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Constraints on Compact Star Radii and the Equation of State From Gravitational Waves, Pulsars and Supernovae J. M. Lattimer Department of Physics & Astronomy Stony Brook University September 13, 2016 Collaborators: E. Brown (MSU), C.

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Constraints on Compact Star Radii and the Equation of State From Gravitational Waves, Pulsars and Supernovae

  • J. M. Lattimer

Department of Physics & Astronomy Stony Brook University

September 13, 2016

Collaborators: E. Brown (MSU), C. Drischler (TU Darmstadt), K. Hebeler (OSU), D. Page (UNAM), C.J. Pethick (NORDITA), M. Prakash (Ohio U), A. Steiner (UTK), A. Schwenk (TU Darmstadt)

Compact Stars and Gravitational Waves Yukawa Institute of Theoretical Physics 1 November, 2016

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Outline

◮ The Dense Matter EOS and Neutron Star Structure

◮ General Causality, Maximum Mass and GR Limits ◮ Neutron Matter and the Nuclear Symmetry Energy ◮ Theoretical and Experimental Constraints on the

Symmetry Energy

◮ Extrapolating to High Densities with Piecewise Polytropes ◮ Radius Constraints Without Radius Observations ◮ Universal Relations ◮ Observational Constraints on Radii

◮ Photospheric Radius Expansion Bursts ◮ Thermal Emission from Quiescent Binary Sources ◮ Ultra-Relativistic Neutron Star Binaries ◮ Neutron Star Mergers ◮ Supernova Neutrinos ◮ X-ray Timing of Bursters and Pulsars ◮ Effects of Systematic Uncertainties

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Dany Page UNAM

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Neutron Star Structure

Tolman-Oppenheimer-Volkov equations dp dr = − G c4 (mc2 + 4πpr 3)(ε + p) r(r − 2Gm/c2) dm dr = 4π ε c2r 2

✲

✲

✲

✲maximum mass

p(ε) M(R)

✲ ✛

small range of radii Observations Equation of State ✛

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Extremal Properties of Neutron Stars

◮ The most compact and massive configurations occur

when the low-density equation of state is ”soft” and the high-density equation of state is ”stiff” (Koranda, Stergioulas & Friedman 1997). soft

⇐ =

stiff

= ⇒ p = ε − εo

causal limit

εo is the only EOS parameter The TOV solutions scale with εo

p = 0◦ εo

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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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. R1.4 > 8.15M⊙ if Mmax ≥ 2.01M⊙. Mmax < 2.4M⊙ if R < 10.3 km. ∂p ∂ε = c2

s

c2 = s M − R curves for minimally compact EOS

= ⇒ = ⇒

s s

quark stars If quark matter exists in the interior, the minimum radii are substantially larger.

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Although simple average mass of w.d. companions is 0.23 M⊙ larger, weighted average is 0.04 M⊙ smaller Champion et al. 2008 Demorest et al. 2010 Fonseca et al. 2016 Antoniadis et al. 2013 Barr et al. 2016 Romani et al. 2012 vanKerkwijk 2010

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Mass-Radius Diagram and Theoretical Constraints

GR: R > 2GM/c2 P < ∞ : R > (9/4)GM/c2 causality: R > ∼ 2.9GM/c2 — normal NS — SQS — R∞ =

R

√

1−2GM/Rc2

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Neutron Star Radii and Nuclear Symmetry Energy

◮ Radii are highly correlated with the neutron star matter

pressure around (1 − 2)ns ≃ (0.16 − 0.32) fm−3. (Lattimer & Prakash 2001)

◮ Neutron star matter is nearly purely neutrons, x ∼ 0.04. ◮ Nuclear symmetry energy

S(n) ≡ E(n, x = 0) − E(n, 1/2) E(n, x) ≃ E(n, 1/2) + S2(n)(1 − 2x)2 + . . . S(n) ≃ S2(n) ≃ Sv + L 3 n − ns ns + Ksym 18 n − ns ns 2 . . .

◮ Sv ∼ 32 MeV; L ∼ 50 MeV from nuclear systematics. ◮ Neutron matter energy and pressure at ns:

E(ns, 0) ≃ Sv + E(ns, 1/2) = Sv − B ∼ 16 MeV p(ns, 0) =

  • n2∂E(n, 0)

∂n

  • ns

≃ Lns 3 ∼ 2.7 MeV fm−3

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Theoretical Neutron Matter Calculations

Nuclei provide information for matter up to ns. Theoretical studies, beginning from fitting low-energy neutron scattering data and few-body calculations of light nuclei, can probe higher densities.

◮ Auxiliary Field Diffusion

Quantum Monte Carlo (Gandolfi & Carlson)

◮ Chiral Lagrangian

Expansion (Drischler, Hebeler & Schwenk; Sammarruca et al.)

2

  • b
  • d

y

  • n

l y Gandolfi et al. (2015) Drischler et al. (2015)

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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◮ Chiral Lagrangian calculations of neutron-rich and

symmetric matter (Drischler, Hebeler & Schwenk 2016) strongly suggest that the quadratic interpolation E(n, x) = E(n, 1/2) + S2(n)(1 − 2x)2 is accurate to within ±0.5 MeV for 0 < n < ∼ (5/4)ns for x << 1. In other words S(n) ≃ S2(n) ≡ 1 8 ∂2E(n, x) ∂x2

  • x=1/2

, and E(ns, 0) = Sv + B, p(ns, 0) = Lns/3.

◮ Experimental constraints on saturation properties (Brown

& Schwenk 2014; Kortelainen et al. 2014, Piekarewicz 2010) B = −15.9 ± 0.4 MeV, ns = 0.164 ± 0.007fm−3, K = 240 ± 20 MeV.

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Unitary Gas Bounds

The assumption that the energy of neutron matter should be larger than the unitary gas, i.e., fermions interacting via pairwise short-range s-wave interaction with an infinite scattering length, which shows a universal behavior, produces strong constraints on the symmetry parameters Sv and L (Kolomeitsev et al. 2016).

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Nuclear Experimental Constraints

The liquid droplet model is a useful frame of reference. Its symmetry parameters Sv and Ss are related to Sv and L: Ss Sv ≃ aL roSv

  • 1 + L

6Sv − Ksym 12L + . . .

  • .

◮ Symmetry contribution to the binding energy:

Esym ≃ SvAI 2

  • 1 +

Ss SvA1/3 −1 .

◮ Giant Dipole Resonance (dipole polarizability)

αD ≃ AR2 20Sv

  • 1 + 5

3 Ss SvA1/3

  • .

◮ Neutron Skin Thickness

rnp ≃

  • 3

5 2roI 3 Ss Sv

  • 1 +

Ss SvA1/3 −1 1 + 10 3 Ss SvA1/3

  • .
  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Theoretical and Experimental Constraints

H Chiral Lagrangian G: Quantum Monte Carlo Sv − L constraints from Hebeler et al. (2012) Experimental constraints are compatible with unitary gas bounds. Neutron matter constraints are compatible with experimental constraints.

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Neutron Star Crusts

The evidence is overwhelming that neutron stars have hadronic crusts.

◮ Neutron star cooling, both long

term (ages up to millions of years) and transient (days to years), supports the existence of ∼ 0.5 − 1 km thick crusts with masses ∼ 0.02 − 0.05M⊙.

◮ Pulsar glitches are best

explained by n 1S0 superfluidity, largely confined to the crust, ∆I/I ∼ 0.01 − 0.05. The crust EOS, dominated by relativistic degenerate electrons, is very well understood.

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Piecewise Polytropes

Crust EOS is known: n < n0 = 0.4ns. Read, Lackey, Owen & Friedman (2009) found high-density EOS can be modeled as piecewise polytropes with 3 segments. They found universal break points (n1 ≃ 1.85ns, n2 ≃ 3.7ns) optimized fits to a wide family of modeled EOSs. For n0 < n < n1, assume neutron matter

  • EOS. Arbitrarily choose n3 = 7.4ns.

For a given p1 (or Γ1): 0 < Γ2 < Γ2c or p1 < p2 < p2c. 0 < Γ3 < Γ3c or p2 < p3 < p3c. Minimum values of p2, p3 set by Mmax; maximum values set by causality.

  • n3, p3

nm

  • n0, p0
  • n1, p1
  • n2, p2

13.8 14.8 14.3 15.3 15.7

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Causality

Even if the EOS becomes acausal at high densities, it may not do so in a neutron star. We automatically reject parameter sets which become acausal for n ≤ n2. We consider two model subsets:

◮ Model A: Reject parameter sets that violate causality in

the maximum mass star.

◮ Model B: If a parameter set results in causality being

violated within the maximum mass star, extrapolate to higher densities assuming cs = c.

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Maximum Mass and Causality Constraints

p

3

< p

2

Model A:

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Radius - p1 Correlation

Model A:

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Mass-Radius Constraints from Causality

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Other Studies

Hebeler, Lattimer & Schwenk 2010 ¨ Ozel & Freire 2016

↑

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Piecewise-Polytrope RM=1.4 Distributions

Model A: Model B:

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Piecewise-Polytrope Average Radius Distributions

Assumes P(M) from observed pulsar-timing masses

Model A: Model B:

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Universal Relations

With these assumptions

◮ Hadronic crust with well-known EOS ◮ Neutron matter constraint (pmin < p1 < pmax) ◮ Two piecewise polytropes for p > p1 ◮ Causality is not violated ◮ Mmax is limited from below from pulsar

  • bservations

model A yields interesting bounds to radius and tight correlations among the compactness, moment

  • f inertia, binding energy and tidal deformability.
  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Moment of Inertia - Compactness Correlations

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Moment of Inertia - Radius Constraints

PSR 0737-3039A

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Binding Energy - Compactness Correlations

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Binding Energy - Mass Correlations

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Tidal Deformatibility - Moment of Inertia

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Tidal Deformatibility - Mass

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Tidal Deformatibility

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Tidal Deformatibility

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Binary Tidal Deformability

In a neutron star merger, both stars are tidally deformed. The most accurately measured deformability parameter is ¯ Λ =16 13 ¯ λ1q4(12q + 1) + ¯ λ2(1 + 12q)

  • where

q = M1 M2 < 1 For S/N ≈ 20 − 30, typical measurement accuracies are expected to be (Rodriguez et al. 2014; Wade et al. 2014): ∆Mchirp ∼ 0.01 − 0.02%, ∆¯ Λ ∼ 20 − 25% ∆(M1 + M2) ∼ 1 − 2%, ∆q ∼ 10 − 15%

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Binary Tidal Deformatibility

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Binary Tidal Deformatibility

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Binary Tidal Deformatibility

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Simultaneous Mass/Radius Measurements

◮ Measurements of flux F∞ = (R∞/D)2 σT 4 eff

and color temperature Tc ∝ λ−1

max yield an

apparent angular size (pseudo-BB): R∞/D = (R/D)/

  • 1 − 2GM/Rc2

◮ Observational uncertainties

include distance D, interstellar absorption NH, atmospheric composition Best chances are:

◮ Isolated neutron stars with parallax (atmosphere ??) ◮ Quiescent low-mass X-ray binaries (QLMXBs) in globular

clusters (reliable distances, low B H-atmosperes)

◮ Bursting sources (XRBs) with peak fluxes close to

Eddington limit (gravity balances radiation pressure) FEdd = cGM κD2

  • 1 − 2GM/Rc2
  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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PRE Burst Models

Observational measurements: FEdd,∞ = GMc κD

  • 1 − 2β,

β = GM Rc2 A = F∞ σT 4

∞

= f −4

c

R∞ D 2 Determine parameters: α = FEdd,∞ √ A κD f 4

c c3 = β(1 − 2β)

γ = Af 4

c c3

κFEdd,∞ = R∞ α . Solution: β = 1 4 ± √1 − 8α 4 , α ≤ 1 8 for real solutions.

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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PRE M − R Estimates

¨ Ozel & Freire (2016) 0.164 ± 0.024 0.153 ± 0.039 0.171 ± 0.042 0.164 ± 0.037 0.167 ± 0.045 0.198 ± 0.047

αmin

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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QLMXB M − R Estimates

¨ Ozel & Freire (2016)

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Combined R fits

Assume P(M) is that measured from pulsar timing ( ¯ M = 1.4M⊙). ¨ Ozel & Freire (2015)

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Folding Observations with Piecewise Polytropes

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Bayesian Analyses

Ozel et al. (2015) Steiner et al. (2012)

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Role of Systematic Uncertainties

Systematic uncertainties plague radius measurements.

◮ Assuming uniform surface temperatures leads to

underestimates in radii.

◮ Uncertainties in amounts of interstellar absorption ◮ Atmospheric composition: In quiescent sources, He or C

atmospheres can produce about 50% larger radii than H atmospheres.

◮ Non-spherical geometries: In bursting sources, the use of

the spherically-symmetric Eddington flux formula leads to underestimate of radii.

◮ Disc shadowing: In burst sources, leads to

underprediction of A = f −4

c

(R∞/D)2, overprediction of α ∝ 1/ √ A, and underprediction of R∞ ∝ √α.

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Additional Proposed Radius and Mass Constraints

◮ Pulse profiles Hot or cold regions

  • n 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 0737+3039) vary i and contribute to ˙ ω: I ∝ MR2.

◮ Supernova neutrinos Millions of

neutrinos detected from a Galactic supernova will measure BE= mBN − M, < Eν >, τν.

◮ QPOs from accreting sources

ISCO and crustal oscillations

NASA Neutron star Interior Composition ExploreR Large Observatory For x-ray Timing ESA/NASA

  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F

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Conclusions

◮ Neutron matter calculations and nuclear experiments are

consistent with each other and set reasonably tight constraints on symmetry energy behavior near the nuclear saturation density.

◮ These constraints, together with assumptions that

neutron stars have hadronic crusts and are causal, predict neutron star radii R1.4 in the range 12.0 ± 1.0 km.

◮ Astronomical observations of photospheric radius

expansion X-ray bursts and quiescent sources in globular clusters suggest R1.4 ∼ 10.5 ± 1 km, unless maximum mass and EOS priors are implemented.

◮ Should observations require smaller or larger neutron star

radii, a strong phase transition in extremely neutron-rich matter just above the nuclear saturation density is

  • suggested. Or should GR be modified?
  • J. M. Lattimer

Constraints on Compact Star Radii and the Equation of State F