Constraints on Neutron Star Sttructure and Equation of State from GW170817

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1 Constraints on Neutron Star Sttructure and Equation of State from GW J. M. Lattimer Department of Physics & Astronomy Stony Brook University March 12, 2018 INT-JINA GW March, 2018

2 GW Source Properties 90% confidence intervals D = Mpc Chirp mass M = M m 1 = M m 2 = M q = m 2 = m 1 The binary tidal deformability Λ < 800

3 Probable Black Hole Formation in GW The GRB suggests a black hole formed within 1.75 s. Large ejected mass estimates imply any black hole formation was not prompt, but delayed by tenths of a second because a substantial disc wind was necessary. Most of the ejecta is inferred to have very high opacity, suggesting synthesis of nuclides between the 2nd and 3rd r-process peak. This implies low electron fractions in most of the ejecta, incompatible with long-term (> 0.3 s) neutrino absorption and a long-lived neutron star. A long-lived but metastable neutron star supported by high rotation would pump large amounts of spin-down energy into the remnant, incompatible with the weak GRB and inferred moderate remant kinetic energy. Simulations show that there was too much angular momentum initially in the remnant for a uniformly-rotating star; it was differentially rotating.

4 Maximum Mass Constraint Pulsar observations imply that slowly rotating neutron stars have a maximum mass M max > 2M. A uniformly rotating star has M max,u M max. Supramassive stars, with M max M M max,u, are metastable but have long t >> 0.1 s lifetimes. A differentially-rotating star likely has M max,d 1.5M max. Hypermassive stars, with M max,u < M < M max,d, are metastable with short t 0.1 s lifetimes. The chirp mass of GW170817, M = M, means the total inspiralling mass M tot = m 1 + m 2 is between 2.72M (q = m 2 /m 1 = 1) and 2.78M (q = 0.7). Corrections for gravitational binding energy and mass loss suggest that 2.28M < M rem < 2.53M. To not initially be stabilized by uniform rotation implies M max < M rem /1.17 < 2.16M.

5 The Effect of Tides credit: Jocelyn Read δφ t = 117 (1+q) q 2 ( πfgw ) GM c 5/3 Λ 3 77

6 Tidal Deformability Tidal deformability λ is the ratio between the induced dipole moment Q ij and the external tidal field E ij, Q ij λe ij,. k 2 is the dimensionless Love number. It is convenient to work with the dimensionless λ λ = λc 10 G 4 M k 2 ( Rc 2 GM For a binary neutron star, the relevant quantity is (q = m 2 /m 1 ) Λ = ) 5 (1 + 12q) λ 1 + (12 + q)q 4 λ2 (1 + q) 5.

7 When We Know What Damour, Nagar and Villaiin (2012) q7 There are also spin-spin and spin-orbit contributions to δφ. For spins aligned with L, they act oppositely to δφ T. In a post-newtonian expansion, δφ S is characterized by a single spin parameter β, primarily determined around 50 Hz.

8 LIGO/VIRGO Parameter Determination Although there are 11 free wave-form parameters to post- Newtonian order, LIGO/VIRGO used 13 to fit their data: Sky location (2) Distance (1) Inclination (1) Coalescence time (1) Coalescence phase (1) Polarization (1) Component masses (2) Spin parameters (2) Tidal parameters (2)

9 GW Tidal Deformability Constraints LIGO/VIRGO (2017)

10 Piecewise Polytropic Equations of State For many reasons, it s believed neutron stars have hadronic crusts; the EOS is well-determined below n 0 0.5n s. n 0 = n s /2.7, p 0 = MeV fm 3, ε 0 = MeV fm 3. Read et al. found that M R is well-approximated with an EOS above n 0 containing as few as 3 polytropic segments. Read et al. found optimal upper boundaries (n 1, n 2, and n 3 = 1.85n s, 3.7n s, and 7.4n s ) globally fit wide varieties of hadronic EOSs, leaving just 3 EOS parameters: p 1, p 2, and p 3. Neutron matter theory, nuclear experiment, and the unitary gas suggest that 8.4 MeV fm 3 < p 1 < 20 MeV fm 3, but we extend the upper limit to 30 MeV fm 3. These limits imply 32 < S v /MeV < 38 and 39 < L/MeV < 85. The parameters p 2 and p 3 are limited from above by causality and below by a maximum mass 1.9M < M max < 2.4M. The parameters p 1, p 2 and p 3 are uniformly sampled.

11 M < M max. causality violated when

12 The Radius-Pressure-M max Correlations upper limit

13 M R and EOS Constraints p 1 < 20 MeV fm 3 ε s

14 Dimensionless Tidal Deformability GW

15 Dimensionless Tidal Deformability 1.4

16 Using the λ β 6 Correlation Given that k 2 β 1 it is inevitable that λ aβ 6. In the GW mass range, 1.1 < M/M < 1.6, piecewise polytropes give a = ± Furthermore, in this mass range, R is insensitive to M. As long as M max > 2M, R = R 1.6 R 1.1 < 0.46 km, < R >= 0.07 km and < R 2 > = 0.11 km (c 2 /G)dR/dM 0.261, and < dr/dm >= 0.134G/c 2. With the assumptions λ = aβ 6 and R M = R 1.4, one finds Λ = 16a ( R1.4 c 2 ) 6 q 8/5 13 GM (1 + q) 26/5 (12 11q + 12q2 ). This is remarkably insensitive to q: Λ q = 16a ( R1.4 c 2 ) 6 (1 q)q 3/5 65 GM (1 + q) 31/5 (96 263q + 96q2 ), which vanishes when q = 1. Λ(q = 0.7)/ Λ(q = 1) = 1.02.

17 Dimensionless Binary Tidal Deformability GW J. M. Lattimer Constraints on Neutron Star Sttructure and Equation of State

18 Dimensionless Binary Tidal Deformability

19 Modified M R and EOS Constraints p 1 < 20 MeV fm 3 ε s

20 Tidal Deformabilities

21 The Bias of Uncorrelated Deformabilities Randomly selecting R 1 and R 2 over a range of 3-4 km is similar to randomly selecting λ 1 and λ 2 within their natural ranges of 1000 or 2000 (model B). Instead, randomly selecting λ 1 and utilizing λ 2 = q 6 λ 1 (model A) decreases the 90% confidence contour of Λ by

22 Conclusions from GW A constraint on Λ corresponds to a constraint on the neutron star radius in the GW mass range: R (3.69 ± 0.04) Λ 1/6 (M/M ) km. dr 0.22(d Λ/100) km This correlation between Λ and R is tight because Λ is insensitive to q, a poorly-determined quantity. The quoted constraint Λ < is not justified by the λ 1 λ 2 constraints; its too small by due to λ 1 λ 2 correlations, even considering hybrid (twin) stars.. Spin priors with negative values correspond to spins anti-aligned with L, which is physically improbable except for systems formed by capture. Such priors overestimate Λ. Failure to include the natural correlation between λ 1 and λ 2, and that λ 2 λ 1, overestimates Λ by An upper limit to M max does not constrain neutron star radii.