Ben Lackey, Leslie Wade
Reconstructing the neutron-star equation of state from
gravitational-wave observations
10 March 2015
Syracuse University
University of Wisconsin-Milwaukee
Physical Review D 91, 043002 (2015)
Second generation gravitational-wave detectors• Will reach design sensitivity around end of decade
• Sensitive to gravitational-waves between ~10 Hz and a few kHzLIGO Hanford
~2019
LIGO Livingston ~2019
Virgo ~2021
LIGO-India ~2022
KAGRA
Stages of BNS coalescence
• Advanced LIGO sensitive to last few minutes of inspiral
• ~104 gravitational-wave cycles
NS NS
Stages of BNS coalescence
• Early inspiral: Evolution depends on chirp mass and symmetric mass ratio ⌘ =
m1m2
(m1 +m2)2
M =(m1m2)3/5
(m1 +m2)1/5
Stages of BNS coalescence
• Late inspiral: EOS-dependent tidal interactions lead to phase shift of ~1 radian up to 400Hz
~3000 GW cycles later at 400Hz
Stages of BNS coalescence
• Last 20-30 cycles: Tidal interactions lead to phase shift of ~1 GW cycle
400Hz up to merger
Stages of BNS coalescence
• Post-merger: Frequencies are a few kHz and depend sensitively on EOS
Hotokezaka et al., arXiv:1307.5888
NR simulation with SACRA code
10 50 100 500 1000 500010-25
10-24
10-23
10-22
10-21
f HHzL
S nHfL
and2Hf»hé HfL»L1ê2
NS-NS EOS HBInitia
l LIGO
AdvancedLIGO
Einstein Telescope
effectively point-particle
tidal effects
AFTER NSNS merger
NS-NS merger
Fourier transform of waveform:
Waveforms from SACRA and WHISKY codes (Credit: Jocelyn Read, arXiv:1306.4065)
Stages of BNS coalescence
Tidal interactions during inspiral• Tidal field of each star induces quadrupole moment
in other star
• Amount of deformation depends on the stiffness of the EOS via the tidal deformability
!
!
• Interaction increases binding energy
• Additional quadrupole moments increase gravitational radiation
NS NS
QijEij
�
Qij = ��(EOS,m)Eij= �⇤(EOS,m)m5Eij
dE
dt= �(1/5)h
...Q
total
ij
...Q
total
ij i
• Post-Newtonian approximation expands solution to Einstein equations in powers of speed of bodies and compactness of the system:
!
• Energy and gravitational-wave luminosity expansions:
!
• Orbital evolution found with energy balance:
• Waveform is then:
x ⌘✓GM⌦
c
3
◆2/3
⇠⇣v
c
⌘2⇠ GM
c
2d
, from GM = ⌦
2d
3
dx
dt
=dE/dt
dE/dx
=�L
dE/dx
d�
dt
⌘ ⌦ =c
3x
3/2
GM
E = �1
2c
2M⌘x [1 + ePP-PN(x; ⌘) + eTidal(x; ⌘,⇤1,⇤2)]
L =32
5
c
5
G
⌘
2x
5 [1 + lPP-PN(x; ⌘) + lTidal(x; ⌘,⇤1,⇤2)]
1PN–4PN 5PN, 6PN
1PN–3.5PN 5PN, 6PN
Tidal interactions during inspiral
h+ + ih⇥ / ⌘M
dLx(t)e2i�(t)
• Tidal parameters encoded in phase evolution of waveform
Tidal interactions during inspiral
h(f) =A(↵, �, ◆, )
dLM5/6f�7/6ei (f)
x = (⇡Mf)2/3 ⇠⇣v
c
⌘2
⇤ =8
13
h(1 + 7⌘ � 31⌘2)(⇤1 + ⇤2) +
p1� 4⌘(1 + 9⌘ � 11⌘2)(⇤1 � ⇤2)
i
5PN 6PNNewtonian
AmplitudePhase
�⇤ =1
2
p1� 4⌘
✓1� 13272
1319⌘ +
8944
1319⌘2◆(⇤1 + ⇤2) +
✓1� 15910
1319⌘ +
32850
1319⌘2 +
3380
1319⌘3◆(⇤1 � ⇤2)
�
(f) = 2⇡ftc � �c �⇡
4+
3
128(⇡Mf)5/3
1 + (PP�PN)(x; ⌘)�
39
2⇤x5 +
✓�3115
64⇤+
6595
364
p1� 4⌘�⇤
◆x
6
�1PN-3.5PN
EOS fit• One-to-one relation between EOS and radius-mass curves
• As well as between EOS and tidal deformability-mass curves
EOS fit• Purely phenomenological EOS with 4 free parameters
• Methods apply to any EOS with free parameters
p(�) =
�⇤
⇥
K1��1 , �0 < � < �1
K2��2 , �1 < � < �2
K3��3 , � > �2
Γ1
Γ2
Γ3
p1
ρ 1 fixed
ρ 2 fixedΓ1
Step 1: Estimate masses and tidal deformability
• Can estimate parameters of each BNS inspiral from Bayes’ Theorem:
!
!
!
•
• : data from nth BNS event
LikelihoodPriorPosterior
Evidence
p(~✓|dn) =p(~✓)p(dn|~✓)
p(dn)
~✓ = {dL,↵, �, , ◆, tc,�c,M, ⌘, ⇤, �⇤}
dn
• Can estimate parameters of each BNS inspiral from Bayes’ Theorem:
!
!
!
• Time series of stationary, Gaussian noise has the distribution
!
!
• Likelihood of observing data d for gravitational wave model with parameters
!
• where (data) = (noise) + (GW signal)
Step 1: Estimate masses and tidal deformability
pn[n(t)] / e�(n,n)/2 (a, b) = 4Re
Z 1
0
a(f)b(f)
Sn(f)df
LikelihoodPriorPosterior
Evidence
p(~✓|dn) =p(~✓)p(dn|~✓)
p(dn)
~✓
p(d|~✓) / e�(d�m,d�m)/2
m(t; ~✓)
Step 1: Estimate masses and tidal deformability
• Can estimate parameters of each BNS inspiral from Bayes’ Theorem:
!
!
!
• Use Markov Chain Monte Carlo (MCMC) to sample posterior and marginalize over nuisance parameters
LikelihoodPriorPosterior
Evidence
p(~✓|dn) =p(~✓)p(dn|~✓)
p(dn)
p(M, ⌘, ⇤|dn) =Z
p(~✓|dn)d~✓nuisance
Step 1: Estimate masses and tidal deformability
68% Credible region 95% 99.7%
3-detector LIGO-Virgo network with network SNR=20 Parameters estimated with LALInferenceMCMC
Step 1: Estimate masses and tidal deformability
68% Credible region 95% 99.7%
3-detector LIGO-Virgo network with network SNR=20 Parameters estimated with LALInferenceMCMC
Step 1: Estimate masses and tidal deformability
68% Credible region 95% 99.7% True EOS
O(10�4)
O(10�2)
O(1)
Step 2: Estimate EOS parameters
• Use Bayes’ theorem again to estimate masses and EOS parameters:
~x = {log(p1),�1,�2,�3,M1, ⌘1, . . . ,MN , ⌘N}
LikelihoodPriorPosterior
Evidence
p(~x|d1 . . . dN ) =p(~x)p(d1 . . . dN |~x)
p(d1 . . . dN )
• Causality: Speed of sound must be less than the speed of light
• Maximum mass: EOS must support observed stars with masses greater than
Step 2: Estimate EOS parameters
• Use Bayes’ theorem again to estimate masses and EOS parameters:
vs =p
dp/d✏ < c
1.93M�
LikelihoodPriorPosterior
Evidence
p(~x|d1 . . . dN ) =p(~x)p(d1 . . . dN |~x)
p(d1 . . . dN )
• Total likelihood is product of likelihoods for each independent event
• Rewritten in terms of the EOS parameters instead of tidal deformability
Step 2: Estimate EOS parameters
• Use Bayes’ theorem again to estimate masses and EOS parameters:
LikelihoodPriorPosterior
Evidence
p(~x|d1 . . . dN ) =p(~x)p(d1 . . . dN |~x)
p(d1 . . . dN )
Marginalized posterior for single eventp(d1, . . . , dN |~x) =
NY
n=1
p(Mn, ⌘n, ⇤n|dn)|⇤n=⇤(Mn,⌘n,EOS)
• EOS parameters found from MCMC simulation for 4+2N parameters by marginalizing over the 2N mass parameters
Simulating a population of BNS events
• Sampled a year of data using the standard “realistic” event rate
• ~40 BNS events/year for single detector with SNR>8
• Masses sampled uniformly in
• Chose MPA1 to be “true” EOS when calculating tidal parameters for these events
• Injected waveforms into simulated noise for the 3-detector LIGO-Virgo network
[1.2M�, 1.6M�]
Results for 1 year of data
Results for 1 year of data
Higher mass NS observations
• Black widow pulsars may have particularly high masses, but large systematic uncertainties
• PSR B1957+20:
• PSR J1311-3430:
• Higher mass NS observations improve the measurability at higher masses
2.40± 0.12M�
2.68± 0.14M�
• Simulated BNS populations where all the masses were fixed at , , or
• Errors are smallest near the masses of the simulated population
• Can still measure NS properties at other masses due to prior constraints on the equation of state
Range of sampled BNS masses
1.0M� 1.4M� 1.8M�
Other EOS models
Systematic errors• Several ways to calculate waveform phase from energy and luminosity
expressions
• Phase difference between 3PN and 3.5PN as big as tidal effect
• Phase difference between TaylorT1 and TaylorT4 as big as tidal effect
100 1000500200 300150 7000
1
2
3
4
5
60.005 0.007 0.01 0.015 0.02 0.03 0.05
f HHzL
Difference
inwavephase
f 3.5,PP-f 3
.5,l
mW
m1=m2=1.4 Mü
l=0.5
l=3
l=9
f3.0,T4-f3.5,T4
f 3.5,T4-f3.5,T1
Hinderer et al. arXiv:0911.3535T1
vs. T
4
3.0PN vs. 3.5PN
Systematic errors!
• Injected TaylorF2, TaylorT1, TaylorT4 waveform models
• Used TaylorF2 as template
• Several ways to improve the waveform model
• Effective one body waveforms
• Reproduce BBH waveforms to high accuracy
• Recent comparisons with BNS simulations are promising
• Numerical simulations are the only solution once NSs are in contact�3.0
�2.5
�2.0
�1.5
�1.0
�0.5
0.0
0.5
1.0EOS : SLy C ' 0.17 T
2 ' 73.545
500 1000 1500 2000u/M
�1.0
�0.5
0.0
0.5
1.0
2200 2300 2400u/M
Sebastiano Bernuzzi et al. arXiv:1412.4553
m1 = m2 = 1.35M�, EOS=SLy
Systematic errors
Conclusions
• The BNS inspiral waveform provides detailed EOS information
• 1 year of data will be sufficient to measure (statistical error):
• Pressure to less than a factor of 2
• Radius to +/- 1 km
• Systematic errors from inexact waveform templates will be primary difficulty in measuring the EOS
• Will be reduced in the near future with improved waveform models
Conclusions
• The BNS inspiral waveform provides detailed EOS information
• 1 year of data will be sufficient to measure (statistical error):
• Pressure to less than a factor of 2
• Radius to +/- 1 km
• Systematic errors from inexact waveform templates will be primary difficulty in measuring the EOS
• Will be reduced in the near future with improved waveform models
Thank you