Tjonnie G. F. Li tgﬂ[email protected] · 2015-06-18 · 18 June 2015. EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS NUCLEAR EQUATION OF STATE ... MS1 MS2 Fig. 4:Hinderer et al

EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS

INFERRING THE NUCLEAR EQUATION OF STATE FROMBINARY NEUTRON STAR MERGERS

Tjonnie G. F. [email protected]

Gravitational Wave Physics and Astronomy WorkshopOsaka, Japan

18 June 2015

mailto:[email protected]


NUCLEAR EQUATION OF STATE

I Behaviour of ultra-densematter highly uncertain

I Complex interplay amongall forces of Nature

I Manifest as relationshipamong pressure, densityand temperature, i.e.equation of state (EOS)

I Observations of neutronstars (NSs) provide a way tostudy the EOS

I Infer by measuring the massand radius (simultaneously) Fig. 1: Özel [1]

Tjonnie Li (Caltech) GWPAW 2015, Osaka 1


CURRENT KNOWLEDGE

I Binary millisecond radio pulsarsystems

I Demorest et al. [2]: EOS mustfacilitate mass up to 2 M

I x-ray binariesI Özel et al. [3]: R = 9− 12 km at

m = 1.4MI Steiner et al. [4]:

R = 10− 13 km at m = 1.4MI Guillot et al. [5]: R = 7− 11 km

assuming R constant Fig. 2: Özel et al. [3]

Model-dependent uncertainties in emission and absorptionmechanisms



EFFECT OF EOS ON GW SIGNALS

I Gravitational waves (GWs) mainly sensitive to density profileI Different model-dependent uncertainties

I Binary NS excellent candidate to study EOS

I Inspiral regime: objects are tidally deformedI EOS-dependent tidal deformability λ(m) = 2/3 k2(m) R5(m)I Modifies the GW phase evolution

I Merger/post-merger: objects are tidally disruptedI Merger remnant GW emission depends on EOS




1.0

0.5

0.0

0.5

1.0

2MΩ

Mω22

|Rh22/(Mν)|R[Rh22/(Mν)]

2000 1000 0 1000 2000(t−tmrg)/M

-30 0 30x

-30

0

30

y

-20 0 20x

-20

0

20

y

-20 0 20x

-20

0

20

y

9

7

5

3log10(ρ)

Fig. 3: Bernuzzi et al. [6]



EARLY STUDIES

I Early studies indicated that asingle loud event is required

I Hinderer et al. [7]: Earlyinspiral (f < 450Hz),signal-to-noise ratio (SNR)ρ > 30

I Damour et al. [8]: Fullinspiral, ρ > 16

I Read et al. [9]: Late inspiral +merger, ρ > 40

0.0 0.5 1.0 1.5 2.0 2.5 3.00

2

4

6

8

10

12

Mass HM

L

ΛH1

036g

cm2s2

L

npeΜ matter only

Adv

. LIG

O

Eins

tein

Tele

scop

e

AP1 AP3

FPSSLy

MPA1

MS1

MS2

Fig. 4: Hinderer et al. [7]

Such loud events are unlikely in Advanced detector era



COMBINING MULTIPLE WEAK EVENTS

Can we learn something from weak events?

I Combine multiple weak eventsI Bayesian study to facilitate combination of eventsI Expand EOS-dependent tidal deformability function λ(m)

λ(m) ≈ c0 + c1

(m−m0

M

)(1)

I Measure λ around cannonical mass of m0 = 1.4M



COMBINING MULTIPLE WEAK EVENTS

0 50 100 150 2000

1

2

3

4

5

c 0[1

0−23s5

]

χ=0Uniform mass distributionUniform mass prior

Injected value SQM3

Injected value H4

Injected value MS1

95% CI SQM3

95% CI H4

95% CI MS1

Fig. 5: Del Pozzo et al. [10] and Agathos et al. [11]Tjonnie Li (Caltech) GWPAW 2015, Osaka 7


INCLUDING SPIN

0 50 100 150 2000

1

2

3

4

5

c 0[1

0−23s5

]

σχ=0.02

Gaussian mass distributionGaussian mass prior

Injected value SQM3

Injected value H4

Injected value MS1

95% CI SQM3

95% CI H4

95% CI MS1

Fig. 6: Agathos et al. [11]Tjonnie Li (Caltech) GWPAW 2015, Osaka 8


EFFECT OF PRIORS

0 50 100 150 2000

1

2

3

4

5

c 0[1

0−23s5

]

σχ=0.02

Gaussian mass distributionUniform mass prior

Injected value SQM3

Injected value H4

Injected value MS1

95% CI SQM3

95% CI H4

95% CI MS1




1.0

0.5

0.0

0.5

1.0

2MΩ

Mω22

|Rh22/(Mν)|R[Rh22/(Mν)]

2000 1000 0 1000 2000(t−tmrg)/M

-30 0 30x

-30

0

30

y

-20 0 20x

-20

0

20

y

-20 0 20x

-20

0

20

y

9

7

5

3log10(ρ)




EOS FROM POST-MERGER SIGNAL

I Stergioulas et al. [12], andBauswein and Janka [13]:Possible to study the EOSthrough the characteristicsof the post-mergerspectrum.

I Relationship among thepeak frequency(associated to m = 2mode) and the radius ofthe NS

I Clark et al. [14] finds thatuseful constraints can beplaced provided the sourceis at 4− 12Mpc

0 1 2 3 4 510

−23

10−22

10−21

f [kHz]

ha

v(2

0 M

pc) 0 5 10 15 20

−1

0

1x 10

−21

h+ a

t 2

0 M

pc

t [ms]

fpeak

10 12 141.5

2

2.5

3

3.5

4

R1.35

[km]

f pe

ak [kH

z]

0.04 0.06 0.08 0.1

(Mtot

/(Rmax

)3)1/2

Fig. 8: Bauswein and Janka [13]



LINKING INSPIRAL TO POST MERGER

1 2 3 4 5f [kHz]

100

101

√5/

(16π

)R|h

22(f

)|/M

MS1b-150100ALF2-140110H4-135135SLy-140120SLy-135135

2.5

3.0

3.5

4.0

4.5

5.0

Mf 2

[×10

2]

Binary Mass M2.4502.5002.5502.6002.650

2.7002.7502.8002.8502.900

EOSMS1bSLyENG2HH4APR4

MS1ALF2MPA1GNH3Γ2

100 200 300 400κT2

2.5

3.0

3.5

4.0

4.5

5.0

Mf 2

[×10

2]

Mass-ratio q1.0001.0771.0801.1541.160

1.1671.2311.2501.2731.500

100 200 300 400κT2

Γth1.6001.750

1.8002.000


I Bernuzzi et al. [6] findsphenomenological relationshipbetween f2 and κT

2 , where

κT2 = 2

(q4

(1 + q)5kA

2

C5A

+q

(1 + q)5kB

2

C5B

)

I But κT2 can be related to λ, where

λ(m) = 2/3 k2(m) R5(m)

I Links EOS information betweeninspiral and post merger



SIMPLE TOY MODEL

102 103

f (Hz)

10−26

10−25

10−24

10−23

10−22|h

(f)|

waveform

f2

Fig. 10: Li et al., in prep.Tjonnie Li (Caltech) GWPAW 2015, Osaka 13


IMPROVEMENTS FROM INCLUDING POST MERGER

0 20 40 60 80 100Sources

10−1

100

101

∆c 0[ 10−

23s5]

Inspiral only

With post merger

Fig. 11: Li et al., in prep.Tjonnie Li (Caltech) GWPAW 2015, Osaka 14


CONCLUDING REMARKS

I Strong EOS constraints possible in Advanced detector eraI Single high-SNR sourceI O(50) low-SNR sources

I Need accurate waveform to mitigate systematic errorsI Effective-One-Body waveforms with spin and tidal effects?

I Possible systematic effects from unknown mass distributionI Improve constraints by including post-merger information

I Need accurate (phenomenological) models of post-merger signal


Appendix References Abstract Acronyms

Thank you



PARAMETERISATION CHOICE

I Lackey and Wade [15]performed similar study withmore physical parameterisation

I Use piecewise polytropes

p(ρ) = KiρΓi (2)

I Assume 4 parameter modelI θ = log p1,Γ1,Γ2,Γ3

I Allows for inclusion of physicalpriors (e.g. thermodynamicalstability)

9

10

11

12

13

14

15

16

R (

km)

1.2M

¯

1.6M

¯

1.93M

¯

0.0 0.5 1.0 1.5 2.0 2.5 3.0M(M¯)

0

1

2

3

4

5

6

7

8

9

λ (

10

36 g

cm

2 s

2)

Loudest 20, 3σ

Loudest 20, 2σ

Loudest 20, 1σ

Fit to MPA1

Fig. 12: Lackey and Wade [15]Tjonnie Li (Caltech) GWPAW 2015, Osaka 17


SYSTEMATIC ERRORS FROM WAVEFORM UNCERTAINTY

0 200 400 600 800 1000Λ

0.000

0.002

0.004

0.006

0.008

0.010

0.012P

rob

abili

tyd

ensi

tym1 = 1.35 M, m2 = 1.35 M

F2 Injection

T1 Injection

T2 Injection

T3 Injection

T4 Injection

Fig. 13: Wade et al. [16]Tjonnie Li (Caltech) GWPAW 2015, Osaka 18


TOWARDS A FAITHFUL WAVEFORM

−1.0

−0.5

0.0

0.5

1.0 SLy135, κT2 ≈ 73.55

<(Rh22)/ν, NR

100 400 800 1200 1600 2000(t− r∗)/M

−2.5

−1.5

−0.5

0.5

∆φEOBNR22

∆AEOBNR22

∆φTT4NR22

NR phase error

2200 2300 2400

Γ2164, κT2 ≈ 75.07

<(Rh22)/ν, TEOBResum

100 400 800 1200 1600(t− r∗)/M

NR merger

TEOBResum merger

TEOBResum LSO

1700 1800

0.06 0.08 0.10 0.12 0.14Mω

20

40

60

80

100

120

140

Qω

MωLSO-TEOBResum

SLy135, κT2 ≈ 73.55 BBH

TT4

TEOBNNLO

TEOBResum

NR




MODEL SELECTION

250 200 150 100 50 0 50 100 150

lnOEOSMS1

0.0

0.2

0.4

0.6

0.8

1.0cu

mula

tive d

istr

ibuti

on

100 sources/catalogue17 catalogues

SQM3

PP

H4



REFERENCES I

[1] F. Özel. “Soft equations of state for neutron-star matter ruled out by EXO 0748 -676”. Nature 441 (June 2006), pp. 1115–1117. eprint:arXiv:astro-ph/0605106.

[2] P. B. Demorest et al. “A two-solar-mass neutron star measured using Shapirodelay”. Nature 467 (Oct. 2010), pp. 1081–1083. arXiv: 1010.5788[astro-ph.HE].

[3] F. Özel et al. “Astrophysical measurement of the equation of state of neutronstar matter”. Phys. Rev. D 82.10, 101301 (Nov. 2010), p. 101301. arXiv:1002.3153 [astro-ph.HE].

[4] A. W. Steiner et al. “The Equation of State from Observed Masses and Radii ofNeutron Stars”. ApJ 722 (Oct. 2010), pp. 33–54. arXiv: 1005.0811[astro-ph.HE].

[5] S. Guillot et al. “Measurement of the Radius of Neutron Stars with HighSignal-to-noise Quiescent Low-mass X-Ray Binaries in Globular Clusters”. ApJ772, 7 (July 2013), p. 7. arXiv: 1302.0023 [astro-ph.HE].

[6] S. Bernuzzi et al. “Towards a description of the complete gravitational wavespectrum of neutron star mergers”. ArXiv e-prints (Apr. 2015). arXiv:1504.01764 [gr-qc].


arXiv:astro-ph/0605106

http://arxiv.org/false/1010.5788








REFERENCES II[7] T. Hinderer et al. “Tidal deformability of neutron stars with realistic equations

of state and their gravitational wave signatures in binary inspiral”. Phys. Rev. D81.12, 123016 (June 2010), p. 123016. arXiv: 0911.3535 [astro-ph.HE].

[8] T. Damour et al. “Measurability of the tidal polarizability of neutron stars inlate-inspiral gravitational-wave signals”. Phys. Rev. D 85.12, 123007 (June 2012),p. 123007. arXiv: 1203.4352 [gr-qc].

[9] J. S. Read et al. “Measuring the neutron star equation of state with gravitationalwave observations”. Phys. Rev. D 79.12, 124033 (June 2009), p. 124033. arXiv:0901.3258 [gr-qc].

[10] W. Del Pozzo et al. “Demonstrating the Feasibility of Probing the Neutron-StarEquation of State with Second-Generation Gravitational-Wave Detectors”.Physical Review Letters 111.7, 071101 (Aug. 2013), p. 071101. arXiv: 1307.8338[gr-qc].

[11] M. Agathos et al. “Constraining the neutron star equation of state withgravitational wave signals from coalescing binary neutron stars”. ArXiv e-prints(Mar. 2015). arXiv: 1503.05405 [gr-qc].

[12] N. Stergioulas et al. “Gravitational waves and non-axisymmetric oscillationmodes in mergers of compact object binaries”. MNRAS 418 (Nov. 2011),pp. 427–436. arXiv: 1105.0368 [gr-qc].










REFERENCES III

[13] A. Bauswein and H.-T. Janka. “Measuring Neutron-Star Properties viaGravitational Waves from Neutron-Star Mergers”. Physical Review Letters 108.1,011101 (Jan. 2012), p. 011101. arXiv: 1106.1616 [astro-ph.SR].

[14] J. Clark et al. “Prospects for high frequency burst searches following binaryneutron star coalescence with advanced gravitational wave detectors”.Phys. Rev. D 90.6, 062004 (Sept. 2014), p. 062004. arXiv: 1406.5444[astro-ph.HE].

[15] B. D. Lackey and L. Wade. “Reconstructing the neutron-star equation of statewith gravitational-wave detectors from a realistic population of inspirallingbinary neutron stars”. Phys. Rev. D 91.4, 043002 (Feb. 2015), p. 043002. arXiv:1410.8866 [gr-qc].

[16] L. Wade et al. “Systematic and statistical errors in a Bayesian approach to theestimation of the neutron-star equation of state using advanced gravitationalwave detectors”. Phys. Rev. D 89.10, 103012 (May 2014), p. 103012. arXiv:1402.5156 [gr-qc].

[17] S. Bernuzzi et al. “Modeling the Dynamics of Tidally Interacting Binary NeutronStars up to the Merger”. Phys. Rev. Lett. 114 (16 2015), p. 161103.








ABSTRACT

Gravitational waves emitted by binary neutron star mergersencode information about the nuclear equation of state. We

present the prospects of Advanced LIGO/Virgo to extract thisinformation. In particular, results from simulations indicatethat one can already distinguish between extreme nuclear

equation of state models within the era of AdvancedLIGO/Virgo. Moreover, we will discuss how these results canbe further improved by including additional information such

as the post-merger behaviour.



ACRONYMS I

EOS Equation Of State

GW Gravitational Wave

NS Neutron Star

SNR Signal-to-noise Ratio


Documents

Tjonnie G. F. Li tgﬂ[email protected] · 2015-06-18 · 18 June 2015. EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS NUCLEAR EQUATION OF STATE ... MS1 MS2 Fig. 4:Hinderer et al