alexandre le tiecletiec.yolasite.com/resources/gphys.pdf · 2016. 3. 21. · damour & nagar...

Gravitational waves from compact object binaries

Alexandre Le Tiec

Laboratoire Univers et TheoriesObservatoire de Paris / CNRS

aLIGO, aVirgo,KAGRA, eLISA,DECIGO, ET,...

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Main sources of gravitational waves (GW)

LIGO/VirgoeLISA

• Binary neutron stars (2× ∼ 1.4M)

• Stellar mass black hole binaries (2× ∼ 10M)

• Supermassive black hole binaries (2× ∼ 106M)

• Extreme mass ratio inspirals (∼ 10M+ ∼ 106M)

Journee GPhys — 27 mai 2014 Alexandre Le Tiec

Methods to compute GW templates for compact binaries

Numerical Relativity

PostNewtonian Theory

log10

(m2 /m

1)

0 1 2 3

0

1

2

3

4

4

log10

(r /m)

Perturbation Theory

(Com

pact

ness

)

Mass Ratio

−1





log10

(m2 /m

1)

0 1 2 3

0

1

2

3

4

4

log10

(r /m)

Perturbation Theory

(Com

pact

ness

)

Mass Ratio

−1m

1

m2

r


v2

c2∼ Gm

c2r 1




log10

(m2 /m

1)

0 1 2 3

0

1

2

3

4

4

log10

(r /m)

Perturbation Theory

(Com

pact

ness

)

Mass Ratio

−1 m1

m2


q ≡ m1

m2 1




log10

(m2 /m

1)

0 1 2 3

0

1

2

3

4

4

log10

(r /m)

Perturbation Theory

(Com

pact

ness

)

Mass Ratio

−1

m1

m2


Comparing the predictions from these various methods

Why?

• Independent checks of long and complicated calculations

• Identify domains of validity of approximation schemes

• Extract information inaccessible to other methods

• Develop a universal model for compact binaries

What?

• Periastron advance K vs orbital frequency Ω

• Binding energy E vs angular momentum J



Paper Year Methods Observable Orbit Spin

Baker et al. 2007 NR/PN waveformBoyle et al. 2007 NR/PN waveformHannam et al. 2007 NR/PN waveformBoyle et al. 2008 NR/PN/EOB energy fluxDamour & Nagar 2008 NR/EOB waveformHannam et al. 2008 NR/PN waveform 3Pan et al. 2008 NR/PN/EOB waveformCampanelli et al. 2009 NR/PN waveform 3Hannam et al. 2010 NR/PN waveform 3Hinder et al. 2010 NR/PN waveform eccentricLousto et al. 2010 NR/BHP waveformSperhake et al. 2011 NR/PN waveformSperhake et al. 2011 NR/BHP waveform head-onLousto & Zlochower 2011 NR/BHP waveformNakano et al. 2011 NR/BHP waveformLousto & Zlochower 2013 NR/PN waveformNagar 2013 NR/BHP recoil velocityHinder et al. 2014 NR/PN/EOB waveform 3



Paper Year Methods Observable Spin

Detweiler 2008 BHP/PN redshift observableBlanchet et al. 2010 BHP/PN redshift observableDamour 2010 BHP/EOB ISCO frequencyMroue et al. 2010 NR/PN periastron advanceBarack et al. 2010 BHP/EOB periastron advanceFavata 2011 BHP/PN/EOB ISCO frequencyLe Tiec et al. 2011 NR/BHP/PN/EOB periastron advanceDamour et al. 2012 NR/EOB binding energyLe Tiec et al. 2012 NR/BHP/PN/EOB binding energyAkcay et al. 2012 BHP/EOB redshift observableHinderer et al. 2013 NR/EOB periastron advance 3Le Tiec et al. 2013 NR/BHP/PN periastron advance 3Bini & DamourShah et al.Blanchet et al.

2014 BHP/PN redshift observable

Dolan et al.Bini & Damour

2014 BHP/PN precession angle 3

Isoyama et al. 2014 BHP/PN/EOB ISCO frequency 3


Relativistic perihelion advance of Mercury

• Observed anomalous advance ofMercury’s perihelion of ∼ 43”/cent.

• Accounted for by the leading-orderrelativistic angular advance per orbit

∆Φ =6πGM

c2a (1− e2)

• Periastron advance of ∼ 4/yr nowmeasured in binary pulsars

M⊙

Mercury


Relativistic perihelion advance of Mercury

• Observed anomalous advance ofMercury’s perihelion of ∼ 43”/cent.

• Accounted for by the leading-orderrelativistic angular advance per orbit

∆Φ =6πGM

c2a (1− e2)

• Periastron advance of ∼ 4/yr nowmeasured in binary pulsars

m2

m1


Periastron advance vs orbital frequency[Le Tiec, Mroue et al. (2011)]

1.2

1.3

1.4

1.5

Schw

EOB

GSFν

PN

GSFq

0.01 0.02 0.03

-0.01

0

0.01

mΩ

KδK/K


K = 1 + ∆Φ/2π

Binding energy vs angular momentum[Le Tiec, Barausse & Buonanno (2012)]

3 3.2 3.4 3.6 3.8

J/(mμ)

-10-4

0

10-4

δE/μ

GSFν

3PN EOB(3PN) Schw

-0.12

-0.10

-0.08

-0.06

-0.04

E/μ GSFν

NR

EOB(3PN)

3PN

Schw

GSFq

GSFν

GSFq

NR

3PN

3.1 3.15 3.2 3.25

-0.078

-0.072

-0.066

-0.060


Perturbation theory for comparable-mass binaries



log10

(m2 /m

1)

0 1 2 3

0

1

2

3

4

4

log10

(r /m)

Perturbation Theory

(Com

pact

ness

)

Mass Ratio

−1

m1

m2


Main shortcoming of current template waveforms

−2000 −1800 −1600 −1400 −1200 −1000 −800 −600 −400 −200 0

−0.2

−0.1

0

0.1

0.2

0.3

h

Jena (η = 0.25)PNNR

Time [M]

[Ajith et al. (2008)]

• PN breaks down during late inspiral

• Modelling error > statistical error

• Bias on parameter estimation

• Science limited by templates!

-2 -1 0 1 2- 3

-2

-1

0

1

2

3

DMM @%DD

ΗΗ@%

D

modelling bias

[Ohme (2012)]


Summary and prospects

• Observing GWs will open a new window on the Universe

• Highly accurate template waveforms are a prerequisite fordoing science with GW observations

• It is crucial to compare the predictions from PN theory,perturbation theory and numerical relativity

• Perturbation theory may prove useful to build templatesfor IMRIs and even comparable-mass binaries

aLIGO, aVirgo,KAGRA, eLISA,DECIGO, ET,...

( (


Additional Material


Periastron advance vs mass ratio[Le Tiec, Mroue et al. (2011)]

0 0.2 0.4 0.6 0.8 1

-0.024

-0.018

-0.012

-0.006

0

EOB

GSFνPN

0 0.5 1

-0.08

-0.04

0

0.04

Schw

GSFq

q

δK

/K


Gravitational waveform for head-on collisions[Sperhake, Cardoso et al. (2011)]

-0.06

-0.04

-0.02

0

0.02

0.04

0.06ψ 20

/η

PP limitL/M = 9.58L/M = 18.53

-50 0 50 100t / M

-0.04

-0.02

0

0.02

0.04

ψ 30/η

PP limitL/M = 9.58L/M = 18.53


1 : 100

Gravitational waveform for head-on collisions[Sperhake, Cardoso et al. (2011)]

-0.06

-0.04

-0.02

0

0.02

0.04

0.06ψ 20

/η

PP limitL/M = 16.28L/M = 24.76

-50 0 50 100t / M

-0.04

-0.02

0

0.02

0.04

ψ 30/η

PP limitL/M = 16.28L/M = 24.76


1 : 10

Recoil velocity for non-spinning binaries[Nagar (2013)]


Why does perturbation theory perform so well?

• In perturbation theory, one traditionally expands as

kmax∑k=0

Ak(m2Ω) qk where q ≡ m1/m2 ∈ [0, 1]

• However, the relations E (Ω;ma), J(Ω;ma), K (Ω;ma)must be symmetric under exchange m1 ←→ m2

• Hence, a better-motivated expansion is

kmax∑k=0

Bk(mΩ) νk where ν ≡ m1m2/m2 ∈ [0, 1/4]

• In a PN expansion, we have Bn = O(1/c2n

)= nPN + · · ·


alexandre le tiecletiec.yolasite.com/resources/gphys.pdf · 2016. 3. 21. · damour & nagar...

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