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

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

Binary Neutron Stars Waveform

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P. Pang T. Li Constraining EOS with GW 20th Feb 2017 3 / 27

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]

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 4 / 27

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

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

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

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

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

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

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

Binary Neutron Stars Waveform

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P. Pang T. Li Constraining EOS with GW 20th Feb 2017 12 / 27

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

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

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

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)

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 16 / 27

Toy Model of Combined Waveform

fcon

fmerg, h(fmerg)

ξσ

fpeak, AAmplitude

Frequency

b

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 17 / 27

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

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 18 / 27

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

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 19 / 27

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)

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 20 / 27

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)

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 21 / 27

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)

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 22 / 27

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)

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 23 / 27

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)

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 24 / 27

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)

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 25 / 27

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)

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 26 / 27

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

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 27 / 27

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)

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 28 / 27

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.

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 29 / 27

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)

P. Pang T. Li Constraining EOS with GW 20th Feb 2017 30 / 27