gr@99: revisiting the classical tests with compact...
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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
 Geodetic spin precession
à Relativistic periastron advance
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Outline
À Gravitational wave source modelling
Á Gravitational redshift
 Geodetic spin precession
à Relativistic periastron advance
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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)
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
(Credit: L. Lindblom)
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
(Credit: L. Lindblom)
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
−1m
1
m2
r
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
v2
c2„
Gm
c2r! 1
Source modelling Gravitational redshift Spin precession Periastron advance
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 m1
m2
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
q ”m1
m2! 1
Source modelling Gravitational redshift Spin precession Periastron advance
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
m1
m2
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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)
(Com
pact
ness
)
Mass Ratio
−1
Perturbation Theory
m2
m = m1 + m
2
EOB
m1
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Comparing the predictions from these various methods
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Comparing the predictions from these various methods
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Outline
À Gravitational wave source modelling
Á Gravitational redshift
 Geodetic spin precession
à Relativistic periastron advance
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Post-Newtonian expansions and black hole perturbations
Numerical Relativity
PostNewtonian Theory
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
PostNewtonian Theory
&Perturbation
Theory
Perturbation Theory
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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 ka:
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Redshift observable vs orbital separation[Blanchet, Detweiler, Le Tiec & Whiting, PRD (2010)]
0.1
0.2
0.3
0.4
0.5
5 6 7 8 9 10
GSF
rΩ / m2
N1PN2PN3PN4PN5PN6PN7PNExact(1
/z1)
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Redshift observable vs orbital separation[Blanchet, Detweiler, Le Tiec & Whiting, PRD (2010)]
0.1
0.2
0.3
0.4
0.5
5 6 7 8 9 10
GSF
rΩ / m2
N1PN2PN3PN4PN5PN6PN7PNExact(1
/z1)
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Cfit3PN “ 27.6879035p4q
C3PN “ 27.68790269 ¨ ¨ ¨
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Redshift observable vs semi-latus rectum[Akcay, Le Tiec, Barack, 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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Redshift observable vs semi-latus rectum[Akcay, Le Tiec, Barack, 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 = 0.2
N
1PN
2PN
3PN
Exact
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Redshift observable vs semi-latus rectum[Akcay, Le Tiec, Barack, 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 = 0.3
N
1PN
2PN
3PN
Exact
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Redshift observable vs semi-latus rectum[Akcay, Le Tiec, Barack, 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 = 0.4
N
1PN
2PN
3PN
Exact
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Redshift observable vs semi-latus rectum[Akcay, Le Tiec, Barack, 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 = 0.5
N
1PN
2PN
3PN
Exact
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Outline
À Gravitational wave source modelling
Á Gravitational redshift
 Geodetic spin precession
à Relativistic periastron advance
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Precession angle vs orbital separation[Dolan, Warburton et al., 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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Outline
À Gravitational wave source modelling
Á Gravitational redshift
 Geodetic spin precession
à Relativistic periastron advance
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Gravitational self-force from numerical relativity[Le Tiec, Buonanno et al., PRD (2013)]
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Gravitational self-force from numerical relativity[Le Tiec, Buonanno et al., PRD (2013)]
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Gravitational self-force from numerical relativity[Le Tiec, Buonanno et al., PRD (2013)]
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Gravitational self-force from numerical relativity[Le Tiec, Buonanno et al., PRD (2013)]
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
1 : 1
Source modelling Gravitational redshift Spin precession Periastron advance
Using perturbation theory for comparable-mass 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
m1
m2
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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., 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
[Ohme, CQG (2012)]
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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]
[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
[Ohme, CQG (2012)]
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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,...
( (
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Additional Material
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Gravitational waveform for head-on collisions[Sperhake, Cardoso et al., PRD (2011)]
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
ψ 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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
1 : 100
Source modelling Gravitational redshift Spin precession Periastron advance
Gravitational waveform for head-on collisions[Sperhake, Cardoso et al., PRD (2011)]
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
ψ 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
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
1 : 10
Source modelling Gravitational redshift Spin precession Periastron advance
Recoil velocity for non-spinning binaries[Nagar, PRD (2013)]
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
Why does perturbation theory perform so well?
• In perturbation theory, one traditionally expands as
kmaxÿ
k“0
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
k“0
BkpmΩϕq νk where ν ” m1m2m
2 P r0, 14s
• In a PN expansion, we have Bn “ O`
1c2n˘
“ nPN` ¨ ¨ ¨
UWM CGCA Seminar — May 16, 2014 Alexandre Le Tiec