Compact Stars in the QCD Phase Diagram III (CSQCD III)December 12-15, 2012, Guarujá, SP, Brazilhttp://www.astro.iag.usp.br/~foton/CSQCD3

Neutron Star Matter

Jochen WambachInstitut für KernphysikTechnische Universität DarmstadtSchlossgarten Str. 2 64289 DarmstadtGermanyEmail: wambach@physik.tu-darmstadt.de

1 Introduction

Neutron stars are the densest stars in the universe. They are the remnants of violentexplosions of massive progenitors in type II supernovae after collapse of the iron core.Neutron-star matter is bound by gravity and the central density can reach severaltimes that in interior of a heavy nucleus. The largest masses currently known arethose of the pulsars PSR J1614-2230 and PSR J0348+043 with 1.97 ± 0.04 [1] and2.01 ± 0.04 [2] solar masses respectively. Both are in binaries with a white dwarfcompanion which allows precise measurements of the pulsar mass.

In General Relativity, the maximum mass of a neutron star is determined by theequation of state (EoS), P (�), which relates pressure and energy density. The EoS isdetermined by the composition of the neutron star. A schematic sketch is given inFig. 1.

Figure 1: Schematic view of the composition of a neutron star [3]

1

arX

iv:1

307.

6714

v1 [

nucl

-th]

25

Jul 2

013

While the properties of the outer crust and parts of the inner crust are fairly well un-derstood there remain a varity of interesting questions for the deeper interior. Theseinclude the possible existence of exotic nuclear shapes (pasta phases) at the interfaceof the inner crust and the outer liquid core. Also the nuclear pairing properties inthe neutron-proton liquid and their relation to the (neutrino) cooling rates remainunder debate. One of the most interesting questions, however, relates to the verydense inner core. Since the density can potentially reach values where nucleons startto overlap, one might expect a transition to deconfined quark matter. Whether sucha new state of hadronic matter is realized, depends on many details of the high-density EoS, which are poorly understood at present. Here, the high-mass pulsarsPSR J1614-2230 and PSRJ0348+043 put severe constraints.

In the following, I will discuss two facets of the high-density EoS. The first relatesto properties of the outer liquid core and the role of the symmetry energy. Thesymmetry energy is one of the terms in the Bethe-Weizsäcker mass formula for anucleus of mass number A with Z protons and N neutrons:

Esym = asym(N − Z)2

A; A = Z +N (1)

and determines the change in nuclear binding energy with proton-neutron asymmetry.For homogeneous nuclear matter with number densities nn and np for neutrons andprotons, the corresponding symmetry energy, S(n), specifies the difference in energycompared to the symmetric case (np = nn) as

∆E ' S(n)(nn − np

n

)2; n = nn + np . (2)

As will be detailed below, its density dependence is decisive for the mass-radiusrelation of neutron stars [4] and precise constraints from nuclear physics are highlydesirable.

The second part of the discussion deals with a more speculative issue relating todeconfined quark matter in the inner core of a neutron star. Quantum Chromodynam-ics (QCD) predicts that the (approximate) chiral symmetry of left- and right-handedquarks is spontaneously broken in the vacuum of the strong interaction. This givesrise to a non-vanishing ground-state expectation value 〈0| qq |0〉 of the scalar quark-field bilinear which is called the chiral condensate (CC). Physically, it is a measurefor the non-perturbative generation of a ’constituent’ quark mass of around 300-500MeV. Due to asymptotic freedom, it is expected that the CC vanishes at high densityand chiral symmetry is restored. The quarks loose their consttuent mass and aquiretheir much smaller ’Higgs’ masses. As will be discussed, the transtion to the restoredphase is likely to proceed through a series of spatially inhomogeneous phases withmodulations in the quark density. This situation is similar to the pasta phases at thecrust-core boundary. If such phases were to occur in the inner core of a neutron star

2

they could have interesting consequences for transport properties and the interactionwith the star’s magnetic field.

2 A new constraint on the Symmetry Energy

Uncertainties in the high-density EoS of neutron-star matter are encoded in the den-sity dependence of the symmetry energy, S(n). A compilation of theoretical extrapo-lations from the properties of known nuclei is shown in the left panel of Fig. 2 whichreveals large uncertainties above nuclear saturation density, n0 = 0.16 fm

−3.

ρδ ∗

0

10

20

30

40

50

E sym

[M

eV]

0 0.5 1! / !

0

Shetty et al., Fe+Fe/Ni+NiShetty et al., Fe+Fe/Fe+NiKhoa et al., p(6He,6Li*)n

0

25

50

75

100

0 1 2 3! / !

0

SKM*SkLyaDBHFvar AV18NL3ChPTDD-TWDD-!"

Esym ≈ 31 ρ0

6 ,6 ∗

Esym

the high-density EOS of cold nuclear matter must come from the observation of massive neutronstars.

In contrast, laboratory experiments may play a critical role in constraining the size of neutronstars. This is because neutron-star radii are controlled by the density dependence of thesymmetry energy in the immediate vicinity of nuclear-matter saturation density [32]. Recallthat the symmetry energy represents the energy cost in converting protons into neutrons (orviceversa) and may be viewed as the difference in the energy between pure neutron matter andsymmetric nuclear matter. A particularly critical property of the symmetry energy is its slope atsaturation density—a quantity customarily denoted by L [33]. Unlike symmetric nuclear matter,the slope of the symmetry does not vanish at saturation density. Indeed, L is simple related tothe pressure of pure neutron matter at saturation density. That is,

P0 =1

3ρ0L . (4)

Although the slope of the symmetry energy is not directly observable, it is strongly correlatedto the thickness of the neutron skin of heavy nuclei [34, 35]. Heavy nuclei develop a neutronskin as a consequence of a large neutron excess and a Coulomb barrier that hinders the protondensity at the surface of the nucleus. The thickness of the neutron skin depends sensitively onthe pressure of neutron-rich matter: the greater the pressure the thicker the neutron skin. Andit is exactly this same pressure that supports neutron stars against gravitational collapse. Thusmodels with thicker neutron skins often produce neutron stars with larger radii [27, 36]. Thus,it is possible to study “data-to-data” relations between the neutron-rich skin of a heavy nucleusand the radius of a neutron star. We illustrate these ideas in Fig. 5 where the neutron-skin

does not. Then, we have to conclude that a 3% accuracy inAPV sets modest constraints on L, implying that some ofthe expectations that this measurement will constrain Lprecisely may have to be revised to some extent. To narrowdown L, though demanding more experimental effort, a!1% measurement of APV should be sought ultimately inPREX. Our approach can support it to yield a new accuracynear !!rnp ! 0:02 fm and !L! 10 MeV, well below anyprevious constraint. Moreover, PREX is unique in that thecentral value of !rnp and L follows from a probe largelyfree of strong force uncertainties.

In summary, PREX ought to be instrumental to pave theway for electroweak studies of neutron densities in heavynuclei [9,10,26]. To accurately extract the neutron radiusand skin of 208Pb from the experiment requires a preciseconnection between the parity-violating asymmetry APVand these properties. We investigated parity-violating elec-tron scattering in nuclear models constrained by availablelaboratory data to support this extraction without specificassumptions on the shape of the nucleon densities. Wedemonstrated a linear correlation, universal in the meanfield framework, between APV and!rnp that has very smallscatter. Because of its high quality, it will not spoil theexperimental accuracy even in improved measurements ofAPV. With a 1% measurement of APV it can allow one toconstrain the slope L of the symmetry energy to near anovel 10 MeV level. A mostly model-independent deter-mination of !rnp of

208Pb and L should have enduringimpact on a variety of fields, including atomic paritynonconservation and low-energy tests of the standardmodel [8,9,32].

We thank G. Colò, A. Polls, P. Schuck, and E. Vivesfor valuable discussions, H. Liang for the densities ofthe RHF-PK and PC-PK models, and K. Kumar for infor-mation on PREX kinematics. Work supported by theConsolider Ingenio Programme CPAN CSD2007 00042

and Grants No. FIS2008-01661 from MEC and FEDER,No. 2009SGR-1289 from Generalitat de Catalunya, andNo. N N202 231137 from Polish MNiSW.

[1] Special issue on The Fifth International Conference onExotic Nuclei and Atomic Masses ENAM’08, edited byM. Pfützner [Eur. Phys. J. A 42, 299 (2009)].

[2] G.W. Hoffmann et al., Phys. Rev. C 21, 1488 (1980).[3] J. Zenihiro et al., Phys. Rev. C 82, 044611 (2010).[4] A. Krasznahorkay et al., Nucl. Phys. A731, 224 (2004).[5] B. Kłos et al., Phys. Rev. C 76, 014311 (2007).[6] E. Friedman, Hyperfine Interact. 193, 33 (2009).[7] T.W. Donnelly, J. Dubach, and Ingo Sick, Nucl. Phys.

A503, 589 (1989).[8] D. Vretenar et al., Phys. Rev. C 61, 064307 (2000).[9] C. J. Horowitz, S. J. Pollock, P. A. Souder, and R.

Michaels, Phys. Rev. C 63, 025501 (2001).[10] K. Kumar, P. A. Souder, R. Michaels, and G.M. Urciuoli,

http://hallaweb.jlab.org/parity/prex (see section ‘‘Statusand Plans’’ for latest updates).

[11] M. Centelles, X. Roca-Maza, X. Viñas, and M. Warda,Phys. Rev. C 82, 054314 (2010).

[12] I. Angeli, At. Data Nucl. Data Tables 87, 185 (2004).[13] B. A. Brown, Phys. Rev. Lett. 85, 5296 (2000); S. Typel

and B.A. Brown, Phys. Rev. C 64, 027302 (2001).[14] R. J. Furnstahl, Nucl. Phys. A706, 85 (2002).[15] A.W. Steiner, M. Prakash, J.M. Lattimer, and P. J. Ellis,

Phys. Rep. 411, 325 (2005).[16] B. G. Todd-Rutel and J. Piekarewicz, Phys. Rev. Lett. 95,

122501 (2005).[17] M. Centelles, X. Roca-Maza, X. Viñas, and M. Warda,

Phys. Rev. Lett. 102, 122502 (2009); M. Warda, X. Viñas,X. Roca-Maza, and M. Centelles, Phys. Rev. C 80, 024316(2009).

[18] A. Carbone et al., Phys. Rev. C 81, 041301(R) (2010).[19] L.W. Chen et al., Phys. Rev. C 82, 024321 (2010).[20] B. A. Li, L.W. Chen, and C.M. Ko, Phys. Rep. 464, 113

(2008).[21] M. B. Tsang et al., Phys. Rev. Lett. 102, 122701 (2009).[22] C. J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. 86,

5647 (2001).[23] J. Xu et al., Astrophys. J. 697, 1549 (2009).[24] A.W. Steiner, J.M. Lattimer, and E. F. Brown, Astrophys.

J. 722, 33 (2010).[25] O. Moreno, E. Moya de Guerra, P. Sarriguren, and J.M.

Udı́as, J. Phys. G 37, 064019 (2010).[26] S. Ban, C. J. Horowitz, and R. Michaels, arXiv:1010.3246.[27] N. R. Draper and H. Smith, Applied Regression Analysis

(Wiley, New York, 1998), 3rd ed.[28] K. Hebeler, J.M. Lattimer, C. J. Pethick, and A. Schwenk,

Phys. Rev. Lett. 105, 161102 (2010).[29] A.W. Steiner and A. L. Watts, Phys. Rev. Lett. 103,

181101 (2009).[30] D. H. Wen, B. A. Li, and P. G. Krastev, Phys. Rev. C 80,

025801 (2009).[31] I. Vidaña, C. Providência, A. Polls, and A. Rios, Phys.

Rev. C 80, 045806 (2009).[32] T. Sil et al., Phys. Rev. C 71, 045502 (2005).

v090M

Sk7

HFB

-8S

kP

HFB

-17S

kM*

Ska

Sk-R

sS

k-T4

DD

-ME

2D

D-M

E1

FSU

Gold

DD

-PC

1P

K1.s24

NL3.s25

G2

NL-S

V2

PK

1N

L3N

L3*

NL2

NL1

0 50 100 150 L (MeV)

0.1

0.15

0.2

0.25

0.3

∆r n

p(f

m)

Linear Fit, r = 0.979Nonrelativistic modelsRelativistic models

D1S

D1N

SG

II

Sk-T6

SkX S

Ly5

SLy4

MS

kAM

SL0

SIV

SkS

M*

SkM

P

SkI2SV

G1

TM1

NL-S

HN

L-RA

1

PC

-F1

BC

P

RH

F-PK

O3

Sk-G

s

RH

F-PK

A1

PC

-PK

1

SkI5

FIG. 3 (color online). Neutron skin of 208Pb against slopeof the symmetry energy. The linear fit is !rnp ¼ 0:101þ0:001 47L. A sample test constraint from a 3% accuracy inAPV is drawn.

PRL 106, 252501 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending24 JUNE 2011

252501-4

0.16 0.20 0.24 0.28rskin[208Pb] (fm)

12

12.5

13

13.5

14

R[1.

4] (k

m)

phase transition?

FSUGold

NL3

Figure 5. (Color online) The left-hand panel displays the correlation between the neutron-skinof 208Pb and the slope of the symmetry energy for a variety of nonrelativistic and relativisticmodels [37]. The right-hand panel shows the correlation between the neutron-skin of 208Pb andthe radius of a 1.4 M⊙ neutron star for two relativistic mean-field models.

thickness of 208Pb (∆rnp) is plotted on the left-hand panel against the slope of the symmetryenergy (L) for a variety of nonrelativistic and relativistic models [37]. The correlation betweenthese two quantities is extremely strong (0.979) and indicates that the neutron skin of 208Pbmay be used as a proxy for the determination of a fundamental property of the EOS. Also shown

Figure 2: Left panel: Density dependence of the nuclear symmetry energy as ex-trapolated from the properties of known nuclei [5] (ρ = n). Right panel: Correlationbetween the skin thickness ∆rnp of

208Pb and the slope L of the symmetry energy atnuclear saturation density, n0 [6].

Of special importance is the slope of S(n) at saturation density: L = 3n0(dS/dn)n0 ,which is directly related to the pressure P0 of pure neutron matter at this densitysince L = 3P0/n0 [7] and hence the neutron-star radius. While L is not directlyobservable, one can use its strong correlation with the neutron skin thickness ∆rnpof a heavy nucleus to obtain experimental constraints. This correlation is displayedin the right panel of Fig. 2 as predicted both in non-relativistic- as well relativisticmean-field models [6]. The precise measurement of ∆rnp of

208Pb is the objective ofthe PREX experiment at the Jefferson Laboratory by using parity-violating electronscattering. The current value of ∆rnp = 0.34

+0.15−0.17 fm [8] still suffers from limited

statistics and is to be improved in the future.An alternative to obtain experimental information on ∆rnp of heavy nuclei and

hence L is through their static electric dipole polarizability, αD. The strong correla-

3

tion between αD and ∆rnp (left panel of Fig. 3) has been established in mean-fieldmodels [9, 10]. The nuclear dipole polarizability is defined through the frequency-dependent dipole strength function SD(ω) =

∑N |〈N |D |0〉|2δ(ω − EN) (or equiva-

lently the photo-absorption cross section σabs) as

αD =8πe2

9

∫dω

SD(ω)

ω=

1

2π2e2

∫dω

σabs(ω)

ω2, (3)

where ω denotes the nuclear excitation (photon) energy.Because of the inverse energy weighting, αD sensitively depends on the E1 strength

at low energies. A complete measurement of the dipole response SD(ω) for208Pb has

recently been achieved through inelastic scattering of polarized protons at very for-ward angles at RCNP in Osaka [11]. The extracted dipole strength, being consistentwith σabs above the neutron-emission threshold, has also allowed to uniquely deducethe sub-threshold strength, which is crucial because of the inverse-energy weightingin Eq. 3.

Symmetry energy and pressure of neutron matter

neutron matter band predicts

symmetry energy Sv and

its density dependence L

comparison to experimental

and observational constraints Lattimer, Lim (2012)

neutron matter constraints H: Hebeler et al. (2010) and in prep.

G: Gandolfi et al. (2011)

predicts correlation

but not range of Sv and L

Figure 3: Left panel: Correlation between ∆rnp and the electric dipole polarizabilityαD in

208Pb established in Ref. [9]. Right panel: astrophysical constraints on the(volume) symmetry energy S(n0) and its slope L [12].

Exploiting the tight correlation between αD and ∆rnp (left panel of Fig. 3), themeasured value of αD = 20.1±0.6 fm3/e2 [11] translates into a skin thickness ∆rnp =0.156±0.021 fm, which is much more precise than the current PREX result. It entersprominently into the current contraints on the slope parameter L [12] (right panel ofFig. 3).

4

3 Inhomogeneous phases in the inner core

There have been many speculations whether, in the inner core of a neutron star, thedensity is sufficiently large to induce a transition to a state in which quarks becomedeconfined. In this novel state a variety of new phases have been predicted, mostprominenty chirally restored phases in which quarks loose their constituent mass aswell as ”color-superconducting” phases where quarks of different flavors appear inpaired states.

In most studies of chiral symmetry restoration in high-density quark matter it istacitly assumed that the chiral order parameters of the various phases are uniformin space. On the other hand, they could be spatially modulated [13]. In the mean-time, several studies in QCD-like models such as the Nambu Jona-Lasinio model orthe chiral quark-meson model have revealed that this is indeed a possibility. Mostinvestigations on inhomogeneous chiral phases assume simplified forms for their spa-tial variation. A popular example is the ’chiral density wave’ in which the (complex)chiral order parameter rotates uniformly in space. This is analogous to the so-calledFulde-Ferrel phases in (color-) superconductivity [14].

More general spatial modulations are more difficult to obtain. By embedding exactone-dimensional solutions [15] of QCD-like models in 3d-space [16] it has becomepossible, however, to find the energetically most favorable 1d-modulations (plates) asa function of temperature T and chemical potential µ. The resulting phase diagram,displayed in the left panel of Fig. 4, indicates that the inhomogeneous phase covers the

0

20

40

60

80

100

240 260 280 300 320 340 360

T (

MeV

)

µ (MeV)

0

20

40

60

80

100

120

140

160

180

0 0.5 1 1.5 2 2.5 3

T (

Me

V)

n/n0

homog. broken

inhomog.

restored

Figure 4: Left panel: region of inhomogeneous 1d-chiral phases in the QCD phasediagram [16]. The solid blue line marks the phase boundary for a homogeneous first-order chiral transition. Right panel: chiral phase diagram in terms of number densityn rather than chemical potential [17].

region where a first-order chiral transition would occur for a spatially homogeneoustransition. The latter case features a line of first-oder transitions which ends in a

5

critical point of second order, much like in a liquid-gas transition. Allowing for spatialinhomogeneities this chiral critical point disappears from the phase diagram, leavingonly a ’Lifshitz point’ in which three second-order lines meet [16, 18]. Moving fromlow to high chemical potential, the spatial profile of the condensate changes graduallyfrom a periodic kink solution to a sinusoidal modulation, whose amplitude decreasescontinuously until the chirally restored homogeneous phase is finally reached. Thetemperature dependence of the Lifschitz point and its density dependence (right panelof Fig. 4) are rather insensitve to the assumptions of effective QCD-like models.

Limiting oneself to one-dimensional structures is a strong assumption. Especiallyat lower temperatures, higher-dimensional modulations are to be expected. In fact,it has long been known that 1d-phases are unstable to thermal fluctuations and truelong-range order cannot exist [19]. It is therefore important to also investigate higher-dimensional modulations of the chiral order parameter.

The implemetation of general periodic structures turns out to be computation-ally demanding. Therefore, sofar, only 2d-structures have been looked at, assumingsquare- and hexagonal arrays of rod-like structures (Fig. 5) with sinusoidal variationof chiral condensate (mass function) in the x- and y-direction, i.e

M(x, y) = M cos(Qx) cos(Qy) square

M(x, y) =M

3

[2 cos(Qx) cos

(1√3Qy

)+ cos

(2√3Qy

)]hexagonal (4)

where M denotes the amplitude and Q the wavevector.

-8 -6 -4 -2 0 2 4 6 8-8

-6-4

-2 0

2 4

6 8

-1

0

1

M(x

,y)

/ M

Q x

Q y

M(x

,y)

/ M

-8 -6 -4 -2 0 2 4 6 8-8

-6-4

-2 0

2 4

6 8

-1

0

1

M(x

,y)

/ M

Q x

Q y

M(x

,y)

/ M

Figure 5: Normalized mass functions M(x, y) for two-dimensional spatial modulationsof the chiral condensate. Left panel: square lattice. Right panel: hexagonal lattice.

The results of the numerical minimization of the thermodynamic potential [20]are presented in Fig. 6. For both lattice geometries a sharp onset of the crystallinephase is observed around µ ≈ 310 MeV followed by a smooth approach to the restoredphase through a continuous decrease in amplitude and an increase in wavenumber Q.

6

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

Figure 6: Amplitude M and wave number Q at T = 0 as functions of the chemicalpotential µ of 2d modulations the mass function [20]. Left panel: square lattice.Right panel: hexagonal lattice.

To find the true ground state one has to compare the free energies of the variousphases with each other. The results are displayed in Fig. 7.

One observes that the one-dimensional plate-like solutions lead to the biggest gainin free energy compared to all the other cases. In particular, the two-dimensional rod-like structures turn out to be energetically disfavored with respect to one-dimensionalsolutions throughout the whole inhomogeneous window.

-4

-3

-2

-1

0

310 320 330 340

Ω -

Ωre

st (M

eV

/fm

3)

µ (MeV)

restoredhomogen. broken

jacobi 1dcos 1d

square 2dhexagon 2d

Figure 7: Thermodynamic potential relative to the restored phase for different 1d-and 2d modulations of the chiral condensate at T = 0 [20].

7

4 Summary and Conclusion

In this contribution I have discussed two aspects in the physics of neutron stars. Thefirst dealt with a recent measurement of the electric dipole polarizabitiy of 208Pb fromwhich the neutron skin thickness can be determined rather precisely. This adds animportant nuclear physics contraint to the symmetry energy and its derivative andhence the EoS of neutron matter. The second, more speculative, aspect focussedon possible inhomogeneous chiral phases in the inner core, provided deconfined quarkmatter would exist in this region. The energetically favored phases are acompanied byperiodic density modulations which may have important implications for the transportproperties of the inner core.

Acknowledgements: I thank M. Buballa and S. Carignano for discussions on the secondtopic. This work has been supported in part by the Helmholtz Alliance EMMI andthe Helmholtz International Center HICforFAIR.

References

[1] P. Demorest et al., Nature 467, 1081 (2010).

[2] J. Antoniadis et al., Science 340, 1233232 (2013).

[3] http://en.wikipedia.org/wiki/file:neutron-star-structure.jpg

[4] J. Lattimer and M. Prakash, Astrophys. J. 550, 426 (2001).

[5] C. Fuchs and H.H. Wolter, Eur. J. Phys. A39, 5 (2006).

[6] X. Rocca-Maza et al., Phys. Rev. Lett. 106, 252501 (2011b).

[7] J. Piekarewicz and M. Centelles, Phys. Rev. C79, 054311 (2009).

[8] S. Abrahamyan et al., Phys. Rev. Lett. 108, 112502 (2012).

[9] P.-G. Reinhard and W. Nazarewicz, Phys. Rev. 81, 051303(R) (2010).

[10] J. Piekarewicz, Phys. Rev. C83, 034319 (2011).

[11] A. Tamii et a., Phys. Rev. Lett. 107, 062502 (2011).

[12] J. Lattimer and Y. Lim, arXiv:1203.4286 [nuc-th].

[13] W. Broniowski, Acta Phys. Pol. B Proc. Suppl. 5/3, 631 (2012).

[14] P. Fulde and R.A. Ferrell, Phys. Rev. 135, A550 (1964).

8

http://arxiv.org/abs/1203.4286

[15] O. Schnetz, M. Thies and K. Urlichs, Annals Phys. 321, 2604 (2006).

[16] D. Nickel, Phys. Rev. D80, 074025 (2009).

[17] S. Carignano, D. Nickel and M. Buballa, Phys. Rev. D82, 054009 (2010).

[18] D. Nickel, Phys. Rev. Lett. 103, 072301 (2009).

[19] G. Baym, B. Friman and G. Grinstein, Nucl. Phys. B210, 193 (1982).

[20] S. Carignano and M. Buballa, Phys. Rev. D86, 074018 (2012).

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1 Introduction2 A new constraint on the Symmetry Energy3 Inhomogeneous phases in the inner core4 Summary and Conclusion