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

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1 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

2 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

3 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 v = 8 h (PP-PN) ( )( )+ p 2 (M,, EOS)v 10 i 1 4 ( )( 1 2 ) 13 h + at 100Mpc 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 = t HsL

4 4-parameter EOS fit 37 Tabulated theoretical models 37 2 examples of fits loghp in dyneêcm 2 L nuclear density loghr in gêcm 3 L loghp in dyneêcm 2 L p 1 low density ρ0 variable ρ1 fixed ρ2 fixed Γ 2 Γ loghr in gêcm 3 L Γ 3 p( )= K 1 1, 0 < < 1 K 2 2, 1 < < 2 K 3 3, > 2

5 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 J )

6 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

7 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

8 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

9 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

10 EOS Parameters True values d marginalized PDFs 1 BNS 5 BNS 20 BNS 50 BNS True EOS d marginalized PDFs log(p 1 ) (dyne/cm 2 )

11 EOS Parameters BNS 5 BNS 20 BNS 50 BNS True EOS log(p 1 ) (dyne/cm 2 )

12 EOS Parameters BNS 5 BNS 20 BNS 50 BNS True EOS log(p 1 ) (dyne/cm 2 )

13 EOS Parameters BNS 5 BNS 20 BNS 50 BNS True EOS log(p 1 ) (dyne/cm 2 )

14 EOS function p( ) log(p) (dyn/cm 2 ) % confidence nuc nuc nuc 1 BNS 5 BNS 20 BNS 50 BNS True EOS p/ptrue log( ) (g/cm 3 )

15 EOS function p( ) log(p) (dyn/cm 2 ) % confidence nuc nuc nuc 1 BNS 5 BNS 20 BNS 50 BNS True EOS p/ptrue log( ) (g/cm 3 )

16 EOS function p( ) log(p) (dyn/cm 2 ) % confidence nuc nuc nuc 1 BNS 5 BNS 20 BNS 50 BNS True EOS p/ptrue log( ) (g/cm 3 )

17 EOS function p( ) log(p) (dyn/cm 2 ) % confidence nuc nuc nuc 1 BNS 5 BNS 20 BNS 50 BNS True EOS p/ptrue log( ) (g/cm 3 )

18 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) Simulated masses uniformly distributed in this range (10 36 g cm 2 s 2 ) BNS 5 BNS 20 BNS 50 BNS True EOS M(M ) M(M )

19 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) Simulated masses uniformly distributed in this range (10 36 g cm 2 s 2 ) BNS 5 BNS 20 BNS 50 BNS True EOS M(M ) M(M )

20 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) Simulated masses uniformly distributed in this range (10 36 g cm 2 s 2 ) BNS 5 BNS 20 BNS 50 BNS True EOS M(M ) M(M )

21 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) Simulated masses uniformly distributed in this range (10 36 g cm 2 s 2 ) BNS 5 BNS 20 BNS 50 BNS True EOS M(M ) M(M )

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

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

24 Higher mass NS observations Black-widow pulsars may have particularly high masses PSR B : PSR J : 2.40 ± 0.12M with lower limit 1.66M 2.68 ± 0.14M with lower limit 2.1M

25 Other EOS models

26 Other EOS models

27 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