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

Kinosaki Camp — February 24, 2014 Alexandre Le Tiec


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

0

1×10-19

2×10-19

t (sec.)

n(t)

h(t)

10 100 1000 1000010-24

10-23

10-22

10-21

f (Hz)

h(f)

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

0

1×10-19

2×10-19

t (sec.)

n(t)

h(t)

100 h(t)

10 100 1000 1000010-24

10-23

10-22

10-21

f (Hz)

h(f)

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

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

[Futamase, Itoh]





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

[Mino, Sago, Sasaki, Tagoshi, Tanaka, . . . ]





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


[Nakamura, Nakano, Shibata, . . . ]





log10

(m2 /m

1)

0 1 2 3

0

1

2

3

4

4

log10

(r /m)

(Com

pact

ness

)

Mass Ratio

−1

Perturbation Theory

m2

m = m1 + m

2

EOB

m1



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

How?

8 Use the same coordinate system in all calculations

4 Using coordinate-invariant relationships

What?

• Gravitational waveforms at null infinity

• Conservative effects on the orbital dynamics



Comparing the predictions from these various methods

Paper Year Methods

Boyle, Brown et al. 2007 NR/PNDetweiler 2008 BHP/PNBlanchet, Detweiler et al. 2010a BHP/PNBlanchet, Detweiler et al. 2010b BHP/PNDamour 2010 BHP/EOBBarack, Damour, Sago 2010 BHP/EOBLousto, Nakano et al. 2010 NR/BHPSperhake, Cardoso et al. 2011 NR/BHPLe Tiec, Mroue et al. 2011 NR/BHP/PN/EOBNakano, Zlochower et al. 2011 NR/BHPDamour, Nagar et al. 2012 NR/EOBLe Tiec, Barausse, Buonanno 2012 NR/BHP/PN/EOBAkcay, Barack et al. 2012 BHP/EOBLousto, Zlochower 2013 NR/PNNagar 2013 NR/BHPHinderer, Buonanno et al. 2013 NR/EOBLe Tiec, Buonanno et al. 2013 NR/BHP/PNDolan, Warburton et al. 2014 BHP/PN

Shah, Friedman, WhitingBlanchet, Faye, Whiting

*

2014 BHP/PN



Revisiting the classical tests of General Relativity

• Gravitational redshift of light

∆τ

∆t“ 1´

G

c2

MdRd

• Perihelion advance of Mercury

∆Φ “G

c2

6πMda p1´ e2q

• Precession of Earth-Moon spin

Ωprec “G

c2v ˆ∇

ˆ

3Md2r

˙

M⊙

τ t

M⊙

R⊙

rM⊙

R

s

prec



Outline







Post-Newtonian expansions and black hole perturbations



log10

(m2 /m

1)

0 1 2 3

0

1

2

3

4

4

log10

(r /m)

(Com

pact

ness

)

Mass Ratio

−1m

1

m2

r


&Perturbation

Theory

Perturbation Theory



The “redshift observable” for circular orbits

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

Eobs

Eem“ppauaqobs

ppauaqem

“ z

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

z “ ´kaua

• In coordinates adapted to the symmetry:

z “dτ

dt“

1

utx

t

ua

m2

m1

x

particle observer

em

uaobs

pa

pa

dt

dτ

obsE

emE



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

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

10 20 30 40 50 60 70 80 90 100

(1/z1)GSF

rΩ / m2

N1PN2PN3PN4PN5PN6PN7PNExact




0.1

0.2

0.3

0.4

0.5

5 6 7 8 9 10

GSF

rΩ / m2

N1PN2PN3PN4PN5PN6PN7PNExact(1

/z1)




0.1

0.2

0.3

0.4

0.5

5 6 7 8 9 10

GSF

rΩ / m2

N1PN2PN3PN4PN5PN6PN7PNExact(1

/z1)


Cfit3PN “ 27.6879035p4q

C3PN “ 27.68790269 ¨ ¨ ¨


Averaged “redshift observable” for eccentric orbits

• Generic eccentric orbit parametrizedby the two invariant frequencies

Ωr “2π

P, Ωϕ “

1

P

ż P

09ϕptq dt

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

x1zy ”1

T

ż T

0

dτ

zpτq“

P

T

m2

m1

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, Barack, Le Tiec, Sago & Warburton (preliminary)]

0

0.1

0.2

0.3

6 8 10 12 14 16 18 20 22

⟨1/z

⟩ GS

F

p

e = 0.1

N

1PN

2PN

3PN

Exact




0

0.1

0.2

0.3

6 8 10 12 14 16 18 20 22

⟨1/z

⟩ GS

F

p

e = 0.2

N

1PN

2PN

3PN

Exact




0

0.1

0.2

0.3

6 8 10 12 14 16 18 20 22

⟨1/z

⟩ GS

F

p

e = 0.3

N

1PN

2PN

3PN

Exact




0

0.1

0.2

0.3

6 8 10 12 14 16 18 20 22

⟨1/z

⟩ GS

F

p

e = 0.4

N

1PN

2PN

3PN

Exact




0

0.1

0.2

0.3

6 8 10 12 14 16 18 20 22

⟨1/z

⟩ GS

F

p

e = 0.5

N

1PN

2PN

3PN

Exact



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 ”

GM

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

r

s

prec

r

prec

v

s

v

rs

prec



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 “

1

2p∇akb∇akbq|γ

• Compute ψ ” 1´ ωprecuφ, the angle of

spin precession per radian of revolutionspace

time ka

s

s

prec

m2

m1

a

ka

sa



Precession angle vs orbital separation[Dolan, Warburton, Harte, Le Tiec, Wardell & Barack, PRD (2014)]

-0.008

-0.004

0

0.004

0.008

5 10 15 20 25 30 35

ψG

SF

rΩ / m2

2PN

3PN

(4PN)

Exact

-8

-7.8

-7.6

-7.4

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

M⊙

Mercury



Periastron advance in black hole binaries

• Generic eccentric orbit parametrized bythe two invariant frequencies

Ωr “2π

P, Ωϕ “

1

P

ż P

09ϕptq dt

• Periastron advance per radial period

K ”Ωϕ

Ωr“ 1`

∆Φ

2π

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

m2

m1

t=0 t=P



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

1.2

1.3

1.4

1.5

SchwEOBGSFνPNGSFq

0.01 0.02 0.03

-0.01

0

0.01

mΩϕ

KδK/K

1:1



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

1.2

1.3

1.4

1.5

SchwEOBGSFνPNGSFq

0.01 0.02 0.03

-0.01

0

0.01

mΩϕ

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

0

EOB

GSFνPN

0 0.5 1

-0.08

-0.04

0

0.04

Schw

GSFq

q

δK

/K



Symmetric background and ν-expansion

m1

m2

S2Ωφ

S1

• 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

mΩϕ

0.02

0.04

0.06

WN

R-W

q = 1.5

q = 1

q = 3

q = 8

q = 5

χ1 = χ

2 = 0




0.1

0.2

0.3(W

NR-W

) /

ν

0.01 0.02 0.03 0.04mΩ

ϕ

0.1

0.2

0.3

(WNR-W

) /

ν χ1 = χ

2 = 0




0.1

0.2

0.3(W

NR-W

) /

ν

0.01 0.02 0.03 0.04mΩ

ϕ

0.1

0.2

0.3

(WNR-W

) /

ν χ1 = χ

2 = 0

GSF

GSF




0.015 0.02 0.025 0.03 0.035

mΩϕ

0.01

0.02

0.03

0.04W

NR-W

q = 1.5

q = 5

q = 3

q = 8

χ1 = −0.5, χ

2 = 0




0.08

0.16

0.24(W

NR-W

) /

ν

0.01 0.015 0.02 0.025 0.03 0.035mΩ

ϕ

0.08

0.16

0.24

(WNR-W

) /

ν χ1 = −0.5, χ

2 = 0



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

-0.04E/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


1 : 1


Using 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]

[P. 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

-2

-1

0

1

2

3

DMM @%DD

ΗΗ@%

D

modelling bias

[F. Ohme, CQG (2012)]



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)PNNRHybrid

Time [M]

[P. 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

-2

-1

0

1

2

3

DMM @%DD

ΗΗ@%

D

modelling bias

[F. 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 comparable-mass binaries

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

( (


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

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