GR@99: Revisiting the Classical Tests with Compact...

GR@99: Revisiting the Classical Tests withCompact Binaries

Alexandre Le Tiec

Laboratoire Univers et TheoriesObservatoire de Paris / CNRS

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

Source modelling Gravitational redshift Spin precession Periastron advance

Outline

À Gravitational wave source modelling

Á Gravitational redshift

Â Geodetic spin precession

Ã Relativistic periastron advance

UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec

Outline

Main sources of gravitational waves (GW)

LIGO/VirgoeLISA

• Binary neutron stars (2ˆ „ 1.4Md)• Stellar mass black hole binaries (2ˆ „ 10Md)• Supermassive black hole binaries (2ˆ „ 106Md)• Extreme mass ratio inspirals („ 10Md` „ 106Md)

Need for highly accurate template waveforms

0 0.1 0.2 0.3 0.4 0.5

-2×10-19

-1×10-19

1×10-19

2×10-19

t (sec.)

10 100 1000 1000010-24

f (Hz)

S1/2(f)

If the expected signal is known in advance then nptq can be filteredand hptq recovered by matched filtering ÝÑ template waveforms

(Credit: L. Lindblom)

Need for highly accurate template waveforms

0 0.1 0.2 0.3 0.4 0.5

-2×10-19

-1×10-19

1×10-19

2×10-19

t (sec.)

100 h(t)

10 100 1000 1000010-24

f (Hz)

S1/2(f)

If the expected signal is known in advance then nptq can be filteredand hptq recovered by matched filtering ÝÑ template waveforms

(Credit: L. Lindblom)

Methods to compute GW templates for compact binaries

Numerical Relativity

PostNewtonian Theory

(m2 /m

0 1 2 3

(r /m)

Perturbation Theory

Mass Ratio

(m2 /m

0 1 2 3

(r /m)

Perturbation Theory

Mass Ratio

c2r! 1

(m2 /m

0 1 2 3

(r /m)

Perturbation Theory

Mass Ratio

−1 m1

q ”m1

(m2 /m

0 1 2 3

(r /m)

Perturbation Theory

Mass Ratio

(m2 /m

0 1 2 3

(r /m)

Mass Ratio

Perturbation Theory

m = m1 + m

Comparing the predictions from these various methods

• 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

8 Use the same coordinate system in all calculations

4 Using coordinate-invariant relationships

• Gravitational waveforms at null infinity

• Conservative effects on the orbital dynamics

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

Revisiting the classical tests of General Relativity

• Gravitational redshift of light

∆t“ 1´

• Perihelion advance of Mercury

∆Φ “G

6πMda p1´ e2q

• Precession of Earth-Moon spin

Ωprec “G

c2v ˆ∇

Outline

Post-Newtonian expansions and black hole perturbations

(m2 /m

0 1 2 3

(r /m)

Mass Ratio

&Perturbation

Theory

Perturbation Theory

The “redshift observable” for circular orbits

• It measures the redshift of light emittedfrom the point particle:

Eem“ppauaqobs

ppauaqem

• It is a constant of the motion associatedwith the helical Killing field ka:

z “ ´kaua

• In coordinates adapted to the symmetry:

z “dτ

particle observer

Redshift observable vs orbital separation[Blanchet, Detweiler, Le Tiec & Whiting, PRD (2010)]

10 20 30 40 50 60 70 80 90 100

(1/z1)GSF

rΩ / m2

N1PN2PN3PN4PN5PN6PN7PNExact

5 6 7 8 9 10

rΩ / m2

N1PN2PN3PN4PN5PN6PN7PNExact(1

5 6 7 8 9 10

rΩ / m2

N1PN2PN3PN4PN5PN6PN7PNExact(1

Cfit3PN “ 27.6879035p4q

C3PN “ 27.68790269 ¨ ¨ ¨

Averaged “redshift observable” for eccentric orbits

• Generic eccentric orbit parametrizedby the two invariant frequencies

Ωr “2π

P, Ωϕ “

09ϕptq dt

• Average of 1z with respect to propertime τ over one radial period:

x1zy ”1

zpτq“

t = 0 = 0

t = P = T

• Relation x1zypΩr ,Ωϕq well defined in both PN and GSFframeworks, and coordinate invariant

Redshift observable vs semi-latus rectum[Akcay, Le Tiec, Barack, Sago & Warburton (preliminary)]

6 8 10 12 14 16 18 20 22

⟨1/z

⟩ GS

e = 0.1

6 8 10 12 14 16 18 20 22

⟨1/z

⟩ GS

e = 0.2

6 8 10 12 14 16 18 20 22

⟨1/z

⟩ GS

e = 0.3

6 8 10 12 14 16 18 20 22

⟨1/z

⟩ GS

e = 0.4

6 8 10 12 14 16 18 20 22

⟨1/z

⟩ GS

e = 0.5

Outline

De Sitter’s spin precession, or geodetic effect

• In 1916 de Sitter showed that, to leadingorder, a test spin s precesses at

Ωprec »3

2v ˆ∇Φ , Φ ”

c2r! 1

• Precession of Earth-Moon spin axis of„ 1.9”cent. confirmed using LLR data

• Precession of test gyro on polar Earthorbit of „ 6.6”yr confirmed by GPB

• Geodetic spin precession of „ 5˝yrmeasured in the double pulsar

Geodetic precession in black hole binaries

• A test spin sa is parallel-transported alonga geodesic γ with unit tangent ua:

ub∇bsa “ 0 ðñds

dτ“ ωprec ˆ s

• For a circular orbit with a helical Killingfield ka such that ka|γ “ ua,

ω2prec “

2p∇akb∇akbq|γ

• Compute ψ ” 1´ ωprecuφ, the angle of

spin precession per radian of revolutionspace

time ka

Precession angle vs orbital separation[Dolan, Warburton et al., PRD (2014)]

-0.008

-0.004

5 10 15 20 25 30 35

rΩ / m2

40 60 80 100 120 140 160 180

Outline

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 “6πGMd

c2a p1´ e2q

• Periastron advance of „ 4˝yr nowmeasured in binary pulsars

Mercury

Periastron advance in black hole binaries

• Generic eccentric orbit parametrized bythe two invariant frequencies

Ωr “2π

P, Ωϕ “

09ϕptq dt

• Periastron advance per radial period

K ”Ωϕ

Ωr“ 1`

• In the circular orbit limit e Ñ 0, therelation K pΩϕq is coordinate-invariant

t=0 t=P

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

SchwEOBGSFνPNGSFq

0.01 0.02 0.03

KδK/K

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

SchwEOBGSFνPNGSFq

0.01 0.02 0.03

KδK/K

q = m1/m2

= m1m2 /m2ν1:1

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

0 0.2 0.4 0.6 0.8 1

-0.024

-0.018

-0.012

-0.006

GSFνPN

0 0.5 1

Symmetric background and ν-expansion

S2Ωφ

• Consider W ” 1K 2 “ f pΩϕ;ma,Saq for quasicircular orbits

• Impose symmetry by exchange 1 Ø 2 on perturbative resultë construct symmetric background W

• Expansion in powers of symmetric mass ratio ν “ m1m2m2:

W “W ` νWp1q ` ν2 Wp2q `Opν3q

Gravitational self-force from numerical relativity[Le Tiec, Buonanno et al., PRD (2013)]

0.01 0.02 0.03 0.04

q = 1.5

χ1 = χ

0.01 0.02 0.03 0.04mΩ

(WNR-W

ν χ1 = χ

0.01 0.02 0.03 0.04mΩ

(WNR-W

ν χ1 = χ

0.015 0.02 0.025 0.03 0.035

q = 1.5

χ1 = −0.5, χ

0.24(W

0.01 0.015 0.02 0.025 0.03 0.035mΩ

(WNR-W

ν χ1 = −0.5, χ

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

3 3.2 3.4 3.6 3.8

J/(mμ)

δE/μ

3PN EOB(3PN) Schw

-0.04E/μ GSFν

EOB(3PN)

3.1 3.15 3.2 3.25

-0.078

-0.072

-0.066

-0.060

Using perturbation theory for comparable-mass binaries

(m2 /m

0 1 2 3

(r /m)

Perturbation Theory

Mass Ratio

Main shortcoming of current template waveforms

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

−0.2

−0.1

Jena (η = 0.25)PNNR

Time [M]

[Ajith et al., PRD (2008)]

• PN breaks down during late inspiral

• Modelling error ą statistical error

• Bias on parameter estimation

• Science limited by templates!

-2 -1 0 1 2- 3

DMM @%DD

ΗΗ@%

modelling bias

[Ohme, CQG (2012)]

Main shortcoming of current template waveforms

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

−0.2

−0.1

Jena (η = 0.25)PNNRHybrid

Time [M]

[Ajith et al., PRD (2008)]

• PN breaks down during late inspiral

• Modelling error ą statistical error

• Bias on parameter estimation

• Science limited by templates!

-2 -1 0 1 2- 3

DMM @%DD

ΗΗ@%

modelling bias

[Ohme, CQG (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

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

ψ 20/η

PP limitL/M = 9.58L/M = 18.53

-50 0 50 100t / M

ψ 30/η

PP limitL/M = 9.58L/M = 18.53

1 : 100

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

ψ 20/η

PP limitL/M = 16.28L/M = 24.76

-50 0 50 100t / M

ψ 30/η

PP limitL/M = 16.28L/M = 24.76

1 : 10

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

Why does perturbation theory perform so well?

• In perturbation theory, one traditionally expands as

kmaxÿ

Akpm2Ωϕq qk where q ” m1m2 P r0, 1s

• However, the relations E pΩϕ;maq, JpΩϕ;maq, K pΩϕ;maq

must be symmetric under exchange m1 ÐÑ m2

• Hence, a better-motivated expansion is

kmaxÿ

BkpmΩϕq νk where ν ” m1m2m

2 P r0, 14s

• In a PN expansion, we have Bn “ O`

1c2n˘

“ nPN` ¨ ¨ ¨

GR@99: Revisiting the Classical Tests with Compact...

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