OOn the Speed of Sound in Hyperonic Stars
T. F. Motta a (cid:63) , P. A. M. Guichon b , and A. W. Thomas aa CSSM and Department of Physics, University of Adelaide, SA 5005 Australia b IRFU-CEA, Universit´e Paris-Saclay, F91191 Gif sur Yvette, FranceSeptember 24, 2020
Abstract
We build upon the remarkable, model independent constraints on the equation of state of densebaryonic matter established recently by Annala et al. [1]. Using the quark-meson coupling model,an approach to nuclear structure based upon the self-consistent adjustment of hadron structureto the local meson fields, we show that, once hyperons are allowed to appear in dense matter in β -equilibrium, the equation of state is consistent with those constraints. As a result, while onecannot rule out the occurence of quark matter in the cores of massive neutron stars, the availableconstraints are also compatible with the presence of hyperons. At the present time there is a great deal of interest in the existence of very heavy neutron stars (NS),with masses of order 2M (cid:12) or larger. Given the extreme conditions at the core of such stars, withdensities much greater than the density of nuclear matter ( n ), many authors have suggested thatthey may contain deconfined quark matter – see Refs. [2, 3, 4, 5] for recent reviews.Amongst the most important achievements in experimentally constraining the equation of state(EoS) of neutron stars was the measurement of the mass of a pulsar equal to 1.97 ± . (cid:12) [6], whichwas later revised to 1.908 ± . (cid:12) [7]. This discovery and the later measurement by Antoniadis etal. [8] and Cromartie et al. [9] provided important new constraints. One, of an instrumental nature,was that models that do not lead to stars with masses at least as large as 1.9M (cid:12) are incorrect andshould be abandoned. Another, more of an epistemic nature, was based on the following puzzle. Astar this massive must have an innermost core that is incredibly dense; most models point, at the veryleast, to 5 times nuclear matter density. However, at such densities the argument for a fundamentalchange in the nature of matter becomes compelling. Indeed, it is possible that neutrons and protonsmight dissolve and the relevant degrees of freedom become quarks and gluons; that is, the core mightconsist of quark matter (QM).An alternative, to which rather less attention has been paid, is that perhaps the Fermi momentumand chemical potential of the nucleons are so high that hyperons can be created, by which point,because of the Pauli blocking, they would be stable (see e.g. [10, 11, 12, 13, 14]). This possibilityshould also be of enormous interest as strangeness is now considered to be one of the new frontiers ofnuclear physics [15, 16, 17]. While rare ion facilities push the boundaries of the known atomic nucleiin N and Z , strangeness may be thought of as a third axis along which much less is known. In thestrangeness minus one ( S = −
1) sector quite a few Λ-hypernuclei have been measured and we knowthat almost no Σ-hypernuclei are bound [18, 19]. However, when it comes to the S = − (cid:12) , which was far too low. Only with the inclusion of three-bodyforces [21], was it found to be possible to generate NS at 2M (cid:12) with hyperons.In a stimulating recent study, Annala et al. [1] used a novel interpolation method based uponthe speed of sound in-medium, c s , to generate a large, model independent set of equations of state1 a r X i v : . [ nu c l - t h ] S e p EoS). These EoS were required to be consistent with the existence of NS up to 1.97 M (cid:12) , as wellas the constraints on tidal deformability deduced from the neutron star merger data obtained fromGW170817 [22]. By identifying regions within massive NS with γ , the polytropic index, below 1.75(conversely c s < ) and realizing that these were inconsistent with an EoS involving only nucleons, itwas deduced that the existence of quark matter cores in massive NS should be considered the standardscenario.Here we explore the consequences for the speed of sound and the polytropic index of includinghyperons and imposing β -equilibrium on the dense matter in the core of massive NS. As noted by anumber of authors, starting with Stone et al. [21], this naturally implies that NS with masses above ∼ (cid:12) must contain hyperons. We find that the softening of the EoS as each new species of hyperonappears lowers the speed of sound below and the polytropic index well below 1.75. Thus, whileone cannot rule out the appearance of quark matter in the cores of such massive stars, the modelindependent analysis of Annala et al. and the current observational constraints on NS properties areboth also consistent with the appearance of hyperons.The structure of this work is as follows: in the next section we review the claims and methodologyof Ref. [1]; in section 2.2 we briefly review the basic assumptions of the QMC model, and finally, insection 3, we compare the two results before proceeding to the conclusions. In Ref. [1] the authors aimed to provide a model independent analysis of the relation between theshape of the EoS of dense matter and the experimental constraints on NS properties. They randomlygenerated EoS that were selected to respect key high and low density limits, namely the perturba-tive, conformal limit of QCD at high density and the low density limit given by chiral effective fieldtheory (CET). Acceptable EoSs within this set were then required to satisfy important astrophysicalconstraints, such as an upper limit on the maximum mass of a NS of at least 1.97M (cid:12) and the tidaldeformability limits from GW170817 [22].The novel, model independent method that was employed to generate such equations of state wasbased upon multiple different interpolations which were compared with each other, namely1. a piecewise polytropic form for the pressure, p i = k i n Γ i ,2. an interpolation of the adiabatic index itself, Γ( p ), through Chebyshev polynomials,3. a piece-wise interpolation of the speed of sound in terms of linear functions of the baryon chemicalpotential, c s ( µ ).The EOS generated in this manner were studied and categorised via the following methodology.A polytropic index, γ = d ln p/d ln ε , within a certain range, namely γ > γ < .
75 was taken todefine a region of quark matter.On the basis of the large family of EoS generated in this model independent way, Annala et al. [1]found that, although stars of the hadronic type, according to the categorisation discussed above, doreproduce well the properties of stars with moderate mass (roughly 1 . (cid:12) ), the highest mass starsare reproduced overwhelmingly by EOS that show evidence of quark matter. For this reason theyconcluded that one should consider the existence of quark matter in the cores of massive NS as thestandard scenario. The quark-meson coupling (QMC) model builds a description of nuclear matter and finite nuclei basedupon the self-consistent modification of the structure of the bound baryons in the relativistic meanfields generated by meson coupling to the confined quarks [23, 24, 25, 26, 27]. The internal structureof the baryons is usually described by the MIT Bag model [28], although the NJL model has also been2mployed [29, 30]. The quarks confined inside their bag interact with quarks in neighbouring baryonsvia the exchange of meson fields. We include one-gluon-exchange (OGE) between the quarks in thebag and obtain an expression for the baryon effective mass M (cid:63)N = Ω u N u + Ω d N d + Ω s N s − z R B + B V B + ∆ E M (1)where Ω q /R B is the energy eigenvalue for the quark q and the OGE hyperfine color interaction con-tribution is ∆ E M . Both of these depend non-linearly on the scalar meson mean field, ¯ σ [31]. Thebag pressure, zero point fluctuations z and strength of the gluon exchange, given in terms of thestrong coupling α c , are all fitted to reproduce the masses of the entire baryon octet in free space. Theresulting expression can be fitted to an analytic form such as M (cid:63)B (¯ σ ) = M B − g σB (¯ σ )¯ σ = M B − g σB ¯ σ + d B g σB ¯ σ ) , (2)where d B is the scalar polarizability of baryon B .The procedure just outlined yields a density dependent mass for every baryon in the octet withoutadding any new parameters. This density dependence is equivalent to the inclusion of many-bodyforces, as the interaction between a pair of baryons is modified by the presence of others [32, 33].The most important effects are related to the effective three-body forces, which are repulsive, dependon the particular baryons and involve no new parameters. The automatic generation of these forceswas the reason that the QMC model was able to predict NS with masses of order 2 M (cid:12) , even whenhyperons were included, before any such stars had been found [21].The free space baryon-meson couplings are fitted to reproduce standard nuclear matter parame-ters (as Ref. [34]), namely, the saturation density, symmetry energy and binding energy per nucleon,respectively n = 0 . − , (3) a s = 30MeV , (4) E = − . . (5)Taking the vector meson masses to have their experimental values [35], there remain just two parame-ters to be fixed, namely R B and the mass of the sigma meson, m σ . We choose these to be, respectively,0 . m σ = 700MeV.The detailed expressions for the energy density are shown in the Supplementary Material. For theintents and purposes of this discussion, suffice it to say that in minimising that energy density withrespect to β -equilibrium conditions we do find that hyperons must be present in the cores of neutronstars with masses above 1.8M (cid:12) . Figure 1 shows the fraction of each species as a function of the baryondensity, noting that the largest mass stars in our model have core densities of roughly 1fm − . As shown in Fig 2, the EoS in the QMC model exhibits a sudden decrease in the speed of sound as thedensity increases above 0.5 fm − . That decrease, which in other models is indeed characteristic of theintroduction of a QM phase, here is a natural consequence of passing the threshold at which hyperonsstart to be generated. As can be seen by comparing Figs. 1 and 2, the fairly dramatic decreases inthe speed of sound coincide with the appearance of first the Λ, then the Ξ − and subsequently the Ξ .At each threshold the appearance of the hyperon softens the EoS, because at threshold one is addingenergy but little momentum, and this in turn reduces the speed of sound and the adiabatic index.As can be seen in Fig. 3, the value of the adiabatic index also shows a sudden decrease whenthe density reaches the first hyperon threshold and again later with the introduction of the Ξ . It3 .0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0n b D e n s i t y F r a c t i o n npe Figure 1: Fraction of number density per baryon density of each baryon species as a function of thetotal baryon density.is especially important to note that the natural appearance of hyperons in the QMC model leads tovalues of the polytropic index, γ , well below 1 . . However, as we have emphasised, the QMC model EoScontains no quark matter but rather hyperons. The model independent constraints on the EoS of dense matter deduced in Ref. [1] are of great interest.It is remarkable that the kink in the plot of pressure versus energy density, at a baryon density oforder 0.5fm − , coincides with that found when hyperons are included. This is particularly interestinggiven that the appearance of hyperons played no role in the work of Ref. [1]. Rather, the connectionto perturbative QCD at very large densities was key to the appearance of the kink in that work.While acknowledging the power of the techniques employed by Annala et al. [1], we have presentedevidence for a different interpretation. Using the QMC model, which not only produces a very gooddescription of the properties of finite nuclei across the periodic table but is consistent with all obser-vational constraints relating to NS, we find that the kink in the EoS and the consequent reduction inthe speed of sound and the polytropic index, γ , is associated with the appearance of hyperons in thedense matter which is in β -equilibrium.In the present work the polytropic index does indeed fall below 1.75 for energy densities above600 MeV/fm . However, this is a signal of the appearance of hyperons rather than quark matter. Ofcourse, the current calculations cannot exclude the possibility that stars with masses beyond about1.8M (cid:12) might contain quark matter. Nevertheless, one equally cannot exclude that the cores of suchstars consist instead of a significant fraction of hyperons. Clearly, it is a fascinating challenge for futurework to find new ways to test which of these two possibilities is chosen in Nature. Note that, in Ref. [1], the low density side of the equation of state consists of neutron matter. For consistency, weshow here our equation of state of infinite nuclear matter in beta equilibrium at all densities, and are not replacing thelow density region with a realistic crust EoS.
100 200 300p(MeV·fm )0.00.10.20.30.4 c ² b (fm ) Figure 2: The speed of sound squared, c s , is illustrated as a function of pressure and baryon numberdensity. )1.01.52.02.53.0 0.00 0.25 0.50 0.75 1.00n b (fm ) Figure 3: We show the polytropic index, γ = d (ln P ) /d (ln (cid:15) ), as a function of pressure and baryonnumber density. 5 (MeV·fm )10 p ( M e V · f m ) c ²<1c ²<.8c ²<.6c ²<.4c ²<.33QMC model Figure 4: Comparison of the EoS in the QMC model, including hyperons as required by β -equilibrium,with the model independent constraints found in Ref. [1] Acknowledgement
This work was supported by the University of Adelaide and by the Australian Research Council throughDiscovery Project DP180100497.
Author Contributions
All authors participated in editing the manuscript as well as in the derivation of the current versionof the model. The numerical calculations were implemented by T.M.
Competing Interests statement
The authors declare no competing interests.
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