Probing the high-density behavior of symmetry energy with gravitational waves

DOI 10.1140/epja/i2014-14045-6

Review

Eur. Phys. J. A (2014) 50: 45 THE EUROPEANPHYSICAL JOURNAL A

Probing the high-density behavior of symmetry energy withgravitational waves�

F.J. Fattoyeva, W.G. Newton, and Bao-An Li

Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, TX 75429, USA

Received: 19 September 2013 / Revised: 4 January 2014Published online: 27 February 2014 – c© Societa Italiana di Fisica / Springer-Verlag 2014Communicated by A. Ramos

Abstract. Gravitational wave (GW) astronomy opens up an entirely new window on the Universe to probethe equations of state (EOS) of neutron-rich matter. With the advent of next-generation GW detectors,measuring the gravitational radiation from coalescing binary neutron star systems, mountains on rotatingneutron stars, and stellar oscillation modes may become possible in the near future. Using a set of modelEOSs satisfying the latest constraints from terrestrial nuclear experiments, state-of-the-art nuclear many-body calculations of the pure neutron matter EOS, and astrophysical observations consistently, we studyvarious GW signatures of the high-density behavior of the nuclear symmetry energy, which is consideredamong the most uncertain properties of dense neutron-rich nucleonic matter. In particular, we find thetidal polarizability of neutron stars, potentially measurable in binary systems just prior to merger, is moresensitive to the high-density component of the nuclear symmetry energy than the symmetry energy atnuclear saturation density. We also find that the upper limit on the GW strain amplitude from ellipticallydeformed stars is very sensitive to the density dependence of the symmetry energy. This suggests that futuredevelopments in modeling of the neutron star crust, and direct gravitational wave signals from accretingbinaries will provide a wealth of information on the EOS of neutron-rich matter. We also review thesensitivity of the r-mode instability window to the density dependence of the symmetry energy. Whereasmodels with larger values of the density slope of the symmetry energy at saturation seem to be disfavoredby the current observational data, within a simple r-mode model, we point out that a subsequent softerbehavior of the symmetry energy at high densities (hinted at by recent observational interpretations) couldrule them in.

1 Introduction

Understanding the nature of the neutron-rich nucleonicmatter is one of the fundamental quests of both nuclearphysics and astrophysics [1]. To fulfill this goal, many ex-periments and observations are being carried out or pro-posed using a wide variety of advanced new facilities, suchas the Facility for Rare Isotope Beams (FRIB), telescopesoperating at a variety of wavelengths, and more sensitiveGW detectors (such as Advanced LIGO-Virgo). The EOSof neutron-rich matter is a vital ingredient in the interpre-tation of the results of these experiments and observations.Within the parabolic approximation the EOS can be writ-ten in terms of the binding energy per nucleon E(ρ, α) as

E(ρ, α) = E(ρ, 0) + S(ρ)α2, (1)

� Contribution to the Topical Issue “Nuclear Symmetry En-ergy” edited by Bao-An Li, Angels Ramos, Giuseppe Verde,Isaac Vidana.

a e-mail: [email protected]

where α = (ρn − ρp)/ρ is the isospin asymmetry, E(ρ, 0)is the binding energy per nucleon in symmetric nuclearmatter (SNM), and S(ρ) is the symmetry energy whichrepresents the energy cost per nucleon of changing all theprotons in SNM into neutrons. Around the saturation den-sity ρ0, one can make further expansions,

E(ρ, 0) = B +12K0χ

2 + O(χ3), (2)

S(ρ) = J + Lχ +12Ksymχ2 + O(χ3), (3)

where χ ≡ (ρ − ρ0) /3ρ0, B and K0 are the binding energyper nucleon and the nuclear incompressibility at satura-tion density, ρ0, while J , L, and Ksym are the correspond-ing magnitude, slope, and curvature of the symmetry en-ergy at saturation density. Whereas significant progresshas been achieved in constraining the EOS of SNM aroundρ ≈ ρ0, its density dependence remains still rather uncer-tain for nuclear matter at high densities and with largeisospin asymmetries. Observationally, the best constrainton the high-density component of the EOS comes from

Page 2 of 13 Eur. Phys. J. A (2014) 50: 45

the largest observed masses of neutron stars that havebeen recently reported by Demorest et al. [2] and An-toniadis et al. [3]. Combining observational results withterrestrial experimental studies of the collective flow andkaon production in relativistic heavy-ion collisions [4] theEOS of SNM has been limited to a relatively small rangeup to about 4.5 ρ0. The main source of uncertainties inthe EOS of neutron-rich matter therefore mostly comesfrom the poorly known density dependence of the nuclearsymmetry energy S(ρ), which is a vital ingredient in de-scribing the structure of rare isotopes and their reactionmechanisms. Moreover, it determines uniquely the pro-ton fraction and therefore the condition for the onset ofthe direct Urca process in neutron stars, affects signifi-cantly structural properties such as the radii, moments ofinertia and the crust thickness, as well as the frequenciesand damping times of various oscillation modes of neutronstars (see refs. [5–7] for a review).

Intensive efforts devoted to constraining S(ρ) usingvarious approaches have recently led to a close conver-gence [7–13] around J ≈ 30MeV and its density slope ofL ≈ 60MeV, although the associated error-bars from dif-ferent approaches may vary broadly. However, the possi-bility that J and L can be significantly different from thesecurrently inferred values cannot be conclusively ruled out(see ref. [14] for a detailed discussion). On the other hand,the high-density behavior of S(ρ) remains quite uncer-tain despite its importance to understanding what hap-pens in the core of neutron stars [15–19] and in reac-tions with high energy radioactive beams [20]. The pre-dictions for the high-density behavior of the symmetry en-ergy from all varieties of nuclear models diverge dramat-ically [21–23]. Some models predict very stiff symmetryenergies that monotonically increase with density [23–28],while others predict relatively soft ones, or an S(ρ) thatfirst increases with density, then saturates and starts de-creasing with increasing density [21, 29–43]. These uncer-tainties can be associated with our poor knowledge aboutthe isospin dependence of the strong interaction in thedense neutron-rich medium, particularly the spin-isospindependence of many-body forces, the short-range behav-ior of the nuclear tensor force and the isospin dependenceof nucleon-nucleon correlations in the dense medium, see,e.g. refs. [44, 45]. The experimental progress in constrain-ing the high density S(ρ) is also limited partially dueto the lack of sensitive probes. Whereas several observ-ables have been proposed [20] and some indications of thehigh-density S(ρ) have been reported recently [46,47], con-clusions based on terrestrial nuclear experiments remainhighly controversial [48]. Interestingly, it was recently pro-posed that the late-time neutrino signal from a core col-lapse supernova [49] and the tidal polarizability [50] ofcanonical neutron stars in coalescing binaries are very sen-sitive probes of the high-density behavior of nuclear sym-metry energy, suggesting that astrophysical measurementsmight bolster experimental results in this area.

In this review we focus on the sensitivity of vari-ous gravitational wave signals from neutron stars to thehigh-density component of the nuclear symmetry energy.Gravitational waves are oscillations of space-time and are

one of the fundamental predictions of the theory of gen-eral relativity. Although they have not been directly de-tected yet, there are strong indirect evidences that grav-itational radiation exists. The highly accurate matchingof the observed decrease in the orbital period of the cele-brated Hulse-Taylor [51] and PSR J0737-3039 binary sys-tems [52, 53] with the predicted value due to the energyloss through gravitational radiation serves as the mostprecise indirect evidence of the existence of gravitationalwaves. Because of their extremely weak interaction withmatter, gravitational waves carry much cleaner informa-tion of their source as opposed to their electromagneticcounterparts, and therefore open an entirely new windowto probe physics that is hidden to current electromagneticobservations. Of course, this property also makes their de-tection one of the most difficult experimental problemsfaced in physics today.

The manuscript has been organized as follows. Insect. 2 we describe the formalism of constraining the EOSof SNM, and discuss the emergence of uncertainties inthe high-density component of the EOS of neutron-richmatter due to the density dependence of symmetry en-ergy. In sect. 3 we present results for the selected gravita-tional wave signatures from neutron stars that are inves-tigated consistently with the same set of EOSs discussedin sect. 2. In particular, we study the sensitivity of grav-itational waves generated by ellipticities in the neutronstar shape generated by tidal polarization and mountainsof accreted material, and by r-mode oscillations, to thehigh-density component of the nuclear symmetry energy.Finally, our concluding remarks are summarized in sect. 4.

2 Constraining EOS of neutron-rich nuclearmatter

We will concentrate on two models of the nuclear en-ergy density functionals (EDF): the relativistic mean-field(RMF) model and the Skyrme Hartree-Fock (SHF) ap-proach [13,50]. For detailed discussion on these EDFs, seethe contributions to this topical issue by Nazarewicz etal. and by Piekarewicz. All of the EOSs used in this workare adjusted to satisfy the following four conditions withintheir respective known uncertain ranges:

1) Reproducing the EOS of pure neutron matter (PNM)at sub-saturation densities predicted by the lateststate-of-the-art microscopic nuclear many-body calcu-lations [30,54–59].

2) Predicting correctly saturation properties of symmet-ric nuclear matter, i.e., nucleon binding energy B =−16±1MeV, incompressibility of nuclear matter K0 =230±20MeV [60,61], and the nucleon (Dirac) effectivemass M∗

D,0 = 0.61 ± 0.03M [62] at saturation densityρ0 = 0.155 ± 0.01 fm−3.

3) Predicting a fiducial value of the symmetry energyJ = 26± 0.5MeV at a subsaturation density of 2ρ0/3.Notice that theoretical microscopic PNM calculationsalso put tight constraints on both the symmetry en-ergy and its density slope at saturation density: J =

Eur. Phys. J. A (2014) 50: 45 Page 3 of 13

0 1 2 3 4 5

0

50

100

150

200 IU-FSU SkIU-FSU

Symmetry Energy

SNM

E/N

(MeV

)

ρ/ρ0

PNM

Fig. 1. (Color online) The EOS of SNM and PNM as well asthe symmetry energy as a function of density obtained withinthe IU-FSU RMF model and the SHF approach using theSkIU-FSU parameter set. Taken from ref. [50].

31 ± 2MeV and the density slope of the symmetryenergy L = 50 ± 10MeV (see ref. [13] and refer-ences therein). Also note that this range is a model-dependent prediction coming from the theoretical con-straints of PNM, that has no experimental analog.

4) Passing through the terrestrial constraints on the EOSof SNM between 2ρ0 and 4.5ρ0 [4] and predicting amaximum observed mass of neutron stars of about2M� assuming they are made of the nucleons (neu-trons and protons) and light leptons (electrons andmuons) only and without considering other degrees offreedom or invoking any exotic mechanism [2,3].

As an example, two EOSs were obtained using the IU-FSU parametrization of the RMF model [63] and its SHFcounterpart dubbed as the SkIU-FSU parameter set [13],which are shown in fig. 1. By design, they both have thesame EOS for SNM and PNM around and below ρ0. Thus,at sub-saturation densities the values of S(ρ) which is ap-proximately the difference between the EOSs for PNMand SNM are almost identical for the two models. How-ever, the values of S(ρ) are significantly different aboveabout 1.5ρ0 with the IU-FSU leading to a much stifferS(ρ) at high densities. As discussed in ref. [13] this is dueto the characteristic differences in the functional forms ofthe symmetry energy and is a generic feature of RMF andSHF models,

SRMF(ρ) = A(ρ)ρ2/3 + B(ρ)ρ, (4)

SSHF(ρ) = aρ2/3 − bρ − cρ5/3 − dρσ+1, (5)

where A(ρ) and B(ρ) are positive-valued functions of den-sity, a, b, c, d and σ are some constant functions that maydepend on Skyrme parameters only (for example, σ is justa Skyrme parameter). The symmetry energy in the RMFmodel is therefore always positive, while certain terms of

Table 1. Predictions for the properties of a 1.4 solar mass neu-tron star using the seventeen EOSs considered in this paper.The properties of nuclear matter around our base parametriza-tion are systematically varied to obtain flexible range of theEOSs, but within the available theoretical, experimental andobservational constraints. The first column reports the nameof the EOS with a particular nuclear property and/or ζ-parameter indicated. The radii are in units of km, the tidalpolarizability in 1036 cm2 g s2.

EOS R k2 λ Δλ/λ

Base 12.88 0.0879 3.115 –

K = 210 MeV 12.82 0.0858 2.974 −4.52%

K = 220 MeV 12.85 0.0869 3.046 −2.19%

K = 240 MeV 12.91 0.0890 3.183 +2.21%

K = 250 MeV 12.94 0.0900 3.258 +4.59%

M∗ = 0.580 M 12.71 0.1033 3.427 +10.01%

M∗ = 0.595 M 12.83 0.0943 3.271 + 5.02%

M∗ = 0.625 M 12.89 0.0831 2.957 − 5.06%

M∗ = 0.640 M 12.88 0.0792 2.800 −10.12%

L = 42 MeV 12.33 0.1677 3.089 −0.83%

L = 46 MeV 12.64 0.1635 3.096 −0.58%

L = 54 MeV 13.07 0.1582 3.140 +0.80%

L = 58 MeV 13.22 0.1564 3.170 +1.79%

ζ = 0.0200 13.01 0.0885 3.302 +6.00%

ζ = 0.0225 12.94 0.0882 3.204 +2.85%

ζ = 0.0275 12.81 0.0876 3.025 −2.90%

ζ = 0.0300 12.75 0.0873 2.938 −5.67%

the symmetry energy in the SHF model can become nega-tive at higher densities. More quantitatively, the S(ρ) withIU-FSU is 40 to 60% higher in the density range of 3ρ0

to 4ρ0 expected to be attained in the core of canonicalneutron stars.

To test the sensitivity of the various gravitational waveobservables to variations of properties of neutron-rich nu-clear matter around ρ0 within the constraints listed above,we also build seventeen different RMF parameterizationsby systematically varying the values of K0, M∗

0 , L, and theζ parameter of the RMF model that controls the quarticomega-meson self interactions [64] and subsequently thehigh-density component of the EOS of SNM (see table 1for their predictions). Besides the constraints listed above,all parametrizations can correctly reproduce the experi-mental values for the binding energy and charge radiusof 208Pb and the ground state properties of other doublymagic nuclei within 2% accuracy [65]. As a reference forcomparisons, we select K0 = 230MeV, M∗

0 = 0.61 M ,L = 50MeV, and ζ = 0.025 which we refer to as ourbaseline model. This model predicts ρ0 = 0.1524 fm−3,B = −16.33MeV and J = 31.64MeV. The representa-tive model EOSs for PNM at sub-saturation densities andthose for SNM at supra-saturation densities are then com-pared with their constraints in figs. 2 and 3, respectively.It is seen that the SkIU-FSU and all the RMF modelswith 42 < L < 58MeV can satisfy the constraint from the


0.2 0.4 0.6 0.8 1.0 1.20

2

4

6

8

L = 58

MeV

Schwenk-Pethick Friedman-Pandharipande HF-Vlow-k (S-P)

Hebeler-Schwenk Gandolfi et al. Gezerlis-Carlson Vidana et al. RMF IU-FSU SkIU-FSUE

/N (M

eV)

kF (fm-1)

Pure Neutron Matter

L =

42 M

eV

Fig. 2. (Color online) Energy per nucleon as a function of theFermi momentum for PNM for selected models described inthe text (taken from ref. [50]).

2 3 4 5

10

100

Symmetric Nuclear Matter

Danielewicz et al. RMF (ζ = 0.020) RMF (ζ = 0.025) IU-FSU (ζ = 0.030) SkIU-FSU

P (M

eV fm

-3)

ρ/ρ0

Fig. 3. (Color online) The pressure of SNM as a function ofbaryon density. Here ρ0 is the nuclear matter saturation densityand the shaded area represents the EOS extracted from theanalysis of [4]. The figure is taken from ref. [50].

PNM EOS. Also, they all simultaneously satisfy the highdensity SNM EOS constraint with 0.02 < ζ < 0.03. More-over, they all give a maximum mass for neutron stars ina range between 1.94M� and 2.07M� [63], the upper endof which is consistent with existing precise observationalmeasurements [2, 3]. While the determination of neutron-star radii from observations is both challenging and hin-dered by uncertainties in models of the stellar atmosphereamongst other things, significant advances in X-ray as-tronomy have allowed the simultaneous determination ofmasses and radii from a systematic study of several X-ray bursters. By assuming that all neutron stars have the

same radius Guillot et al. [66] have recently analyzed thethermal spectra of 5 quiescent low-mass X-ray binariesin globular clusters to find that neutron star have radiiof R = 9.1+1.3

−1.5 km at a 90% confidence level. A subse-quent re-analysis using Bayesian approach by Lattimerand Steiner [67] suggests that different interpretations ofthe data is strongly favored and they find much largerneutron star radii of R = 12.1+0.7

−0.7 km for a 1.4 solar massneutron star; neither range can currently be conclusivelyruled out. The models discussed in our text predict canon-ical neutron star radii between 12.33 km and 13.22 km inthe range of currently observed values [66–70].

3 Gravitational waves signals

3.1 Tidal Love number and polarizability

Coalescing binary neutron stars are among the mostpromising sources of laser-interferometric gravitationalwaves. One of the most important features of binary merg-ers is the tidal deformation neutron stars experience asthey approach each other prior to merger. The strengthof the tidal deformation can give us invaluable infor-mation about the neutron-star matter EOS [71–82]. Atthe early stage of an inspiral tidal effects may be effec-tively described through the tidal polarizability parameterλ [71, 74–76] defined via

Qij = −λEij , (6)

where Qij is the induced quadrupole moment of a star inbinary due to the static external tidal field of the compan-ion star Eij . The tidal polarizability can be expressed interms of the dimensionless quadrupolar tidal Love numberk2,

λ =23R5k2, (7)

where R is the radius of a neutron star in isolation, i.e.long before the merger. The tidal Love number k2 dependson the stellar structure and can be calculated using thefollowing expression [72,77]:

k2 =120

(Rs

R

)5 (1 − Rs

R

)2 [2 − yR + (yR − 1)

Rs

R

]

×{

Rs

R

(6 − 3yR +

3Rs

2R(5yR − 8) +

14

(Rs

R

)2 [26

−22yR +(

Rs

R

)(3yR − 2) +

(Rs

R

)2

(1 + yR)])

+3(

1 − Rs

R

)2 [2 − yR + (yR − 1)

Rs

R

]

× log(

1 − Rs

R

)}−1

, (8)

where Rs ≡ 2M is the Schwarzschild radius of the star,and yR ≡ y(R) can be calculated by solving the followingfirst-order differential equation,

rdy(r)dr

+ y(r)2 + y(r)F (r) + r2Q(r) = 0, (9)


-10

-5

0

5

10210 220 230 240 250

(d)(b)

(a)

K0 (MeV)

(λ−λ

0)/λ0 (%

)

(c)

40 44 48 52 56 60

-10

-5

0

5

10

L (MeV)

0.58 0.60 0.62 0.64-10

-5

0

5

10

(λ−λ

0)/λ0 (%

)

M*/M0.020 0.025 0.030

-10

-5

0

5

10

ζ

Fig. 4. (Color online) Percentage changes in the tidal polariz-ability of a canonical 1.4 solar mass neutron star by individu-ally varying properties of nuclear matter K0 (a), M∗ (b), L (c),and the ζ parameter (d) of the RMF model with respect to thevalue using the base model. The figure is taken from ref. [50].

with

F (r) =r − 4πr3 (E(r) − P (r))

r − 2M(r), (10)

Q(r) =4πr

(5E(r) + 9P (r) + E(r)+P (r)

∂P (r)/∂E(r) −6

4πr2

)r − 2M(r)

−4[M(r) + 4πr3P (r)r2 (1 − 2M(r)/r)

]2

. (11)

Equation (9) must be integrated together with theTolman-Oppenheimer-Volkoff (TOV) equation. That is,

dP (r)dr

= −

(E(r) + P (r)

)(M(r) + 4πr3P (r)

)

r2(1 − 2M(r)/r

) , (12)

dM(r)dr

= 4πr2E(r). (13)

Given the boundary conditions in terms of y(0) = 2,P (0) = Pc and M(0) = 0, the tidal Love number can beobtained once an EOS is supplied. Previous studies haveused both polytropic EOSs and several popular nuclearEOSs available in the literature [71–82]. While other par-ticles may be present, it is sufficient to assume that neu-tron stars consist of only neutrons (n), protons (p), elec-trons (e) and muons (μ) in β-equilibrium [50].

In ref. [50], we examined sensitivity of the tidal po-larizability λ of a 1.4M� neutron star to the variationsof SNM properties and the slope of the symmetry energyaround ρ0 as shown in fig. 4 and table 1. The changesof λ relative to the values for our base RMF model areshown for the remaining RMF EOSs. It is very interestingto see that the tidal polarizability is rather insensitive tothe variation of L within the constrained range, although

0.3 0.6 0.9 1.2 1.5 1.80

1

2

3

4

5

Advanced LIGO Einstein Telescope IU-FSU SkIU-FSU L = 40 MeV L = 60 MeV

λ (x

1036

cm

2 g s

2 )

M (MSun)

D = 100 Mpc

Fig. 5. (Color online) Tidal polarizability λ of a single neu-tron star as a function of neutron-star mass for a range ofEOS that allow various stiffness of symmetry energies. A crudeestimate of uncertainties in measuring λ for equal mass bi-naries at a distance of D = 100 Mpc is shown for the Ad-vanced LIGO [84] (shaded light-grey area) and the EinsteinTelescope [85] (shaded dark-grey area). This result was firstreported in ref. [50].

it changes up to ±10% with K0, M∗ and ζ within theirindividual uncertain ranges.

While the averaged mass is M = 1.33± 0.05M�, neu-tron stars in binaries have a broad mass distribution [83].It is thus necessary to investigate the mass dependence ofthe tidal polarizability. Whereas what can be measuredfor a neutron star binary of mass M1 and M2 is the mass-weighted tidal polarizability [76],

λ =126

[M1 + 12M2

M1λ1 +

M2 + 12M1

M2λ2

], (14)

for the purpose of this study it is sufficient to consider bi-naries consisting of two neutron stars with equal masses.In fig. 5 the tidal polarizability λ as a function of neutron-star mass for a range of EOSs is shown. Very interestingly,it is seen that the IU-FSU and SkIU-FSU models whichare different only in their predictions for the nuclear sym-metry energy above about 1.5ρ0 (see fig. 1) lead to sig-nificantly different λ values in a broad mass range from0.5 to 2 M�. More quantitatively, a 41% change in λ from2.828 × 1036 (IU-FSU) to 1.657 × 1036 (SkIU-FSU) is ob-served for a canonical neutron star of 1.4 M� (see table 2).For a comparison, we notice that this effect is as strong asthe symmetry energy effect on the late time neutrino fluxfrom the cooling of proto-neutron stars [49]. However, weshould note that the SNM components of the EOSs usedin ref. [49] are also significantly different and therefore afurther systematic test may be desirable. Moreover, it isshown that the variation of L on 40 to 60MeV as allowedby the PNM constraints has a very small effect on the tidalpolarizability λ of massive neutron stars, which is consis-tent with the results shown in fig. 4. On the other hand,


Table 2. Predictions for the properties of a 1.4 solar massneutron star using the IU-FSU EOS with difference densitydependence of the symmetry energy. The slopes of the symme-try energy are in units of MeV, radii are in units of km, andthe tidal polarizability in 1036 cm2 g s2. The relative percentageerror Δλ/λ is calculated with respect to the original IU-FSUparametrization [63].

EOS L R k2 λ Δλ/λ

IU-FSU 47.2 12.49 0.0930 2.828 –IU-FSU-min 40.0 12.20 0.1054 2.841 + 0.46%IU-FSU-max 60.0 13.07 0.0761 2.906 + 2.76%SkIU-FSU 47.2 11.71 0.0753 1.657 −41.41%

the L parameter affects significantly the tidal polarizabil-ity of neutron stars with M ≤ 1.2M�. These observationscan be easily understood. From eq. (8) the Love numberk2 is essentially determined by the compactness parameterM/R and the function y(R). Both of them are obtainedby integrating the EOS all the way from the core to thesurface. Since the saturation density approximately cor-responds to the central density of a 0.3M� neutron star,one thus should expect that only the Love number of low-mass neutron stars to be sensitive to the EOS around thesaturation density. However, for canonical and more mas-sive neutron stars, the central density is higher than 3 to 4saturation density ρ0, and therefore both the compactnessM/R and y(R) show stronger sensitivity to the variationof EOS at supra-saturation densities. Since all the EOSsfor SNM at supra-saturation densities have already beenconstrained by the terrestrial nuclear physics data and re-quired to give a maximum mass about 2M� for neutronstars, the strongest effect on calculating the tidal polar-izability of massive neutron stars should therefore comefrom the high-density behavior of the symmetry energy.

It has been suggested that the Advanced LIGO-Virgodetector may potentially measure the tidal polarizabilityof binary neutron stars with a moderate accuracy. To testwhether planned GW detectors are sensitive enough tomeasure the predicted effects of high-density symmetryenergy on the tidal polarizability, as an example, we esti-mate uncertainties in measuring λ for equal mass binariesat an optimally-oriented distance of D = 100 Mpc [76,86]using the same approach as detailed in refs. [76,82]. For ex-ample, for point-particle models of binary inspiral, ref. [71]showed that one can obtain analytical gravitational waveform accurate to 2–3 post-Newtonian (PN) order. Thetidal contribution to the GW signal is then found to beaccurate to less than 10%. Current uncertainties in thedetermination of λ is estimated as [76]

Δλ = α

(M

M�

)2.5 (M2

M1

)0.1 (fend

Hz

)−2.2 (D

100Mpc

),

(15)where M = M1 + M2 is the total mass of the binary,α = 1.0× 1042 cm2 g s2 for a single Advanced LIGO-Virgodetector, and α = 8.4 × 1040 cm2 g s2 for a single EinsteinTelescope detector. These uncertainties are shown for theAdvanced LIGO-Virgo (shaded light-grey area) and the

Einstein Telescope (shaded dark-grey area) in fig. 5 as-suming the IU-FSU as the base model.

It is seen that discerning between high-density symme-try energy behaviors is at the limit of Advanced LIGO-Virgo’s sensitivity for stars of mass 1.4M� and belowbased on the currently estimated uncertainty, and it is pos-sible that a rare but nearby binary system may be foundand provide a much more tighter constraint [76]. Whilediscoveries of binary neutron star systems PSR B1913+16and PSR J0737-3039 at a nearby location (6.4 kpc and0.6 kpc, respectively) reminds us that such a possibilitymay be likely, the rate estimates for detection of binaryneutron stars are found to be very small for a single Ad-vanced LIGO-Virgo interferometer [76]. Nonetheless, mea-surements for binaries consisting of light neutron stars canstill help further constrain the symmetry energy aroundthe saturation density. On the other hand, given that themasses of neutron stars in the binary can be measured ac-curately, it is noteworthy that the narrow uncertain rangefor the proposed Einstein Telescope even at very largedistances will enable it to constrain the high-density com-ponent of the symmetry energy at the same level —ofabout 10%— as the accuracy of tidal contribution to thegravitational wave form. We should mention once againthat these conclusions are solely based on our composi-tion model, and there could be degeneracies involved withother effects, such as the softening of the EOS due to pos-sible exotic degrees of freedom that might be present inthe core of neutron stars.

3.2 Gravitational waves from elliptically deformedpulsars

Rotating neutron stars are major candidates for sources ofcontinuous gravitational waves in the frequency bandwithof the current laser interferometric detectors [86]. Accord-ing to general relativity any rotating axial-asymmetricobjects should radiate gravitationally. There are severalmechanisms that may lead to an axial asymmetry in neu-tron stars [87]:

a) Anisotropic stress built up during the crystallizationperiod of the solid neutron star crust may supportstatic “mountains” on the surface of neutron stars [88].

b) Due to its violent formation in supernova the rota-tional axis of a neutron star may not necessarily coin-cide with the principal axis of the moment of inertia,which results in a neutron star precession [89,90].

c) Since neutron stars possess strong magnetic fields theycan create a magnetic pressure, which in turn may dis-tort the star. This is possible only if the magnetic fieldaxis is not aligned with the axis of rotation, which isalways the case for pulsars [91].

These processes generally result in a triaxial neutronstar configuration. Gravitational waves are then character-ized by a tiny dimensionless strain amplitude, h0, whichdepends on the degree to which the neutron star is de-


formed from axial symmetry [92],

h0 =16π2G

c4

εIzzf2

r, (16)

where f is the rotation frequency of the neutron star, Izz

is its moment of inertia around the principal axis, ε =(Ixx − Iyy)/Izz is its equatorial ellipticity as defined inthe literature on gravitational waves [86], and r is thedistance from the source to the observer. The ellipticity isrelated to the neutron star maximum quadrupole momentthrough [93]

ε =

√8π

15Q22,max

Izz. (17)

Notice that the gravitational wave strain amplitude doesnot depend on the neutron star moment of inertia Izz, andthe total dependence upon the underlying EOS is entirelydue to the maximum quadrupole moment, Q22,max. Usinga chemically detailed model of the crust, ref. [94] computesthe maximum quadrupole moment for a neutron star,

Q22,max =

2.4×1038 g cm2(σmax

10−2

)(R

10 km

)6.26(1.4M�M

)1.2

, (18)

where σmax is the breaking strain of the crust. Noticethat this is an approximate formula, and a more com-plete relation for the equation of quadrupole moment re-quires the proper knowledge of both the crust-core tran-sition density and the crust composition, as given in theeq. (69) of ref. [94]. Although earlier studies have esti-mated the value of the breaking strain to be in the rangeof σmax =

[10−5–10−2

][92], more recently using molecular

dynamics simulations it was estimated that the breakingstrain can be as large as σmax ≈ 0.1, which is consider-ably larger than the previous findings [88]. Using a ratherconservative value of σmax = 0.01, in a work involving oneof us [87] we reported the first direct nuclear constraintson the gravitational wave signals to be expected fromseveral pulsars. Particularly, it has been found that forseveral millisecond pulsars located at distances 0.18 kpcto 0.35 kpc from Earth, the maximal gravitational wavestrain amplitude is in the range of h0 ∼ [0.4–1.5]× 10−24.The EOS used in ref. [87] was calculated using the mo-mentum dependent interaction (MDI). The high densitycomponent of the symmetry energy in MDI is controlledby a single parameter x that is introduced in the single-particle potential of the MDI EOS. In calculating bound-aries of the possible neutron star configurations, onlyEOSs that are consistent with the isospin diffusion labora-tory data and measurements of the neutron skin thicknessin 208Pb [95,96] were used that suggest the x-parameter isin the range of x = −1 (stiff) and x = 0 (soft). It was thenshown that EOSs with stiff symmetry energy, such as MDIwith x = −1, results in less compact stellar models, andhence more deformed neutron stars. It is of course reason-able to expect that more compact stellar configurationsas predicted by the models with softer symmetry energywould be more resistant to any kind of deformation [87].

1.0 1.2 1.4 1.6 1.8 2.00.5

1.0

1.5

2.0

2.5

IU-FSU SkIU-FSU L = 40 MeV L = 60 MeV

I (x

1045

g c

m2 )

M (MSun)

Fig. 6. (Color online) Neutron star moment of inertia as calcu-lated using eq. (19) using various EOSs with different stiffnessof symmetry energies.

1.0 1.2 1.4 1.6 1.8 2.00

5

10

15

20

25


Q22

, max

(x 1

039 g

cm

2 )

M (MSun)

Fig. 7. (Color online) Neutron star maximum quadrupole mo-ment as calculated using eq. (18).

Also, in a work involving two of us [97] we have demon-strated that pasta phases can play essential role on themaximum quadrupole ellipticity sustainable by the crust.In particular, for EOSs with density slope of the symme-try energy of L < 70MeV, depending on the pasta phasesthe effect on the maximum quadrupole ellipticity can beas large as an order of magnitude.

In fig. 6 we display the neutron star moment of inertiacalculated using a simple empirical relationship proposedby ref. [98],

I � 0.237MR2

[1 + 4.2

(M kmM�R

)+ 90

(M kmM�R

)4]

.

(19)This expression is shown to hold for a wide class of equa-tions of state that do not exhibit considerable softeningand for neutron star models above 1 M�. In fig. 7 wedisplay the maximum quadrupole moment calculated viaeq. (18), where we chose the value of the breaking strainas σmax = 0.1. We notice that both moment of inertia andthe maximum quadrupole moment are sensitive to thedensity dependence of the symmetry energy. Particularly,


Table 3. Properties of the nearby pulsars considered in this study. The first column identifies the pulsar. The remaining columnsare rotational frequency, distance to Earth, the observed [99] and the calculated upper limits on the gravitational wave strainamplitude. Notice that the masses of most of these pulsars are presently unknown and therefore we have adopted a canonical1.4M� mass neutron star in calculating the gravitational wave strain amplitudes, which are all given in units of 1.0 × 10−24.

Pulsar f (Hz) r (kpc) hob0 hth

0 (IU-FSU) hth0 (IU-FSU-min) hth

0 (IU-FSU-max) hth0 (SkIU-FSU)

J0437−4715 173.69 0.1 0.5730 15.9590 13.7350 21.2256 10.6567J0613−0200 326.60 0.5 0.1110 11.2855 9.7127 15.0097 7.5359J0751+1807 287.46 0.6 0.1640 7.2855 6.2702 9.6898 4.8649J1012+5307 190.27 0.5 0.0694 3.8303 3.2965 5.0943 2.5577J1022+1001 60.78 0.4 0.0444 0.4886 0.4205 0.6498 0.3262J1024−0719 193.72 0.5 0.0501 3.9704 3.4171 5.2807 2.6513J1455−3330 125.20 0.7 0.0515 1.1846 1.0195 1.5755 0.7910J1730−2304 123.11 0.5 0.0593 1.6035 1.3801 2.1327 1.0708J1744−1134 245.43 0.5 0.1100 6.3730 5.4848 8.4761 4.2556J1857+0943 186.49 0.9 0.0727 2.0442 1.7593 2.7188 1.3650J2019+2425 254.16 0.9 0.0923 3.7969 3.2678 5.0499 2.5354J2124−3358 202.79 0.2 0.0485 10.8773 9.3614 14.4668 7.2634J2145−0750 62.30 0.5 0.0383 0.4106 0.3534 0.5462 0.2742J2322+2057 207.97 0.8 0.1120 2.8600 2.4614 3.8038 1.9098

1.0 1.2 1.4 1.6 1.8 2.00

5

10

15

20

25

30


ε (x

10-6

)

M (MSun)

Fig. 8. (Color online) Ellipticity as a function of the neutronstar mass.

low-mass neutron stars exhibit a strong sensitivity tothe density dependence of the nuclear symmetry energyaround saturation density, while massive neutron stars aremore sensitive to the high-density component of the sym-metry energy. When the general relativistic expression forthe moment of inertia is used the result remains the samequalitatively, although it slightly changes quantitatively.We emphasize that the gravitational strain amplitude ascalculated above is independent of the moment of inertia,and strongly depends on the axial-asymmetric quadrupolemoment. In fig. 8 we show the maximum ellipticity thatcan be supported by the crust as a function of thestellar mass. Since ε is proportional to the quadrupolemoment scaled by the moment of inertia, it decreaseswith increasing stellar mass. It is seen that models withstiff symmetry energy favor the larger crust “mountains”.There is also a significant difference for the calculateddeformations using models with the same symmetry en-ergy at saturation, but different high-density component(IU-FSU and SkIU-FSU) of the symmetry energy.

1.0 1.2 1.4 1.6 1.8 2.00.0

0.2

0.4

0.6

0.8

1.0


h 0 (x

10-2

4 )

M (MSun)

LIGO Sensitivity

PSR J2145-0750f = 62.30 Hzr = 0.5 kpc

Fig. 9. (Color online) Gravitational wave strain amplitude asa function of the neutron star mass. The result is shown forpulsar J2145−0750.

Finally, in table 3 we report calculated upper limitson gravitational wave strain amplitude for various pulsarsand compare the results with observational upper limits.The results illustrate the relationships discussed above. Itis shown that the gravitational wave strain amplitude isvery sensitive to the density dependence of the symmetryenergy. This sensitivity is most apparent in the eq. (18)of the maximum quadrupole moment that depends on the6.26-th power of the stellar radius. While gravitationalwaves for low-mass neutron stars are sensitive to the den-sity slope of the nuclear symmetry energy, this sensitiv-ity weakens as the neutron star becomes more massive.For massive neutron stars the effect of the high-densitycomponent of the symmetry energy is dominant (also seefig. 9). The results in table 3 suggest that at presentthe gravitational radiation from these pulsars should bewithin the detection capabilities of LIGO [99]. However,no signal detection from any of these targets were re-ported. The fact that such a detection has not been made


yet deserves a few comments. First and most importantly,recall that in computing the upper limits on h0 we haveused the maximum value for the breaking strain of theneutron star crust σ = 0.1 [88], which is very optimistic.Theoretical calculations of σ are different by at least fourorders of magnitude, where it could take values as low asσ = 10−5 [92]. Even if the maximum value chosen in thiswork has been found to be correct, it is very importantto understand that this does not suggest that neutronstars will have deformations of such magnitude. In fact,the main problem is to provide a reasonable scenario thatleads to the development of large deformations [100]. Inthis regard, accretion-powered pulsars could be promisingsources due to the expected asymmetry of the accretionflow near the stellar surface. However the required mod-eling for accreting systems have not been easy because oftheir complicated dynamics. Besides, none of the pulsarspresented in table 3 are known to be in accreting sys-tems [101]. Second, we have assumed a fixed 1.4M� neu-tron star, while h0 depends on the neutron star mass andgets much smaller for massive neutron stars (see fig. 9).Third, we have assumed a very simple approximate for-mula for the maximum quadrupole moment, and ignoredthe effects that may appear from the properties of thecrust, whose uncertainties are likely much larger than thesymmetry energy effect that we found. Last, distance es-timates that are based on dispersion measurement couldalso be off by a factor of 2–3 [86]. Obviously, much workremains to be done in both the observational and theo-retical front, in order to extract information on the den-sity dependence of the nuclear symmetry energy using thismethod.

3.3 Neutron star oscillations

In a similar way that helioseismology studies the internalstructure and dynamics of our Sun, neutron star seismol-ogy can be used to obtain information on various proper-ties of neutron stars, and therefore to extract the EOS ofneutron-rich matter [102]. Depending on their character-istics, neutron-star oscillation modes are usually dividedinto toroidal and spherical modes. Further, based on thenature of the restoring force these modes are classified asfollows:

1) f undamental modes, or f -modes, associated with theglobal oscillation of the fluid, whose frequencies are inthe range of 1–10 kHz;

2) gravity modes, or g-modes associated with the fluidbuoyancy, whose frequencies are in the range of 2–100Hz;

3) pressure modes, or p-modes, associated with pressuregradient and whose frequencies lie in the range of fewkHz;

4) rotational modes (also known as Rossby modes), orr-modes, associated with the Coriolis force whichacts as a restoring force along the surface, and whosefrequency depends on the stars rotational frequency;

5) purely general-relativistic gravitational wave modes,or w-modes, associated with the curvature of space-time and typically have a very high frequency of above7 kHz.

There has recently been considerable interest and effortin extracting information on the density dependence ofthe symmetry energy from analyzing various neutron starobservations associated with oscillation modes. In partic-ular, it was shown in refs. [97, 103–105] that by identify-ing quasiperiodic oscillation (QPO) following giant mag-netar flares in soft gamma repeaters with the torsionaloscillations of the crust it may be possible to extract thedensity dependence of the symmetry energy (see also thecontributions to this topical issue by Iida and Oyamatsu).For example, the authors of ref. [103] have calculated thefrequency of shear oscillations of the neutron star crust,and showed that they depend sensitively on the slope ofthe symmetry energy at saturation. By using a consistenttreatment of the EOS of crust and core, but an approx-imate method for the description of the torsional modesin a work involving two of us [97] we showed that one canassociate the observed QPO frequencies with the funda-mental shear mode, if the density slope of the symmetryenergy is L < 65MeV. Soon after, using a more sophis-ticated calculation for the crustal frequencies, but an in-consistent EOS of the crust and core, ref. [104] arrived ata conclusion that the value of density slope should lie inthe range of 100 < L < 130MeV. In their later analysis,the authors of ref. [105] also reported a more conservativerange of L ≥ 47.4MeV, depending on the identification ofthe fundamental crust modes. These results suggest thatsuch oscillations could be a powerful probe of the symme-try energy at saturation density, but that it is importantto incorporate a sophisticated model of the crust, consis-tent calculations of the EOS of crust and core and theeffects of the high magnetic fields into the analysis.

Due to their purely general-relativistic nature, w-modes have also become the focal point of many inves-tigations [106, 107]. In particular, it was shown that thedensity dependence of the nuclear symmetry energy has aclear imprint on both the frequency and the damping timeof the axial w-modes [106]. Although the w-mode frequen-cies are outside the bandwidth of the current gravitationalwave detectors, major upgrades will be completed over thecoming years that will significantly improve sensitivity re-quired to detect gravitational waves over a much broaderband [84,85].

In this contribution we will concentrate on the r-mode of the oscillation, whose study has increased sig-nificant attention soon after their first relativistic calcula-tions [108,109].

3.3.1 The r-mode instability in neutron stars

Theoretically, rotating neutron stars cannot have a spinfrequency larger then their Kepler frequency, ΩK. This isan absolute upper limit on the stars rotation rate abovewhich the matter gets ejected from the star’s equator.


However, the available observational data suggests thatthe spin-up of neutron stars from accretion is limited tofrequencies much lower Ω/ΩK < 0.1. One possible expla-nation is the Chandrasekhar-Friedmann-Schutz (CFS) in-stability of the r-mode oscillations. This instability mightplay an important role in generating gravitational wavesfrom the accretion-powered millisecond pulsars in low-mass X-ray binaries (LMXBs). The CFS instability setsan upper limit on the rotation frequency, above whichr-modes are unstable to gravitational radiation. The r-mode instability window depends on the competition be-tween the gravitational radiation and the viscous dissipa-tion timescales, where gravitational radiation makes ther-mode amplitude grow and viscosity in the fluid dampsthe amplitude, stabilizing the mode. It was first shown inrefs. [108, 110] that for all slowly rotating neutron stars,gravitational radiation timescales exceed the one due toviscous damping. In what follows we will use the same ap-proach as discussed in refs. [111–114]. By assuming thatthe crust of the neutrons star is perfectly rigid, one canfind an upper limit on the instability window. This is be-cause the viscous boundary layer between the fluid coreand the rigid crust is maximally dissipative. In the real-istic case of more elastic crust, the dissipation from thecore-crust boundary decreases and therefore widens theinstability window. In a recent work involving some ofus [111], we studied the sensitivity of the r-mode insta-bility to the density dependence of the symmetry energy.In particular, by employing a simple model of a neutronstar with a perfectly rigid crust constructed with a set ofcrust and core EOS that span the range of nuclear experi-mental uncertainly in the symmetry energy, we found thatEOSs characterized by the density slope L of smaller than65MeV are more consistent with the observed frequenciesin LMXBs. In this work the electron-electron scatteringwas considered as the main dissipative mechanism. How-ever, considering a different approach in which the vis-cous dissipation was assumed to operate throughout thewhole core of the star instead of at the crust-core boundarylayer, and using both microscopic and phenomenologicalapproaches to the nuclear EOS, ref. [115] concluded thatobservational data seem to favor values of density slopelarger than 50MeV. We should make the caveat that bothstudies use greatly simplified models of the instability, andrely on an interpretation of observations that is only oneof several possible.

According to ref. [110], the gravitational radiationtimescale can be evaluated using the following expression:

1τGR

=32πGΩ2l+2

c2l+5

(l − 1)2l

[(2l + 1)!!]2

(l + 2l + 1

)2l+2

×∫ Rt

0

Er2l+2dr, (20)

where Ω is the stellar angular frequency and E ≡ E(r) isthe energy density profile of the star. The viscous damp-ing timescale due to viscous dissipation at the core-crustboundary layer assuming a perfectly rigid crust and fluid

1.0 1.2 1.4 1.6 1.8 2.00

2

4

6

8

10

(b)

τ GR (s

)

M (MSun)

(a)

1.0 1.2 1.4 1.6 1.8 2.015

20

25

30

35

40 IU-FSU SkIU-FSU L = 40 MeV L = 60 MeV

τee vv (s

)

Ω = ΩK; T = 108 K

Fig. 10. (Color online) Timescales for the (a) gravitationalradiation driven r-mode instability and (b) corresponding vis-cous dissipation due to the electron-electron scattering at thecore-crust interface as a function of stellar mass.

core can be calculated using [114]

τv =2l+1(l + 1)!

l(2l + 1)!!IlcR2l+2t

√EtΩηt

∫ Rt

0

Er2l+2dr. (21)

where the integral

Il =∫ π

0

sin2l−1 θ(1+cos θ)2√

| cos θ − 1/(l + 1)|dθ, (22)

and the subscript “t” is used to identify quantities atthe core-crust interface. Here we only consider quadrupolemodes of l = 2, for which I2 ≈ 0.80411. For hot neutronstars with temperature above T = 109 K, it is expectedthat the neutron-neutron scattering becomes the domi-nant dissipation mechanism, and its viscosity is expressedby

ηnn = knnρ9/4T−2, (23)

whereknn = 347 g−

54 cm

234 K2 s−1 (24)

and ρ is the mass density. As the temperature drops be-low about 109 K, the neutron star shear viscosity is thendominated by the electron-electron scattering, which alsodepends on density and temperature,

ηee = keeρ2T−2, (25)

wherekee = 6.0 × 106 g−1 cm5 K2 s−1. (26)

The critical rotation frequency Ωc is then defined whenτGR = τv. In this work we do not consider new-born stars,whose temperatures are usually above T > 109 K and forwhich the bulk viscosity would be the dominant dissipa-tion mechanism.

In fig. 10 the timescales for the gravitational radia-tion (left window) and the shear viscosity (right window),which dissipates the r-mode instability are shown as a


1 10

500

600

700

800

900

1000

1100

Instability Region

Stable r-modes

1.9 MSun

f (H

z)

T (108 K)

1.4 MSun

1 10

500

600

700

800

900

1000

1100 IU-FSU SkIU-FSU L = 40 MeV L = 60 MeV LMXB sLXMB

f (H

z)

Fig. 11. (Color online) The r-mode instability lines for a1.4 solar mass (left windows) and a 1.9 solar mass neutronstars are shown for various EOSs discussed in the text. Alsothe location of the observed low mass X-ray binaries (LMXBs)and short recurrence time LMXBs (sLMXB) are shown [116].

function of stellar mass. The stellar structure is modeledwith a rigid crust [111] for the adopted EOSs, and theshear viscosity of the boundary layer is taken to be domi-nated by electron-electron scattering. Both timescales de-pend on the stellar mass with gravitational timescale beinga decreasing function of the neutron star mass, whereasthe viscous damping timescale is an increasing function ofthe stellar mass. Although both the gravitational radiationtimescale and the viscous damping timescale seems to bemore sensitive to the density dependence of the symme-try energy at saturation, an apparent sensitivity emergesfor the high-density component of the symmetry energyin the viscous damping timescale only. Indeed, we shouldnote that these timescales strongly depend on the core-to-crust transition properties, which occurs at about half sat-uration density. The sensitivity to the high density com-ponent of the symmetry energy only enters through theintegrals shown in eqs. (20) and (21) which are evaluatednumerically.

Since the masses of the neutron stars in low-mass X-ray binaries (LMXBs) are not measured as accurately asthose in certain binary NS-NS or NS-White Dwarf (WD)systems, we explore the position of LMXBs in the r-modeinstability window, defined as the region in frequency-temperature space above the critical frequency, in bothcanonical and massive neutron stars. The r-mode insta-bility lines for neutron stars of 1.4M� and 1.9M� aredisplayed in the left and right windows of fig. 11, respec-tively. The core temperatures T of the LMXBs are de-rived from their observed accretion luminosity assumingthe cooling is either dominated by the modified Urca pro-cess for normal nucleons (left stars) or by the modifiedUrca process for neutrons being superfluid and protonsbeing superconducting (right stars) in the core [116]. Itis shown that for a canonical neutron star, all consideredLMXBs lie outside the instability window and is consistentwith the observation that no r-mode is currently excited

in LMXBs, due to the shortness in duration of the unsta-ble r-mode activity [117, 118]. However, for the massiveneutron stars the instability window shifts lower in fre-quencies as shown in the case of a 1.9 solar mass neutronstar. In particular, the EOS with L = 60MeV predictsan unstable point that is at the margin of the observa-tional result. Thus, if the observed stars are very massive—i.e. more massive then 1.9 M�— some of them mayfall within the instability region constrained by the EOSs.Considering that LMXBs should fall below the instabilitywindow and assuming that current observational interpre-tation is correct, then within our simple model one canconclude from fig. 11 that one of the following must hold:a) stars in LMXBs are not so massive; and either b) thehigh-density component of the symmetry energy is soft, orc) the density slope of the symmetry energy at saturationis L < 60MeV. Suppose that the short recurrence timeLMXBs (sLMXBs) presented in fig. 11 are identified tobe massive, and the density slope of the symmetry energyL constrained by terrestrial experiments is equal or largerthen 60MeV, then one would conclude that the symmetryenergy must be soft at higher densities.

4 Conclusions

In this survey we have examined the sensitivity of vari-ous neutron star-sourced gravitational wave signals to thedensity dependence of symmetry energy, with a specialemphasis on the high density behavior of the symmetryenergy. Specifically, we have addressed the sensitivity ofgravitational wave signals from tidally polarized neutronstars, accretion-induced neutron star ellipticities, and r-mode oscillations to the EOSs of neutron-rich matter.

To study this sensitivity we have used various EOSsfor neutron-rich nucleonic matter satisfying the latest con-straints from terrestrial nuclear experiments, state-of-the-art nuclear many-body calculations for EOS of PNM, andastrophysical observation. We have used this same set ofEOSs to test each of the different GW sources, so thattheir relative sensitivities to dense nuclear matter proper-ties may be consistently compared.

We have found that among various gravitational wavesignatures the tidal polarizability of neutron stars in coa-lescing binaries is particularly sensitive to the high-densitybehavior of the nuclear symmetry energy. Moreover, tidalpolarizability is relatively insensitive to variations of theEOS of SNM and the density dependence of the symmetryenergy around saturation density within their remainingexperimental and model uncertainty ranges.

Next, we calculated the gravitational wave strain am-plitude from neutron stars elliptically deformed by crustalmountains. The exact calculation of the neutron starquadrupole moment (and therefore the gravitational wavestrain amplitude) requires a detailed knowledge of theneutron star crust. Within a simple approximation, ourestimations show that it may be possible to disentanglethe effect of the high-density component of the symme-try energy from its density dependence around saturationdensity for the very massive neutron stars only. Much work


needs to be done to completely understand the physicsof neutron star crust, in particular its ability to supportlarge ellipticities, but it is clear that observations of gravi-tational waves from accreting neutron stars is a promisingavenue towards constraining the EOS of neutron-rich nu-cleonic matter.

Lastly, we have explored the dependence of the r-modeinstability window on the high-density component of thesymmetry energy. We have analyzed and confirmed theprevious findings that the instability window is mostlysensitive to the EOS at around the crust-core transitiondensity, and therefore to the density dependence of thesymmetry energy around saturation density. The knowl-edge of high-density behavior of the symmetry energy be-comes very important, especially when one combines fu-ture stringent experimental constraints on the value of Lwith existing and future observations of LMXBs. Withinthis simple model, we have found that EOSs character-ized by L value smaller then 60MeV are consistent withthe observations. In particular, if large values of L are fa-vored from forthcoming terrestrial experiments, then mod-els with stiff symmetry energy at high densities would beruled out.

Future measurements of the gravitational wave signalswill therefore be utterly important in constraining strin-gently the high-density behavior of nuclear symmetry en-ergy, and thus the nature of dense neutron-rich nucleonicmatter.

This work was supported in part by the National Aeronauticsand Space Administration under Grant No. NNX11AC41G is-sued through the Science Mission Directorate, and the NationalScience Foundation under Grant No. PHY-1068022.

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Probing the high-density behavior of symmetry energy with gravitational waves