EOS Constraints From Neutron Stars

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1 EOS Constraints From Neutron Stars J. M. Lattimer Department of Physics & Astronomy Stony Brook University January 17, 2016 Bridging Nuclear and Gravitational Physics: the Dense Matter Equation of State ECT, Trento, Italy 5 9 June, 2017

2 Outline The Unitary Gas Constraint on the Symmetry Energy Neutron Star Universal Relations Using Tidal Deformabilities to Infer the Equation of State New Kinds of Observations

3 vankerkwijk 2010 Romani et al Although simple average mass of w.d. companions is 0.23 M larger, weighted average is 0.04 M smaller Demorest et al Fonseca et al Antoniadis et al Barr et al Champion et al. 2008

4 Causality + GR Limits and the Maximum Mass A lower limit to the maximum mass sets a lower limit to the radius for a given mass. Similarly, a precision upper limit to R sets an upper limit to the maximum mass. R 1.4 > 8.15 km if M max 2.01M. = M R curves for minimally compact EOS = quark stars p ε = c2 s c 2 = s M max < 2.4M if R < 10.3 km. If quark matter exists in the interior, the minimum radii are substantially larger.

5 The Unitary Gas The unitary gas consists of fermions interacting via pairwise zero-range s-wave interaction with an infinite scattering length: ak F 1 0, n = k 3 F /(3π 2 ). As long as the scattering length a >> k 1 F (interparticle spacing), and the range of the interaction R << k 1 F, the properties of the gas are universal in the sense they don t depend on the interaction details. k F is the only length scale, so the unitary gas energy scale is a constant times the Fermi energy ɛ F = ( k F ) 2 /(2m): E UG = ξ 0 E F = ξ 3 2 k 2 F 10m. ξ is known as the Bertsch parameter, measured in cold-atom experiments.

6 Pure neutron matter (PNM) differs from the unitary gas: a 18.5 fm; ak F for n = n s, ξ ξ ak F R 2.7 fm; Rk F 4.5 for n = n s, ξ ξ (Rk F ) +... Repulsive 3-body interactions required to fit energies of light nuclei. Each implies (u = n/n s ) E PNM (u) E UG,0 u 2/3 12.6u 2/3 MeV S(u) = E PNM E(u, Y p = 1/2) S 0 + L (u 1) + [ 3 E UG,0 u 2/3 B + K ] s 18 (u 1)2 + S LB Thus, S(u = 1) = S 0 S0 LB = E UG,0 + B 28.5 MeV ( L(u = 1) = L 0 = 3 u ds ) = 2E UG,0 25.2MeV du u=1

7 S 0 > S LB 0 : ( S = S LB ) u t, ( ds/du = ds LB /du ) u t Allowed u t = S(u) (MeV) u t =1 (S 0 LB,L0 ) 10 S LB (u) Excluded u=n/n 0 Kolomeitsev et al. (2017)

8 Symmetry Parameter Exclusions L (MeV) Excluded TMA NLρδ NL3 STOS,TM1 NLρ LS220 u t =1/2 KVOR FSUgold DBHF TKHS KVR DD2, IUFSU DD,D 3 C,DD-F SFHo GCR LB (S 0,L0 ) u t =1 HS MKVOR SFHx Allowed Kolomeitsev et al. (2017) S 0 (MeV)

9 Symmetry Parameter Correlations Compilations from Dutra et al. (2012, 2014)

10 More Realistic Exclusion Region

11 An Analytic Approximation gives S(u t ) = S LB (u t ), S 0 + L 3 (u t 1) = E UG,0 u 2/3 t ( ) ds = du u t ( ) ds LB du E 0 K n 18 (u t 1) 2 Q n 162 (u t 1) 3 L = 2E UG,0 u 1/3 t K n 3 (u t 1) Q n 18 (u t 1) 2 Assume Q n = 0 and K n = 3L (i.e., K sym 3L K s ). Then S 0 = E UG,0 3u 4/3 (1 + 2ut 2 ) E 0, L = 2E UG,0 t u 4/3 t or after eliminating u t, [ ) ] 3/2 S 0 = L ( 2EUG,0 L E 0 u t

12 Experimental Constraints Isovector Skins and Isobaric Analog States from Danielewicz et al. (2017) Other experimental constraints from Lattimer & Lim (2013) Unitary gas constraints from Kolomeitsev et al. (2016) Experimental and neutron matter constraints are compatible with unitary gas bounds.

13 Piecewise Polytropes Crust EOS is known: n < n 0 = 0.4n s. Read, Lackey, Owen & Friedman (2009) found high-density EOS can be modeled as piecewise polytropes with 3 segments. They found universal break points (n n s, n 2 3.7n s ) optimized fits to a wide family of modeled EOSs. For n 0 < n < n 1, assume neutron matter EOS. Arbitrarily choose n 3 = 7.4n s. For a given p 1 (or Γ 1 ): 0 < Γ 2 < Γ 2c or p 1 < p 2 < p 2c. 0 < Γ 3 < Γ 3c or p 2 < p 3 < p 3c. Minimum values of p 2, p 3 set by M max ; maximum values set by causality. nm n 0, p 0 n1, p 1 n2, p 2 n 3, p

14 Maximum Mass and Causality Constraints p 3 < p 2 Model A:

15 Mass-Radius Constraints from Causality

16 PRE M R Estimates α min ± Özel & Freire (2016)

17 QLMXB M R Estimates Özel & Freire (2016)

18 Combined R fits Assume P(M) is that measured from pulsar timing ( M = 1.4M ). Özel & Freire (2015)

19 Piecewise-Polytrope Average Radius Distributions Assumes P(M) from observed pulsar-timing masses

20 Folding Observations with Piecewise Polytropes

21 Other Studies R 1.4,min = 9.25 km M max = 1.66M R 1.4,min = 10.6 km M max = 1.97M Hebeler, Lattimer, Pethick & Schwenk 2010 Özel & Freire 2016

22 Constraints From Above Kurkela et al. (2014)

23 pqcd + Neutron Matter Constraints Kurkela et al. (2014)

24 Radius - p 1 Correlation

25 Moment of Inertia - Radius Constraints PSR A

26 Binding Energy - Mass Correlations

27 Tidal Deformatibility - Moment of Inertia See also Yagi & Yunes (2013)

28 Tidal Deformatibility - Mass

29 Tidal Deformatibility

30 Binary Tidal Deformability In a neutron star merger, both stars are tidally deformed. The most accurately measured deformability parameter is where Λ = 16 [ λ 1 q 4 (12q + 1) + λ 2 (1 + 12q) ] 13 q = M 1 M 2 < 1 For S/N 20 30, typical measurement accuracies are expected to be (Rodriguez et al. 2014; Wade et al. 2014): M chirp %, Λ 20 25% (M 1 + M 2 ) 1 2%, q 10 15%

31 Binary Tidal Deformatibility - Λ

32 Binary Tidal Deformatibility

33 Binary Tidal Deformatibility 2.0M + 2.0M 1.4M + 1.4M 2.0M + 1.2M 1.2M + 1.2M 1.6M + 1.6M

34 Additional Proposed Radius and Mass Constraints Pulse profiles Hot or cold regions on rotating neutron stars alter pulse shapes: NICER and LOFT will enable X-ray timing and spectroscopy of thermal and non-thermal emissions. Light curve modeling M/R; phase-resolved spectroscopy R. Moment of inertia Spin-orbit coupling of ultra- relativistic binary pulsars (e.g., PSR ) vary i and contribute to ω: I MR 2. Supernova neutrinos Millions of neutrinos detected from a Galactic supernova will measure BE= m B N M, < E ν >, τ ν. QPOs from accreting sources ISCO and crustal oscillations Neutron star Interior Composition ExploreR NASA Large Observatory For x-ray Timing ESA/NASA

35 Is grandpa in the rocket ship? NICER successfully launched aboard a Falcon rocket, June 3. Will be powered up June 13.

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