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Breaking the EOS-gravity degeneracy with masses and pulsating frequencies of neutron stars

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2014 J. Phys. G: Nucl. Part. Phys. 41 075203

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Journal of Physics G: Nuclear and Particle Physics

J. Phys. G: Nucl. Part. Phys. 41 (2014) 075203 (11pp) doi:10.1088/0954-3899/41/7/075203

Breaking the EOS-gravity degeneracy withmasses and pulsating frequencies ofneutron stars

Weikang Lin1,6, Bao-An Li1, Lie-Wen Chen2,3, De-Hua Wen4

and Jun Xu5

1 Department of Physics and Astronomy, Texas A&M University-Commerce,Commerce, TX 75429-3011, USA2 Department of Physics and Astronomy and Shanghai Key Laboratory for ParticlePhysics and Cosmology, Shanghai Jiao Tong University, Shanghai 200240,People’s Republic of China3 Center of Theoretical Nuclear Physics, National Laboratory of Heavy IonAccelerator, Lanzhou 730000, People’s Republic of China4 Department of Physics, South China University of Technology, Guangzhou 510641,People’s Republic of China5 Shanghai Institute of Applied Physics, Chinese Academy of Sciences,Shanghai 201800, People’s Republic of China

E-mail: Bao-An.Li@Tamuc.edu

Received 21 December 2013, revised 22 February 2014Accepted for publication 20 March 2014Published 17 April 2014

AbstractA thorough understanding of many astrophysical phenomena associated withcompact objects requires reliable knowledge about both the equation of state(EOS) of super-dense nuclear matter and the theory of strong-field gravitysimultaneously because of the EOS-gravity degeneracy. Currently, variationsof the neutron star (NS) mass–radius correlation from using alternative gravitytheories are much larger than those from changing the NS matter EOS withinknown constraints. At least two independent observables are required tobreak the EOS-gravity degeneracy. Using model EOSs for hybrid stars anda Yukawa-type non-Newtonian gravity, we investigate both the mass–radiuscorrelation and pulsating frequencies of NSs. While the maximum mass of NSsincreases, the frequencies of the f , p1, p2, and wI pulsating modes are foundto decrease with the increasing strength of the Yukawa-type non-Newtoniangravity, providing a useful reference for future determination simultaneously

6 Present address: Department of Physics, The University of Texas at Dallas, Richardson, TX 75080, USA.

0954-3899/14/075203+11$33.00 © 2014 IOP Publishing Ltd Printed in the UK 1

J. Phys. G: Nucl. Part. Phys. 41 (2014) 075203 W Lin et al

of both the strong-field gravity and the supranuclear EOS by combining dataof x-ray and gravitational wave emissions of NSs.

Keywords: equation of state, dense matter, gravity, neutron stars

(Some figures may appear in colour only in the online journal)

1. Introduction

Neutron stars (NSs) are natural testing grounds of grand unification theories of fundamentalforces as their compositions are determined mainly by the weak and electromagnetic forcesthrough the β equilibrium and charge neutrality conditions while their mechanical stabilityis maintained by the balance between the strong and gravitational forces. Moreover, NSs areamong the densest objects with the strongest gravity in the Universe, making them ideal placesto test simultaneously our understanding about the nature of super-dense nuclear matter andthe strong-field predictions of general relativity (GR). Many fundamental questions about boththe strong-field gravity and properties of super-dense nuclear matter remain to be answered[1]. Tied to the fundamental strong equivalence principle [2], the degeneracy between the NSmatter content and its gravity requires a simultaneous understanding about both the strong-field gravity and the equation of state (EOS) of super-dense nuclear matter. For example, themasses and radii of NSs are determined by the balance between the strong-field behavior ofgravity and the EOS of dense nuclear matter. While the nature of super-dense matter in NSsand its EOS are currently largely unknown, there is also no fundamental reason to chooseEinstein’s GR over other alternatives and it is known that the GR theory itself may breakdown at the strong-field limit [3]. Indeed, it is very interesting to note that alternative gravitytheories, which have all passed low-field tests but diverge from GR in the strong-field regime,predict NSs with significantly different properties than their GR counterparts [2]. In fact, thevariations in the predicted NS mass–radius correlation from modifying gravity are larger thanthose from varying the EOS of super-dense matter in NSs [4]. Therefore, the EOS of super-dense matter and the strong-field gravity have to be studied simultaneously to understandproperties of NSs accurately. Moreover, the gravity-EOS degeneracy can only be broken byusing at least two independent observables. We notice that effects of modified gravity (MOG)on properties of NSs have been under intense investigation. The conclusions are, however,strongly model-dependent, see, e.g., [5–10]. In particular, it is interesting to mention thatthe very recent test of strong-field gravity using the binding energy of the most massive NSobserved so far, the PSR J0348+0432 of mass 2.01 ± 0.04M�, found no evidence of anydeviation from the GR prediction [11]. However, it is worth noting that this analysis used avery stiff EOS with an incompressibility of K0 = 546 MeV at normal nuclear matter density[12, 13]. Such a super-stiff incompressibility is more than twice the value of K0 = 240 ±20 MeV extracted from analyzing various experiments in terrestrial nuclear laboratories duringthe last 30 years [14].

Many kinds of modified gravity theories have been published ever since Einstein publishedhis GR in 1915, see, e.g., [15, 16] and references therein for a very recent reviews. Besides theNewtonian potential, an extra Yukawa term naturally arise in the gravitational potential at theweak-field limit in some of these theories, e.g., f(R) [17, 18], the nonsymmetric gravitationaltheory [19] and MOG [20], see [21] for a very recent review. Such theories are able to explainsuccessfully the well-known astrophysical observations, such as the flat galaxy rotation curves[19, 22] and the Bullet Cluster 1E0657-558 observations [23] in the absence of dark matter.Basically, this is because particles with masses, besides the presumably massless graviton,

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may also carry the gravitational force [22]. Following Fujii [24–26], we model effects of theMOG by adding a Yukawa term to the conventional gravitational potential.

To our best knowledge, while various properties of NSs and gravitational waves havebeen used to probe the EOS and gravity often individually, it is not so clear what observablesshould be used to effectively break the EOS-gravity degeneracy. In this work, we show that thesimultaneous measurement of the NS mass and its pulsating frequencies is useful for breakingthe EOS-gravity degeneracy. Model EOSs for super-dense matter with or without the hadron–quark phase transition are considered. For hybrid stars, the EOS is constructed by using anextended isospin- and momentum-dependent nuclear effective interaction for the hyperonicphase, the MIT bag model for the quark phase, and the Gibbs conditions for the mixedphase. Model parameters are constrained by existing experimental data whenever possible inconstructing the EOS. By turning on or off the hadron–quark phase transition, varying theMIT bag constant or the strength of the Yukawa term, we explore the possibility of breakingthe EOS-gravity degeneracy by using the mass and the frequencies of the f , p1, p2, and wI

pulsating modes of NSs. Since effects of the EOS on properties of both NSs and gravitationalwaves are better known as a result of extensive previous studies in the literature, we focus oneffects of the Yukawa-type non-Newtonian gravity on both the maximum mass and pulsatingfrequencies of NSs. While the NS maximum mass is found to increase, the frequencies of thef , p1, p2, and wI pulsating modes decrease with the increasing strength of the Yukawa term.A simultaneous measurement of the NS mass and the frequency of one of its pulsating modes,preferably the f -model, can thus readily break the EOS-gravity degeneracy, providing a usefulreference for testing simultaneously both the strong-field gravity and supranuclear EOS usingx-rays and gravitational waves hopefully in the near future.

2. Yukawa-type non-Newtonian gravity and model EOS of hybrid stars

Despite the fact that gravity is the first force discovered in nature, the quest to unify it withother fundamental forces remains elusive because of its apparent weakness at short distance,see, e.g., [27–34]. In developing grand unification theories, the conventional inverse-square-law (ISL) of Newtonian gravitational force has to be modified due to either the geometricaleffect of the extra space-time dimensions predicted by string theories and/or the exchange ofweakly interacting bosons, such as the neutral spin-1 vector U-boson [35], proposed in thesuper-symmetric extension of the Standard Model, see, e.g., [36–40] for recent reviews. Asan Ansatz, Fujii [24, 26] first proposed that the non-Newtonian gravity can be described byadding a Yukawa term to the conventional gravitational potential between two objects of massm1 and m2, i.e.,

V (r) = −Gm1m2

r(1 + αe−r/λ), (1)

where α is a dimensionless strength parameter, λ is the length scale, and G is the gravitationalconstant. In the boson exchange picture, the strength of the Yukawa potential between twobaryons can be expressed as α = ±g2/(4πGm2

b) where ± stands for scalar/vector bosons,mb is the baryon mass, and λ = 1/μ (in natural units), with g and μ being the boson-baryoncoupling constant and the boson mass, respectively. As mentioned earlier, the Yukawa termappears naturally at the weak-field limit in several modern MOG theories and has been used toexplain successfully the flat galaxy rotation curves without invoking dark matter [17–23]. Thelight and weakly interacting U-boson is a favorite candidate mediating the extra interaction[8, 35, 41], and various new experiments in terrestrial laboratories have been proposed to searchfor the U-boson, see, e.g., [42] and references therein. The search for evidence of the MOG is atthe forefront of research in several sub-fields of natural sciences including geophysics, nuclear

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Figure 1. Left: constraints on the strength and range of the Yukawa term from terrestrialnuclear experiments in comparison with g2/μ2 ≈ 40–50 GeV−2. Right: model EOSsfor hybrid stars with and without the Yukawa contribution using MIT bag constantB1/4 = 170 MeV and B1/4 = 180 MeV, respectively. The soft and stiff EOSs for npematter are taken from [69, 70].

and particle physics, as well as astrophysics and cosmology, see, e.g., [7, 24, 27–30, 43–52].Various upper limits on the deviation from the ISL has been put forward down to femtometerrange using different experiments [53]. These experiments have established a clear trend ofincreased strength α at shorter length λ. In the short range down to λ ≈ 10−15–10−8 m, groundstate properties of nuclei, neutron–proton and neutron-lead scattering data as well as thespectroscopy of antiproton atoms have been used to set upper limits on the value of g2/μ2 (orequivalently the |α|vsλ). Shown in the left window of figure 1 is a comparison of several recentconstraints [47, 48, 53–55] in the |α|vsλ plane in the range of λ ≈ 10−15–10−8 m obtained fromnuclear laboratory experiments. Extensive reviews on constraints at longer length scales areavailable in the literature, see, e.g., [36, 37]. We emphasize that below the upper limits shownhere, the Yukawa term has no effect on properties of finite nuclei and nuclear scatterings. Forexample, the upper limits on |α| in the femtometer scale were obtained by requiring explicitlythat the binding energies and charge radii of nuclei do not deviate from their experimentalvales by more than 2% [53]. As a reference, the straight lines are the |α|vsλ correlation forg2/μ2 ≈ 40–50 GeV−2.

Some justifications and cautions about using the Yukawa term to model MOG in NSs arein order here. On one hand, the Yukawa term only appears in some of the MOG theories atthe weak-field limit while NSs have strong gravity fields. Indeed, in this sense it is highlyquestionable whether one can use the Fujii ansatz in studying NSs. On the other hand,considered as the fifth force due to the exchange of the U-boson, a Yukawa term is possibleindependent of the strength of gravity. As mentioned earlier, to our best knowledge, currentlythere is no fundamental reason to select one theory over others for describing the strong-fieldgravity as long as the theory can explain all known facts from observations. Our reason ofchoosing the Fujii ansatz is mostly technical rather than physical. To our best knowledge, theYukawa-like non-Newtonian gravity is the only form that has its strength and range parametersboth limited using experimental data as illustrated in figure 1 for the short-range part whilethe constraints for longer ranges can be found in the literature, see, e.g., [47, 48, 54, 55].Moreover, as we shall discuss in detail in the following, it is relatively easier to implement theFujii ansatz and study its effects on both static and oscillating properties of NSs, although itspossible origins as a prediction of either the string theory or super-symmetric extension of the

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standard model may not be widely accepted. There are many different kinds of MOG theories,and all of them have their own field equations governing static properties of NSs. Moreover, todetermine the pulsating frequencies of NSs one has to use the first-order perturbations of thefield equations. Such kinds of much more complicated studies are interesting and are beingpursued by us and others using some of the MOG theories. However, they are beyond thescope of the present work.

Because of the degeneracy between the matter content and gravity in NSs, a negativeYukawa term can be considered effectively either as an anti-gravity [22] or a stiffening of theEOS [8]. According to Fujii [25], the Yukawa term in the boson exchange picture is simplypart of the matter system in GR. Therefore, effectively one can adjust the EOS only whilekeep the same TOV equation for the static equilibrium for NSs. For infinite nuclear matter, theextra energy density due to the Yukawa term is [8, 30]

εUB

= 1

2V

∫ρ(�x1)

g2

4π

e−μr

rρ(�x2) d�x1 d�x2 = 1

2

g2

μ2ρ2, (2)

where V is the normalization volume, ρ is the baryon number density, and r = |�x1 − �x2|.Assuming a constant boson mass independent of the density, one obtains the additional pressurePUB = 1

2g2

μ2 ρ2, which is just equal to the additional energy density. Including effects of the

reduced gravity through the Yukawa term, the total pressure becomes P = P0 +PUB where P0

is the nuclear pressure inside NSs. Within this framework, the task of breaking the EOS-gravitydegeneracy has effectively become a question of separating effects of the P0 and PUB.

As it was emphasized by Fujii [25], since the new vector boson contributes to the nuclearmatter EOS only through the combination g2/μ2, while both the coupling constant g andthe mass μ of the light and weakly interacting bosons are small, the value of g2/μ2 canbe large. On the other hand, by comparing the g2/μ2 value with that of the ordinary vectorboson ω, Krivoruchenko et al have pointed out that as long as the g2/μ2 value of the U-bosonis less than about 200 GeV−2 the internal structures of both finite nuclei and NSs will notchange [8]. However, global properties of NSs can be significantly modified [8, 9]. One of thekey characteristics of the Yukawa correction is its composition dependence, unlike Einstein’sgravity. Thus, ideally one needs to use different coupling constants for various baryons existingin NSs. Moreover, to our best knowledge, it is unknown if there is any and what might be theform and strength of the Yukawa term in the hadron–quark mixed phase and the pure quarkphase. Thus, instead of introducing more parameters, for the purpose of this exploratory study,we assume that the PUB term is an effective Yukawa correction existing in all phases with theg being an averaged coupling constant.

For the EOSs of hybrid stars, we consider a quark core covered by hyperons and leptonsusing models developed by us in several recent studies of NS properties [56–59]. Morespecifically, the quark matter is described by the MIT bag model with reasonable parameterswidely used in the literature [60, 61]. The hyperonic EOS is modeled by using an extendedisospin- and momentum-dependent effective interaction (MDI) [62] for the baryon octet withparameters constrained by empirical properties of symmetric nuclear matter, hyper-nuclei, andheavy-ion reactions [58]. In particular, the underlying EOS of symmetric nuclear matter isconstrained by comparing transport model predictions with data on collective flow and kaonproduction in high-energy heavy-ion collisions [63, 64]. Moreover, the nuclear symmetryenergy Esym(ρ) with this interaction is chosen to increase approximately linearly with density(i.e., the MDI interaction with a symmetry energy parameter x = 0 [58]) in agreement withavailable constraints around and below the saturation density [65]. The Gibbs construction wasadopted to describe the hadron–quark phase transition [66]. As known, the Gibbs constructionrequires that both the baryon and charge chemical potentials are in equilibrium, and it leads

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to a smooth phase transition. On the other hand, the Maxwell construction only requires thebaryon chemical equilibrium, and the phase transition is first order. As mentioned in [58], amore realistic EOS of the mixed phase can be obtained from the Wigner–Seitz cell calculationby taking into account the Coulomb and surface effects. The resulting EOS lies between thosefrom the Gibbs and Maxwell constructions. The latter can thus be viewed as two extremecases corresponding to certain values of the surface tension parameter in the Wigner–Seitz cellcalculation. For the bag constant, the original MIT bag model gives a value of B = 55 MeV/fm3

[60], while from lattice calculations it is about B = 210 MeV/fm3 [67]. Thus, the bag constantcan be viewed as a free parameter in a reasonable range. In [58], we compared the mass–radius relation with different values of the bag constant. It was shown that the phase transitionhappens at a higher density with a larger bag constant, while the maximum mass is insensitiveto the value of the bag constant. To give a reasonable value of the transition density, we chooseB1/4 = 170 MeV (i.e., B ≈ 109 MeV/fm3) or 180 MeV (i.e., B ≈ 137 MeV/fm3) in the presentstudy. The u and d quarks are taken as massless and the mass of the strange quark is set to be150 MeV.

Similar to the previous work in the literature, the hybrid star is divided into the liquidcore, the inner crust, and the outer crust from the center to surface. For the inner crust, aparameterized EOS of P0 = a + bε4/3 is adopted as in [56, 57]. For the outer crust, the BPSEOS [68] is adopted. As an example, shown in the right panel of figure 1 are the EOSs forhybrid stars with MIT bag constant B1/4 = 170 MeV and B1/4 = 180 MeV, respectively,with and without the Yukawa contribution. The corresponding hadron–quark mixed phaseabove/below the pure hadron/quark phase covers the density range of ρ/ρ0 = 1.31–6.56 andρ/ρ0 = 2.19–8.63, respectively. Including the Yukawa term the EOS stiffens as the value ofg2/μ2 increases. It is seen that the two sets of parameters, B1/4 = 170 MeV and g2/μ2 =50 GeV−2 or B1/4 = 180 MeV and g2/μ2 = 40 GeV−2, lead to approximately the same totalpressure. As a reference, the MDI EOS for the npeμ matter without hyperons and quarks isalso shown. In addition, for comparisons we also included in the right window of figure 1 thesoft and stiff EOSs for npe matter constructed by Lattimer [69] and Hebeler et al [70]. TheseEOSs are constrained at sub-saturation densities by the EOS of pure neutron matter calculatedby a chiral effective field theory and at supra-saturation densities by the observed maximummasses of NSs. It is interesting to note that our EOSs for the npeμ matter are very close totheir EOSs for the npe matter at low densities and are inside their soft-stiff EOS band at highdensities. We remark that our hadronic EOS, as shown in [71], is constrained by relativisticheavy-ion reaction data (flow and kaon production) in the density range of approximately2–4.5ρ0 [63] which falls below the stiff EOS necessary to predict a 2.0M� NS. However, theconstraints from relativistic heavy-ion reactions were not considered in [69, 70]. Interestingly,it is obvious that the soft-stiff EOS band from [69, 70] leaves a large room for varying gravitytheories at high densities.

3. Breaking the EOS-gravity degeneracy with neutron star masses andpulsating frequencies

Stellar oscillations of non-rotating compact stars in GR have been extensively studiedespecially since the late sixties, see, e.g., [72–80]. Responding to the strong promise ofdetecting gravitational waves in the near future and the outstanding scientific opportunitiesprovided by the advanced detectors, the NS oscillating frequencies, especially the fundamental( f ) mode, the first few pressure (p) modes, and the first pure space-time oscillation (wI) havebeen studied with much interest and fruitfully. In particular, using various model EOSs the

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J. Phys. G: Nucl. Part. Phys. 41 (2014) 075203 W Lin et al

Figure 2. Left: the mass–radius relation of static NSs with B1/4 = 170 MeV and variousvalues of g2/μ2 in units of GeV−2 indicted using numbers above the lines; results usingthe soft and stiff EOSs are taken from [69, 70]. Right: the maximum mass of NSs as afunction of g2/μ2 with B1/4 = 170 and 180 MeV, respectively.

inverse problem of extracting information about the EOS of supranuclear matter from futuremeasurement of gravitational waves from pulsating NSs in our own galaxy has been addressedin a number of works, see, e.g., [81–84]. We examine here simultaneously effects of theYukawa-type non-Newtonian gravity and the nuclear EOS on the frequencies of the f , p1,p2, and wI pulsating modes and the mass–radius correlation of NSs. In studying the f , p1,and p2 modes, we used and compared numerical algorithms developed in both [77, 78] and[79, 80], and have reproduced the results in [77] and [85] within 1%. For the wI mode, as inour previous work [59, 86], we use the continuous fraction method [87].

Before trying to break the EOS-gravity degeneracy, we first examine effects of the Yukawa-type non-Newtonian gravity on the mass–radius correlation. Shown in the left panel of figure 2is the mass–radius relation of hybrid stars with the bag constant B1/4 = 170 MeV and varyingvalues of g2/μ2. First of all, without the Yukawa contribution (black solid line) the maximumstellar mass supported is only about 1.46M�. Including the Yukawa term, as the EOSs areincreasingly stiffened with larger values of g2/μ2, the maximum stellar mass increases. Withg2/μ2 = 50 GeV−2 the maximum mass of 1.97M� is just in the middle of the measured massband of 1.97 ± 0.04M� for PSR J1614-2230 [88]. The corresponding radius is about 12.4 km.Moreover, one can see that the maximum mass for the NS containing only the npeμ matteris similar to that for hybrid stars calculated with B1/4 = 170 MeV and g2/μ2 = 40 GeV−2,although the radius is smaller for the npeμ star. Compared to the latter, the softening of theEOS due to the hyperonization and hadron–quark phase transition reduces, while the effectivestiffening of the EOS due to the reduced gravity increases both the NS maximum mass and thecorresponding radius. To see more clearly relative effects of the bag constant B and the Yukawaterm, shown in the right panel of figure 2 are the maximum stellar masses as a function of g2/μ2

with B1/4 = 170 MeV and B1/4 = 180 MeV, respectively. As expected, with B1/4 = 180 MeVa smaller value of g2/μ2 = 40 GeV−2 is needed to obtain a maximum mass consistent with theobserved masses of PSR J1614-2230 and PSR J0348+0432. It is worth noting that the massesof these NSs have ruled out many soft EOSs and a quark core. Many possible stiffeningmechanisms have been proposed during the last few years to reproduce a maximum massabout 2M�. The effective stiffening of the nuclear EOS due to reduced gravity is one extremebut a possible mechanism. The results shown above indicate that measuring the NS masses

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J. Phys. G: Nucl. Part. Phys. 41 (2014) 075203 W Lin et al

Figure 3. Left: the f -mode frequency versus the NS mass. The upper two curves areresults without including the Yukawa potential while the lower two are obtained withg2/μ2 = 40 and 50 GeV−2 (for hybrid stars with B1/4 = 170 MeV). The soft-stiffresults are obtained using the EOSs of [69, 70]; Right: the f -mode frequency versusthe strength of Yukawa-type non-Newtonian gravity g2/μ2 for NSs of mass 1.4M� and1.2M�.

alone is insufficient to break the EOS-gravity degeneracy. The simultaneous measurement ofboth the masses and radii can remedy the situation. Unfortunately, it is currently under hotdebate whether the radii are around 8–10 km [89, 90], 10–12 km [91], or larger than 14 km[92] because of the well-known difficulties and model dependence in extracting the radii ofNSs. Thus, alternative observables should be investigated.

We now turn to the pulsating frequencies of NSs and their dependence on both the nuclearEOS and the strength g2/μ2 of the Yukawa-type non-Newtonian gravity. Shown in the leftpanel of figure 3 is the f -mode frequency versus NS mass. The upper two curves are resultswithout including the Yukawa potential while the lower two are obtained with g2/μ2 = 40and 50 GeV−2 (for hybrid stars with B1/4 = 170 MeV) necessary to obtain a maximum massof about 2M�. Generally, the f -mode frequency is higher with a softer EOS consistent withfindings in earlier studies, see, e.g., [83, 84]. Results using the soft and stiff EOSs of [69, 70]are also shown. It is interesting to see that the soft-stiff band only allows possible variationof gravity above about 1.4M�. Within the band, the frequency of the f -mode is sensitive tothe variation of g2/μ2. With the Yukawa-type non-Newtonian gravity the f -mode frequencydecreases significantly while the maximum mass increases. It is also interesting to see thatalthough the maximum masses are almost the same for the NS containing npeμ matter onlyand the hybrid star with B1/4 = 170 MeV and g2/μ2 = 40 GeV−2, their f -mode frequenciesare rather different. To be more quantitative about effects of the non-Newtonian gravity,shown in the right panel of figure 3 are the f -mode frequency versus the strength g2/μ2 forNSs of mass 1.4M� and 1.2M�, respectively. Combining the results shown in both panelsof figure 3, it is obvious that if both the mass and the f -mode frequency are measured, theEOS-gravity degeneracy is readily broken. Similar gravity effects were found on the p1, p2,and wI-modes. Depending on the frequency, it requires different amount of energy to excitevarious pulsating modes of NSs. Shown in figure 4 is a comparison of the f , p1, p2, and wI

pulsating frequencies for a NS of mass 1.4M� with or without the Yukawa term. While effectsof the Yukawa term on the p1 and p2 modes are relatively larger, these modes are harderto excite compared to the f -mode. We notice also that frequencies of the f and p1 modesare within reach by the advanced gravitational detectors available although not at their peak

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J. Phys. G: Nucl. Part. Phys. 41 (2014) 075203 W Lin et al

Figure 4. Comparison of the f , p1, p2, and wI pulsating frequencies for a NS of mass1.4M�. The vertical scale is arbitrary.

sensitivities, while the detection of the wI mode requires new detectors capable of measuringgravitational at high frequencies. Thus, the f -mode is currently the most promising one forbreaking the EOS-gravity degeneracy. The pulsating modes also have their respective dampingtimes characterized by the imaginary parts of their complex frequencies. We have also studiedeffects of both the nuclear EOS and gravity on the damping times, but did not find anythingparticularly useful for the purpose of this work.

4. Conclusions

Both the EOS of super-dense nuclear matter and the strong-field gravity are poorly known andtheir determinations are at the forefronts of nuclear physics and astrophysics, respectively. Acomprehensive and simultaneous understanding of both of them are required to interpretunambiguously observed properties of neutron stars (NSs) because of the EOS-gravitydegeneracy. To break the degeneracy, at least two independent observables are required. Inthis work, to explore potentially useful observables for breaking the EOS-gravity degeneracy,we investigated effects of the nuclear EOS and Yukawa-type non-Newtonian Gravity on themass–radius correlation, the f -mode, p-mode, and w-mode stellar oscillations of NSs. Wefound that all the NS pulsating frequencies are significantly lowered while their masses areincreased by the Yukawa term for a given nuclear EOS. A simultaneous measurement ofthe NS mass and the frequency of one of its pulsating modes, preferably the f -mode, canthus readily break the EOS-gravity degeneracy. With the already available high-quality x-rayobservational data of NSs and the various ongoing upgrades of advanced gravitational wavedetectors, we hope such a combination of observables will be possible in the near future.

Acknowledgments

We would like to thank F Fattoyev, WZ Jiang, and WG Newton for useful discussions. Thiswork was supported in part by the US National Science Foundation under grant no. PHY-1068022, the US National Aeronautics and Space Administration under grant NNX11AC41Gissued through the Science Mission Directorate, the CUSTIPEN (China-US Theory Institutefor Physics with Exotic Nuclei) under US DOE grant number DE-FG02-13ER42025,the National Natural Science Foundation of China under grant no. 10947023, 11275125,

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J. Phys. G: Nucl. Part. Phys. 41 (2014) 075203 W Lin et al

11135011, 11175085, 11235001, 11035001, 11275073 and 11320101004, the ShanghaiRising-Star Program under grant no. 11QH1401100, the ‘Shu Guang’ project supportedby Shanghai Municipal Education Commission and Shanghai Education DevelopmentFoundation, the Program for Professor of Special Appointment (Eastern Scholar) at ShanghaiInstitutions of Higher Learning, the Science and Technology Commission of ShanghaiMunicipality (11DZ2260700), the ‘100-talent plan’ of Shanghai Institute of Applied Physicsunder grant no. Y290061011 from the Chinese Academy of Sciences, Shanghai PujiangProgram under grant no. 13PJ1410600, and the Chinese Fundamental Research Funds for theCentral Universities (grant no. 2014ZG0036).

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