What can we learn about the neutron-star equation of state from inspiralling binary neutron stars?

What can we learn about the neutron-star equation of state from inspiralling binary neutron stars? Ben Lackey, Les Wade University of Washington, Seattle Washington, 1 July 2014

What can we measure from the waveform? Phase shift during the inspiral from tidal interactions Tidal field E ij of one NS induces quadrupole moment Q ij in other NS Q ij = (EOS,M NS )E ij (EOS,M NS )MNSE 5 ij Increased quadrupole moment leads to more tightly bound system and additional quadrupole radiation

What can we measure from the waveform? Tidal effects first appear at same order as 5PN point-particle terms Leading term is a linear combination of the tidal deformabilities for each NS Results in a phase shift of ~1 cycle up to ISCO depending on the EOS h(f) / M5/6 f 7/6 e i (f) D L apple (f) =2 ft c c 4 + 3 39 128 v = 8 h 5 1 + (PP-PN) (1 + 7 31 2 )( 1 + 2 )+ p 2 (M,, EOS)v 10 i 1 4 (1 + 9 11 2 )( 1 2 ) 13 h + at 100Mpc 3. 10-22 2. 10-22 1. 10-22 -1. 10-22 0-2. 10-22 No tidal, m 1 =1.35, m 2 =1.35 5PN tidal, m 1 =1.35, m 2 =1.35, L 1 =591, L 2 =591-3. 10-22 26.940 26.945 26.950 26.955 26.960 26.965 t HsL

4-parameter EOS fit 37 Tabulated theoretical models 37 2 examples of fits loghp in dyneêcm 2 L 36 35 34 33 nuclear density 32 13.5 1 14.5 15.0 15.5 16.0 loghr in gêcm 3 L loghp in dyneêcm 2 L 36 35 34 33 p 1 low density ρ0 variable ρ1 fixed ρ2 fixed Γ 2 Γ 1 32 13.5 1 14.5 15.0 15.5 16.0 loghr in gêcm 3 L Γ 3 p( )= K 1 1, 0 < < 1 K 2 2, 1 < < 2 K 3 3, > 2

Current EOS constraints Causality: Speed of sound must be less than the speed of light in a stable neutron star v s = dp/d <c Maximum mass: EOS must be able to support the observed star with mass greater than 1.93M (lower limit on J0348+0432)

Estimating EOS parameters from LIGO data Analogue of 2-step procedure described by Steiner, Lattimer, Brown (Astrophys. J. 722, 33) for mass-radius measurements They combined several mass-radius measurements from accreting neutron stars to estimate EOS parameters We will use estimates of M (M,, EOS) inspiral events to estimate EOS parameters from several BNS

Step 1: Estimate M (M,, EOS). Can estimate BNS parameters from Bayes theorem: Prior Likelihood Posterior p( d ~ n, H, I) = p(~ H, I)p(d n, ~ H, I) p(d n H, I) Evidence ~ = {,,,,D L,t c, c, M,, } d n : gravitational wave data from nth BNS system H: waveform model I : prior information about the parameters Posterior calculated with MCMC or estimated with Fisher matrix which assumes Gaussian likelihood and prior Marginalized distribution trivial Z to compute with MCMC or Fisher p(m,, d n, H, I) = p( d ~ n, H, I)d ~ marg

Step 2: Estimate EOS parameters Can estimate EOS parameters from Bayes theorem: Prior Likelihood Posterior p(~x d 1...d N, H, I) = p(~x H, I)p(d 1...d N ~x, H, I) p(d 1...d N H, I) Evidence ~x = {log(p 1 ), 1, 2, 3, M 1, 1,...,M N, N } d 1...d N : gravitational wave data from all N BNS events prior: Flat in EOS parameters except and M max 1.93M Likelihood: p(d 1,...,d N ~x, H, I) = NY n=1 v s = p dp/d apple c Marginalized posterior/quasilikelihood from single BNS event p(m n, n, n d n, H, I) n = (M n, n,eos) Perform MCMC simulation over the 4+2N parameters, then marginalize over the 2N mass parameters

Simulating a population of BNS events We chose the true EOS to be MPA1 Moderate EOS in middle of parameter space R(1.4M ) 12.5km, and M max 2.5M Sampled 50 BNS systems with SNR > 8 Individual masses distributed uniformly in (1.2M, 1.6M ) Sky position and distance distributed uniformly in volume Orientation distributed uniformly on unit sphere then calculated from masses and true EOS

EOS Parameters True values 1 5.0 4.5 3.5 2.5 1.5 1-d marginalized PDFs 1 BNS 5 BNS 20 BNS 50 BNS True EOS 5.0 2-d marginalized PDFs 2 1.0 5.0 3 1.0 33.5 3 34.5 35.0 35.5 log(p 1 ) (dyne/cm 2 ) 1.5 2.5 3.5 4.5 5.0 1 1.0 5.0 2 1.01.52.53.54.55.0 3

EOS Parameters 1 5.0 4.5 3.5 2.5 1.5 1 BNS 5 BNS 20 BNS 50 BNS True EOS 5.0 2 1.0 5.0 3 1.0 33.5 3 34.5 35.0 35.5 log(p 1 ) (dyne/cm 2 ) 1.5 2.5 3.5 4.5 5.0 1 1.0 5.0 2 1.01.52.53.54.55.0 3

EOS function p( ) 38.0 37.0 log(p) (dyn/cm 2 ) 36.0 35.0 3 3 3 95% confidence nuc 1 1.8 nuc 2 3.6 nuc 1 BNS 5 BNS 20 BNS 50 BNS True EOS 3.5 2.5 p/ptrue 1.5 1.0 0.5 0.0 1 14.2 14.4 14.6 14.8 15.0 15.2 15.4 log( ) (g/cm 3 )

NS Radius and Tidal Deformability Del Pozzo et al. found (1.4M ) can be measured to +/-10% with 50 sources but the slope of (M) cannot be measured Fitting the EOS function p( ) instead of (M), and using known EOS information dramatically improves the measurement of (M) R (km) 20.0 18.0 16.0 1 1 10.0 Simulated masses uniformly distributed in this range (10 36 g cm 2 s 2 ) 1 10.0 8.0 6.0 1 BNS 5 BNS 20 BNS 50 BNS True EOS 8.0 6.0 0.0 0.5 1.0 1.5 2.5 M(M ) 0.0 0.0 0.5 1.0 1.5 2.5 M(M )

Fisher matrix versus MCMC Fisher matrix gives comparable results to lalinference_mcmc for loudest signals

Systematic errors in point particle waveform Injected TaylorF2, TaylorT1, TaylorT4 waveforms and used TaylorF2 as template

Higher mass NS observations Black-widow pulsars may have particularly high masses PSR B1957+20: PSR J1311-3430: 2.40 ± 0.12M with lower limit 1.66M 2.68 ± 0.14M with lower limit 2.1M

Other EOS models

Conclusion Detailed EOS information can be found from the inspiral of BNS systems with aligo Results are roughly equivalent to mass-radius observations Systematic errors from uncertainty in point-particle waveform are significant Unrelated to those in mass-radius observations