Tjonnie G. F. Li tgfl[email protected] · 2015-06-18 · 18 June 2015. EQUATION OF STATE INSPIRAL...
Transcript of Tjonnie G. F. Li tgfl[email protected] · 2015-06-18 · 18 June 2015. EQUATION OF STATE INSPIRAL...
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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
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
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]
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
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
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
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
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
EFFECT OF EOS ON GW SIGNALS
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]
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
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
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
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
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
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
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
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
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
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
Fig. 7: Agathos et al. [11]Tjonnie Li (Caltech) GWPAW 2015, Osaka 9
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
EFFECT OF EOS ON GW SIGNALS
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]
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
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]
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
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
Fig. 9: Bernuzzi et al. [6]
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
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
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
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
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
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EQUATION OF STATE INSPIRAL POST MERGER CONCLUSIONS
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
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Appendix References Abstract Acronyms
Thank you
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Appendix References Abstract Acronyms
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
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Appendix References Abstract Acronyms
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
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Appendix References Abstract Acronyms
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
Fig. 14: Bernuzzi et al. [17]
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Appendix References Abstract Acronyms
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
Fig. 15: Agathos et al. [11]Tjonnie Li (Caltech) GWPAW 2015, Osaka 20
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Appendix References Abstract Acronyms
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].
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Appendix References Abstract Acronyms
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].
Tjonnie Li (Caltech) GWPAW 2015, Osaka 22
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Appendix References Abstract Acronyms
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].
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Appendix References Abstract Acronyms
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.
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Appendix References Abstract Acronyms
ACRONYMS I
EOS Equation Of State
GW Gravitational Wave
NS Neutron Star
SNR Signal-to-noise Ratio
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