthe values for ˜Λ even further. The measurement doesnot rule out the model with double the delta coupling2 G δ but, as we can see in Fig. 2, it does tend to givevalues for the radius which are in some tension with thepreferred region. CONCLUSIONThe QMC model, where the quark structure of thebaryons adjusts self-consistently to the mean scalarfields generated in medium, has been extended to in-clude the isovector-scalar meson, δ . The self-consistentchange in structure leads to repulsive three-body forcesbetween all baryons (nucleons and hyperons), with-out additional parameters. Because of these forces themodel still (c.f. Ref. Rikovska-Stone et al. (2007)) yieldsmaximum neutron star masses of the order (cid:39) . (cid:12) .The major effect of the δ meson on NS properties isto increase their radii (c.f. Fig. 2), with the effect be-ing considerably larger than the effect of varying thesaturation properties of symmetric nuclear matter (c.f.Fig. 3).We have studied the moment of inertia ( I ) and tidaldeformability (Λ) of NS over a wide range of masses. Inthe case of the moment of inertia our results are consis-tent with the constraint suggested by Zhao and Lattimer(Zhao and Lattimer (2018)) on the variation of I/M R versus M/R . Our results also support the universal re-lation between I , the Love number and Λ, suggestingthat it is indeed independent of the EoS used.Following the neutron star merger observed recentlyby LIGO and Virgo, we also explored the dependenceof the binary tidal deformability, ˜Λ, on the parametersof the model as well as the masses of the stars involved.The results of our analysis tend to favour the larger ra-dius end of the constrained region in Fig. 2, while be-ing consistent with the binary tidal deformability con-straints reported in Refs. Abbott et al. (2018); Lattimerand Schutz (2005).ACKNOWLEDGEMENTSThis work was also supported in part by the Univer-sity of Adelaide and by the Australian Research Coun-cil through the ARC Centre of Excellence for ParticlePhysics at the Terascale (CE110001104) and DiscoveryProjects DP150103101 and DP180100497.REFERENCES B. Abbott et al. GW170817: Observation of GravitationalWaves from a Binary Neutron Star Inspiral.
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