Reconstructing the neutron-star equation of state from ... · •Several ways to improve the waveform model! • Effective one body waveforms ! • Reproduce BBH waveforms to high

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


• Early inspiral: Evolution depends on chirp mass and symmetric mass ratio ⌘ =

m1m2

(m1 +m2)2

M =(m1m2)3/5

(m1 +m2)1/5


• Late inspiral: EOS-dependent tidal interactions lead to phase shift of ~1 radian up to 400Hz

~3000 GW cycles later at 400Hz


• Last 20-30 cycles: Tidal interactions lead to phase shift of ~1 GW cycle

400Hz up to merger


• 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)


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


!

!

!

• 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)


pn[n(t)] / e�(n,n)/2 (a, b) = 4Re

Z 1

0

a(f)b(f)

Sn(f)df


Evidence

p(~✓|dn) =p(~✓)p(dn|~✓)

p(dn)

~✓

p(d|~✓) / e�(d�m,d�m)/2

m(t; ~✓)



!

!

!

• Use Markov Chain Monte Carlo (MCMC) to sample posterior and marginalize over nuisance parameters


Evidence

p(~✓|dn) =p(~✓)p(dn|~✓)

p(dn)

p(M, ⌘, ⇤|dn) =Z

p(~✓|dn)d~✓nuisance


68% Credible region 95% 99.7%

3-detector LIGO-Virgo network with network SNR=20 Parameters estimated with LALInferenceMCMC


68% Credible region 95% 99.7%

3-detector LIGO-Virgo network with network SNR=20 Parameters estimated with LALInferenceMCMC


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}


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



vs =p

dp/d✏ < c

1.93M�


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




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

Documents

Reconstructing the neutron-star equation of state from ... · •Several ways to improve the waveform model! • Effective one body waveforms ! • Reproduce BBH waveforms to high