Phases of cold nuclear matter and the equation of state of neutron stars

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1 Phases of cold nuclear matter and the equation of state of neutron stars Gordon Baym University of Illinois, Urbana New perspectives on Neutron Star Interiors ECT*, Trento 10 October 2017

2 Better understanding of dense nuclear matter: Driven by: - observations of two heavy neutron stars with M 2 M - ongoing observational simultaneous determinations of neutron star masses and radii in low mass x-ray binaries, and by NICER on International Space Station - emerging understanding in QCD of how nuclear matter can turn into deconfined quark matter in the interior - creation of quark gluon plasmas in ultrarelativistic heavy ion collisions - creation of Bose and BCS superfluids in ultracold atoms. New sources of gravitational radiation events from merging of neutron stars with black holes or neutron stars. New window on dense matter. 16 Oct. 2017

3 High mass neutron star, PSR J in neutron star-white dwarf binary Demorest et al., Nature 467, 1081 (2010); E. Fonseca et al., ApJ. 832, 16 (2016). Spin period = 3.15 ms; orbital period = 8.7 day Inclination = 89:17 o ± 0:02 o : edge on M neutron star = 1.928±0.017M ; M white dwarf = ±0.006M (Gravitational) Shapiro delay of light from pulsar when passing the companion white dwarf

4 Second high mass neutron star, PSRJ in neutron star-white dwarf binary Antonidas et al., Science (2013) Spin period = 39 ms; orbital period = 2.46 hours Inclination =40.2 o M neutron star =2.01 ± 0.04M ; M white dwarf = ±0.003M Significant gravitational radiation Ṗ/Ṗ GR = 1.05 ± Myr to coalescence! AAAALigo

5 Possible high mass neutron stars (< 2.7M ) in extreme black widow pulsars J , B , J (neutron star - He star binaries) Romani et al. ( ), Ap. J. Lett., 760:L36 (2012), Ap. J. 804:115R (2015), van Kerkwijk, Breton, &. Kulkarni (B ), Ap.J. 728, 95 (2011), Schroeder & Halpern (J ), Ap. J. 793, 78 (2014). M ns-j1311 ~ M ; M companion ~ 0.01M Uncertainties arise from incomplete modeling of heating of the companions by the neutron stars.

6 Neutron star interior Mass ~ M sun Radius ~ km Temperature ~ K Surface gravity ~10 14 that of Earth Surface binding ~ 1/10 mc 2

7 e - +p n Nuclei before neutron drip n + n : makes nuclei neutron rich _ as electron Fermi energy increases with depth p+ e - + n : not allowed if e - state already occupied Beta equilibrium: µ n = µ p + µ e Fermi seas Shell structure (spin-orbit forces) for very neutron rich nuclei? Do N=50, 82 remain neutron magic numbers? Proton shell structure? Being explored at rare isotope accelerators: RIKEN Rare Ion Beam Facility, and later GSI (MINOS), FRIB, RAON (KoRIA)

8 Modification of shell structure for N >> Z Usual shell closings (N ~ Z) at 20, 28, 50, 82, 126 even No shell effect for Mg(Z=12), Si(14), S(16), Ar(18) at N=20 and 28 Bastin. et al. PRL (2007,... Oxygen has new shell closure at N=16 Otsuka et al PRL (2005) Calcium has new shell closure at N=34 D. Steppenbeck et al. Nature (2013) Spin-orbit forces and hence shell structure modified by tensor and 3-body forces in neutron rich nuclei

9 The liquid interior Neutrons (likely superfluid) ~ 95% Non-relativistic Protons (likely superconducting) ~ 5% Non-relativistic Electrons (normal, T c ~ T f e -137 ) ~ 5% Fully relativistic Eventually muons, hyperons??, quark matter and possible exotica: pion condensation kaon condensation quark droplets Phase transition from crust to liquid at n b 0.7 n fm -3 (mass density ~ 2 X10 14 g/cm 3 ). 10% uncertainty! n fm -3 Uncertainities in nuclear matter liquid: interpolations between pure neutron matter and symmetric nuclear matter.

10 Standard construction of neutron star models 1) Compute energy per nucleon in neutron matter (pure or in beta equilibrium: µ n = µ p + µ e ). Include 2 and 3 body forces between nucleons Akmal, Pandharipande & Ravenhall, Phys. Rev. C58 (1998) 1804 p 0 condensate

11 Neutron star models using static interactions between nucleons E = energy density = r c 2 n b = baryon density P(r) = pressure = n b2 (E/n b )/ n b TOV equation Maximum neutron star mass Mass vs. central density Mass vs. radius Akmal, Pandharipande and Ravenhall, 1998 APR equation of state

12 Neutron stars: cold quark matter Fundamental limitations of eq. of state based on NN interactions alone Accurate for n~ n 0. But for n >> n 0 : -can forces be described with static few-body potentials? -Force range ~ 1/2m p => relative importance of 3 (and higher) body forces ~ n/(2m p ) 3 ~ 0.4n fm -3. -No well defined expansion in terms of 2,3,4,...body forces. -Can one even describe system in terms of well-defined ``asymptotic'' laboratory particles? Early percolation of nucleonic volumes! Role of strangeness hyperons (very model dependent!) How can quark matter give stiff eq. of state, to explain large masses?

13 Learning about dense matter from neutron star observations Masses and radii of neutron stars Binary systems: stiff e.o.s Thermonuclear bursts in X-ray binaries => Mass vs. Radius, strongly constrains eq.of state. NICER to measure M and R directly Glitches: probe n,p superfluidity and crust Cooling of n-stars: search for exotica Measuring equation of state in crust (Andrew Cumming s talk)

14 The equation of state is very stiff Mass stiff e.o.s. softf e.o.s. Central density Softer equation of state => lower maximum mass and higher central density Binary neutron stars ~ 1.4 M : consistent with soft eq. of state PSR J : M neutron star = 1.93 ± 0.02M PSR J : M neutron star = 2.01 ± 0.04M require very stiff equation of state! How possible?

15 Neutron star masses binary neutron stars Õzel & Freire, Annu Rev AA (2016) Hulse-Taylor PSR J : Hulse-Taylor with high mass companions M nstar = ± 0.017M PSR J : M nstar = 2.01 ± 0.04M Galactic black hole masses ns +white dwarf ns +white dwarf Wiktorowicz & Belcynski optically observed w.d companion accreting bursters

16 Measuring masses and radii of neutron stars in thermonuclear bursts in X-ray binaries F. Özel, GB., T. Güver, PRD 82, (2010); J.M. Lattimer and A. W. Steiner, Ap.J, 784, 123 (2014). F. Özel, D. Psaltis, T. Guver, GB, C. Heinke, and S. Guillot, Ap. J. 820, 28:1 (2016). Eddington Luminosity Apparent Radius Time (s) Measurements of apparent surface area, flux at Eddington limit (radiation pressure = gravity), combined with distance to star, constrains M and R.

17 Mass vs. radius determination of neutron stars in burst sources (low mass x-ray binaries) F. Özel, GB., T. Güver, PRD 82, (2010); J.M. Lattimer and A. W. Steiner, Ap.J, 784, 123 (2014). F. Özel, D. Psaltis, T. Guver, GB, C. Heinke, and S. Guillot, Ap. J. 820, 28:1 (2016).

18 1983

19 Modern phase diagram Deconfined quarks and gluons ç Asakawa-Yazaki critical point (1989) Search in RHIC & SPS energy scans. Quarks confined States of color superconductivity diquark BCS pairing 2SC / Color flavor locked (Alford, Rajagopal, Wilczek,...)

20 Crossover at zero net density: see no evidence of phase transition in pressure, entropy, or energy density. Wuppertal-Budapest lattice collaboration WB: S. Borsanyi et al., PLB (2014) HotQCD: A. Bazavov et al., PRD (2014) Lattice gauge theory not yet well implemented for finite baryon density!! Fermion sign problem

21 Crossover at zero net baryon density ç QCD lattice gauge theory -- for finite light quark masses -- predicts crossover from confined phase at lower T to deconfined phase at higher T. Do quarks roam freely in the deconfined phase? If so, they must also roam freely at lower T. Are there really quarks running about freely in this room?

22 No free quarks even above the crossover! In confined region quarks are inside hadrons. Also have quarks and antiquarks in the QCD forces between hadrons. With higher density or temperature, form larger clusters, which percolate at the crossover. In deconfined regime clusters extending across all of space. Percolation of clusters along the density axis, at zero temperature. Quarks can still be bound even if deconfined. n perc ~ 0.34 (3/4p r n3 ) fm -3 r n = nucleon radius n 0 = density of matter inside large nucleus.

23 T Asakawa-Yazaki critical pt. QGP SU(3) C x U(1) B SU(3 )V X SU(3) A Hadronic B A Diquark pairing U(1) B Possible new critical pt. μ B Critical points similar to those in liquid-gas phase diagram (H 2 O) Can go continuously from A to B around the upper critical point. Liquid-gas phase transition. In lower shaded region have BCS pairing of nucleons, of quarks, and possibly other states (meson condensates). Different symmetry structure than at higher T.

24 Phase diagram of ultracold atomic fermion gases: in T and strength of the particle interactions Unitary regime (Feshbach resonance) BEC-BCS crossover. No phase transition through crossover

25 Phase diagram of ultracold atomic fermion gases: in T and strength of the particle interactions

26 Smooth evolution of states in atomic clouds and nuclear matter GB, T.Hatsuda, M.Tachibana, & Yamamoto. J. Phys. G: Nucl. Part. 35, (2008) H. Abuki, GB, T. Hatsuda, & N. Yamamoto,Phys. Rev. D81, (2010) Evolution of Fermi atoms with weakening attraction between atoms: Similarly, as nuclear matter becomes denser can one expect continuous evolution from hadrons (nucleons) to quark pairs (diquarks)? Masuda, Hatsuda, & Takatsuka, Ap. J. (2013) denser à Quark hadron continuity

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28 Have good idea of equation of state at nuclear densities and at high densities. Look at pressure vs. baryon chemical potential P quark (NJL) g V =0 => too soft to give 2M neutron stars nuclear APR In between?? ~ 2 to 8 n 0. Quarks in Nambu-Jona-Lasinio (NJL) model with universal repulsive short-range qq coupling (Kunihiro) μ APR = Akmal, Pandharipande, Ravenhall nucleonic equation of state with nucleonic potentials (2 and 3 body) fit to NN scattering and light nuclei

29 Have good idea of equation of state at nuclear densities and at high densities. Look at pressure vs. baryon chemical potential P interpolate smoothly! (Hatsuda et a!) nuclear APR quark (NJL) Quarks in Nambu-Jona-Lasinio (NJL) model with universal repulsive short-range qq coupling (Kunihiro) g V =0 => too soft to give 2M neutron stars μ APR = Akmal, Pandharipande, Ravenhall nucleonic equation of state with nucleonic potentials (2 and 3 body) fit to NN scattering and light nuclei

30 Quark matter cores in neutron stars Canonical picture: compare calculations of eqs. of state of hadronic matter and quark matter. GB & S.A. Chin (1976) Crossing of thermodynamic potentials => first order phase transition. ex. nuclear matter using 2 & 3 body interactions, vs. perturbative expansion or bag models. E/A ground state (E/A) H (E/A) Q soft 1 st order -- Maxwell construction Assumes hadronic state at high densities not possible when hadrons substantially overlap Allows only quark equations of state lying under hadronic at high density. Soft only and therefore can t support two solar mass stars. n b Typically conclude transition at n~10n nm -- would not be reached even in high mass neutron stars => at most small quark matter cores

31 How can QCD give large mass neutron stars? Pressure P is a continuous function of baryon chemical potential μ P P * μ * μ Stiff equation of state has high pressure for given mass (or energy) density or equivalently low energy density for given pressure. T. Kojo, P. D. Powell, Y. Song, & GB, PR D 91, (2015)

32 Stiffer equations of state given more massive neutron stars, with lower central densities P ρ Green equation of state is stiffer than red. Has larger pressure for given mass density ρ, and has smaller ρ for given pressure P

33 How can QCD give large mass neutron stars? Energy or mass density ε = ρ c 2 = μ n P P μ * n * P * μ * μ ε * slope : smaller for stiffer equation of state

34 Hybrid eqs. of state are intrinsically softer i Continuous eqs. of state can be much stiffer Phase with larger P at given μ thermodynamically preferred P H extrapolated P P Q soft P stiff P Q ground state P H density discontinuity at phase transition ground state μ μ P H Assumes hadronic state at high densities not possible when hadrons substantially overlap Hadrons only at low density and quark matter at high density. In between???

35 Model calculations of neutron star matter within NJL model NJL Lagrangian chiral interactionsl BCS pairing interactions = Kobayashi-Maskawa- t Hooft six quark axial anomaly plus universal repulsive quark-quark vector coupling T. Kunihiro Include u,d, and s quarks K. Masuda, T. Hatsuda, & T. Takatsuka, Ap. J.764, 12 (2013) GB, T. Kojo, T. Hatsuda, T. Takatsuka, & Y. Song arxiv: ROPP pressure g V = G g V /G = baryon density mass density

36 Minimal model: g V = 0 P interpolated quarks NJL soft nuclear APR μ Soft quark equation of state does not allow high mass neutron stars

37 Vector interaction stiffens eq. of state P increase g V g V =0 stiffens eq. of state nuclear APR interpolate μ Shift of pressure in quark phase towards higher μ

38 Vector interaction stiffens eq. of state P unstable region nuclear APR μ Larger g V leads to unphysical thermodynamic instability

39 Restore stability with increased BCS (diquark) pairing interaction, H P NJL H=1.5 G, g v =G Overall shift Nuclear APR NJL H=0, g v =G μ Increased BCS pairing (onset of stronger 2-body correlations) as quark matter comes nearer to becoming confined

40 Sample unified equation of state: HQC17 (= hadron-quark crossover) T. Kojo, T. Hatsuda, GB, et al. HQC17 HQC17 Consistent with eq. of state inferred from M vs. R observations Quark eqs. of state can be stiffer than previously thought: allow for n.s. masses > 2 M, and with substantial quark cores in neutron stars!!!

41 Masses and radii of neutron stars vs. central mass density from integrating TOV equation with HQC17 T. Kojo, P.D. Powell, Y. Song, & GB, PR D91, (2015); GB et al. arxiv: , ROPP (in press) Include stronger corelations between quarks by increasing the 5 effective pairing interaction H between quarks beyond standard NJL H ~ 1.5 G 1.5 Increased vector repulsion between quarks: g V ~ G g v /G= 0 HQC17 HQC17 Mass vs. central baryon density: Mass vs. radius:

42 Summary For 2 n 0 < n B < 7-8 n 0 mattter is intermediate between purely hadronic and purely quark Quark model eqs. of state can be stiffer than previously thought, allowing for neutron star masses > 2 M Interaction parameters of order vacuum values H g v G s vac Much more to do: Uncertainties in nuclear matter equation of state (APR, etc.) Uncertainties in interpolating from nuclear matter to quark matter lead to errors in maximum neutron star masses and radii Uncertainties in the vector coupling and pairing forces; Going beyond the NJL model -- running g v (Fukushima-Kojo) Need to produce finite temperature equation of state ( 50 MeV) for modelling neutron star -- neutron star (or black hole) mergers as sources of gravitational radiation. Masuda et al. PTEPr. 2016, 021D01 Cooling and transport properties???

43 Calculating neutron star tidal deformability with HQC17 Metric tensor with neutron star of mass M and quadrupole moment Q ij E ij = external tidal force Tidal deformability λ = response of Q to E: Dimensionless deformability Λ: = neutron star Schwarzschild radius C. Markakis and M. O Boyle (UIUC) SLy4 Stiffer equation of state => larger star => larger λ HQC17

44 C. Markakis and M. O Boyle (UIUC) HQC17 SLy4 Sekiguchi, QM2017 Lower λ (~ 0.9 x ) for HQC17 => smaller effect of tidal deformation on rise of chirp frequency Chris Pankow s talk Hinderer, Lackey, Lang, & Read, PR D 81, 1 (2010).