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Prospects of constraining the nuclear equation ofstate with gravitational-wave signals in the Advanced
detector era and beyond
Peter T. H. PangTjonnie G. F. Li
The Chinese University of Hong Kong
20th Feb 2017
P. Pang T. Li Constraining EOS with GW 20th Feb 2017 1 / 27
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Motivation
2 Binary blackhole merger are observed
No BNS in O1 still support 1000 Gpc−3yr−1 ∼ O(10)yr−1 [1, 2]
Observation on BNS is promising
P. Pang T. Li Constraining EOS with GW 20th Feb 2017 2 / 27
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Binary Neutron Stars Waveform
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P. Pang T. Li Constraining EOS with GW 20th Feb 2017 3 / 27
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Tidal Parameters
The inspiral waveform goes ash(f) = A(f)eiΨ(f) = A(f)eiΨPP(f)+iΨTidal(f ;Λi)
The dimensionless tidal deformability Λ:Λ = 2
3k2R5/m5 = λ/m5
Mass weighted dimensionless tidal deformability κT2
κT2 = 3(
q4
(1+q)5 Λ1 + q(1+q)5 Λ2
)q = m1/m2
The tidal deformability λ derivate a lot between EOS [12]
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Matching Estimation
Single source can constraint the EOS strongly when the source isrelatively close [3]
See also: [4, 5, 6, 7, 8]
P. Pang T. Li Constraining EOS with GW 20th Feb 2017 5 / 27
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Bayesian Analysis
Parameter estimation relies on Bayes’ theorem
P (~θ|d,H, I) =P (d|~θ,H, I)P (~θ|H, I)
P (d|H, I)(1)
d is the data
~θ is the parameters of the waveform
H is the model
I is the background information
Markov chain Monte Carlo (MCMC) is used for calculating P (d|~θ,H, I)
P. Pang T. Li Constraining EOS with GW 20th Feb 2017 6 / 27
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Bayesian Analysis
Ingredients for the posterior
The ”inner product”
〈a|b〉 = 4<∫ fhigh
flow
ab∗df
Sn(f)where Sn(f) is the power spectral density (2)
The likelihood
P (d|~θ,H, I) = P (n = d− h|I) where P (n|I) ∝ e(−12〈n|n〉) (3)
The evidence
P (d|H, I) =
∫all space
P (d|~θ,H, I)P (~θ|H, I)d~θ (4)
P. Pang T. Li Constraining EOS with GW 20th Feb 2017 7 / 27
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Bayesian Analysis
Model selection:
P (Hi|d, I)
P (Hj |d, I)=P (d|Hi, I)
P (d|Hj , I)
P (Hi|I)
P (Hj |I)
Oij = Bij
P (Hi|I)
P (Hj |I)
(5)
Oij is the odd ratio
Bij is the Bayes factor
P. Pang T. Li Constraining EOS with GW 20th Feb 2017 8 / 27
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Bayesian Analysis
Multiple detections:
Parameter Estimation
P (~θ|d1, d2, d3, · · · dN ,H, I) =P (dN |~θ,H, I)P (~θ|d1, d2, d3, · · · dN−1,H, I)
P (dN |H, I)
=
N∏i=1
P (~θ|di,H, I)P (~θ|H, I)1−N
(6)Models Selection
P (Hi|d1, d2, d3, · · · dN , I)
P (Hj |d1, d2, d3, · · · dN , I)=P (Hi|I)
P (Hj |I)
N∏l=1
P (dl|Hi, I)
P (dl|Hj , I)(7)
P. Pang T. Li Constraining EOS with GW 20th Feb 2017 9 / 27
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Parameter Estimation
About O(10) source in realistic distance range (100Mpc to 250Mpc)are sufficient for distinguishing EOS [9]
See also: [10, 11]
P. Pang T. Li Constraining EOS with GW 20th Feb 2017 10 / 27
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Parameter Estimation
Reconstruction of EOS base on piecewise polytrope, physical constraintand BNS observation [12]
See also: [13]P. Pang T. Li Constraining EOS with GW 20th Feb 2017 11 / 27
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Binary Neutron Stars Waveform
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P. Pang T. Li Constraining EOS with GW 20th Feb 2017 12 / 27
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Post-merger Phase
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Distinct peaks in the gravitational wave spectrum are observed amongdifferent EOS [14]
See also: [15, 16, 18]P. Pang T. Li Constraining EOS with GW 20th Feb 2017 13 / 27
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High Frequency Burst Searches
Short Time Fourier Transform is used instead of Fourier Tansform
Correlation between detectors
Time-frequency series of waveform is reconstructed
The short time Fourier Transform of a simulated signal [17]
P. Pang T. Li Constraining EOS with GW 20th Feb 2017 14 / 27
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High Frequency Burst Searches
Estimation of fpeak can be used for EOS constraint
Values of fpeak can be inferred for relatively close sources [19]
See also: [20, 21, 22]
P. Pang T. Li Constraining EOS with GW 20th Feb 2017 15 / 27
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Combining Inspiral and Post-merger Waveform
fpeak shows a similar relation with κT2 among difference EOS [14]
fpeak =0.053850
M
1 + 0.00087434κT21 + 0.00455κT2
(8)
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Toy Model of Combined Waveform
fcon
fmerg, h(fmerg)
ξσ
fpeak, AAmplitude
Frequency
b
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Preliminary Results
0 10 20 30 40
Number of Events
1
2
3
4
5
6
7
8
α0
(1036
gcm
2s2
)
Median of α0 agasint Number of Event Stacked of ALF2PM
No PM
Actual value
The inclusion of post-merger stage shows improvement compare to theabsent of it
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Summary and beyond
Summary:
Inspiral
Constrain EOS with close single sourceConstrain EOS with O(10) sources in realistic distance
Post-merger
Reconstruction of waveforms with single close source
Beyond:
Combination of inspiral and post-merger waveform
Shows possibility of improvement
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Reference I
B. P. Abbott et. al. Upper Limits on the Rates of Binary Neutron Starand Neutron-Star-Black-Hole Mergers from Advanced LIGO’s FirstObserving Run AJL, 832, 2 (2016)
J. Abadie et. al. Predictions for the Rates of Compact BinaryCoalescences Observable by Ground-based Gravitational-wave DetectorsClass. Quant. Grav., 27, 17 (2010)
J.Read, L. Baiotti, J. Creighton, J. Friedman, B. Giacomazzo, K.Kyutoku, C. Markakis, L. Rezzolla, M. Shibata, and K. Taniguchi Mattereffects on binary neutron star waveforms Phys. Rev. D 88, 044042 (2013)
J. Read, C. Markakis, M. Shibata, K. Uryu, J. Creighton, and J.FriedmanMeasuring the neutron star equation of state with gravitational waveobservations. Phys. Rev. D 79, 124033 (2009)
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Reference II
T. Hinderer, B. ,R. Lang and J. Read Tidal deformability of neutronstars with realistic equations of state and their gravitational wavesignatures in binary inspiral Phys. Rev. D 81, 123016 (2010)
E. Flanagan and T. Hinderer Constraining neutron-star tidal Lovenumbers with gravitational-wave detectors Phys. Rev. D 77, 021502(R)(2008)
B. Lackey, K. Kyutoku, M. Shibata, P. Brady and J. Friedman Extractingequation of state parameters from black hole-neutron star mergers:Nonspinning black holes Phys. Rev. D 85, 044061 (2012)
T. Damour, A. Nagar and L. Villain Measurability of the tidalpolarizability of neutron stars in late-inspiral gravitational-wave signalsPhys. Rev. D 85, 123007 (2012)
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Reference III
Walter Del Pozzo, Tjonnie G.F. Li, Michalis Agathos, Chris Van DenBroeckDemonstrating the feasibility of probing the neutron star equation of statewith second-generation gravitational wave detectors. PRL 111, 071101(2013). http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.071101
M. Agathos, J. Meidam, W. Del Pozzo, T.G.F. Li, M. Tompitak, J.Veitch, S. Vitale, and C. Van Den Broeck Constraining the neutron starequation of state with gravitational wave signals from coalescing binaryneutron stars Phys. Rev. D 92, 023012 (2015)
P. Kumar, M. Purrer and H. P. Pfeiffer Measuring neutron star tidaldeformability with Advanced LIGO: a Bayesian analysis of neutron star -black hole binary observations arXiv:1610.06155 (2017)
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Reference IV
Benjamin D. Lackey and Leslie WadeReconstructing the neutron-star equation of state with gravitational-wavedetectors from a realistic population of inspiralling binary neutron stars.Phys. Rev. D 91, 043002 (2015)
J. Read,1 B. Lackey, B. Owen and J. Friedman Constraints on aphenomenologically parametrized neutron-star equation of state Phys.Rev. D 79, 124032 (2009)
S. Bernuzzi, T. Dietrich, and A. NagarModeling the Complete Gravitational Wave Spectrum of Neutron StarMergers. PRL 115, 091101 (2015). http://authors.library.caltech.edu/60334/1/PhysRevLett.115.091101.pdf
S. Bernuzzi, A. Nagar, T. Dietrich, and T. Damour Modeling theDynamics of Tidally Interacting Binary Neutron Stars up to the MergerPRL 114, 161103 (2015)
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Reference V
S. Bernuzzi, D. Radice, C. D. Ott, L. F. Roberts, P. Mosta and F.Galeazzi How Loud Are Neutron Star Mergers? Phys. Rev. D 94, 024023(2016)
J. Clark Postmerger Burst Activities https://dcc.ligo.org/DocDB/
0139/G1602372/001/pmns_bursts_valencia-Dec-2016.pdf
K. Hotokezaka, K. Kyutoku, H. Okawa, M. Shibata and K. Kiuchi BinaryNeutron Star Mergers: Dependence on the Nuclear Equation of StatePhys. Rev. D 83.124008 (2011)
J. Clark, A. Bauswein, L. Cadonati, H.-T. Janka, C. Pankow, and N.Stergioulas Prospects For High Frequency Burst Searches FollowingBinary Neutron Star Coalescence With Advanced Gravitational WaveDetectors Phys. Rev. D 90 ,062004 (2014)
J. Clark, A. Bauswein, N. Stergioulas and D. Shoemaker Observinggravitational waves from the post-merger phase of binary neutron starcoalescence Class. Quant. Grav. 33, 085003 (2016)
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Reference VI
S.Klimenko, I.Yakushin, A.Mercer, G.Mitselmakher Coherent method fordetection of gravitational wave bursts Class. Quant. Grav. 25, 114029(2008)
K. Hayama, S. D. Mohanty, M. Rakhmanov, S. Desai Coherent network
analysis for triggered gravitational wave burst searches Class. Quant.
Grav. 24: S681-S688 (2007)
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Bayesian Analysis
Ingredients for the posterior
The ”inner product”
〈a|b〉 = 4<∫ fhigh
flow
ab∗df
Sn(f)where Sn(f) is the power spectral density (9)
The likelihood
P (d|~θ,H, I) = P (n = d− h|I) where P (n|I) ∝ e(−12〈n|n〉) (10)
The evidence
P (d|H, I) =
∫all space
P (d|~θ,H, I)P (~θ|H, I)d~θ (11)
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Bayesian Analysis
Model selection:
P (Hi|d, I)
P (Hj |d, I)=P (d|Hi, I)
P (d|Hj , I)
P (Hi|I)
P (Hj |I)
Oij = Bij
P (Hi|I)
P (Hj |I)
(12)
Oij is the odd ratio
Bij is the Bayes factor
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Bayesian Analysis
Multiple detections:
Parameter Estimation
P (~θ|d1, d2, d3, · · · dN ,H, I) =P (dN |~θ,H, I)P (~θ|d1, d2, d3, · · · dN−1,H, I)
P (dN |H, I)
=
N∏i=1
P (~θ|di,H, I)P (~θ|H, I)1−N
(13)Models Selection
P (Hi|d1, d2, d3, · · · dN , I)
P (Hj |d1, d2, d3, · · · dN , I)=P (Hi|I)
P (Hj |I)
N∏l=1
P (dl|Hi, I)
P (dl|Hj , I)(14)
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High Frequency Burst Searches
Power Spectrum Density Reconstruction:
Pi =1
Ndet
Ndet∑j=1
ρjmaxk(ρk)
Pij (15)
where Pij is the psd in i-th frequency band of j-th detector, ρj is theSNR is the j-th detector.
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High Frequency Burst Searches
Assumed PSD of PMNS and BH
SNS(f) = A0 exp
(−(f − f ′peak
2σ
)2)
+A1
(f
flow
)αSBH(f) = A1
(f
flow
)α (16)
Bayesian Information Criterion (BIC)
BIC = n logχ2min + k log n, (17)
χ2min =
1
n− 1
n∑i=1
(Pi − Si)2 (18)
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