Josefina Lindblom European Commission DG Research - Unit T4: SMEs
GR@99: Revisiting the Classical Tests with Compact...
Transcript of 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
 Geodetic spin precession
à Relativistic periastron advance
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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)
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 2014 Alexandre Le Tiec
v2
c2„
Gm
c2r! 1
[Futamase, Itoh]
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
Kinosaki Camp — February 24, 2014 Alexandre Le Tiec
q ”m1
m2! 1
[Mino, Sago, Sasaki, Tagoshi, Tanaka, . . . ]
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
Kinosaki Camp — February 24, 2014 Alexandre Le Tiec
[Nakamura, Nakano, Shibata, . . . ]
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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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:
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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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)
Kinosaki Camp — February 24, 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)
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
Kinosaki Camp — February 24, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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 = 0.2
N
1PN
2PN
3PN
Exact
Kinosaki Camp — February 24, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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 = 0.3
N
1PN
2PN
3PN
Exact
Kinosaki Camp — February 24, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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 = 0.4
N
1PN
2PN
3PN
Exact
Kinosaki Camp — February 24, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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 = 0.5
N
1PN
2PN
3PN
Exact
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 2014 Alexandre Le Tiec
Source modelling Gravitational redshift Spin precession Periastron advance
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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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
Kinosaki Camp — February 24, 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]
[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)]
Kinosaki Camp — February 24, 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]
[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)]
Kinosaki Camp — February 24, 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 comparable-mass binaries
aLIGO, aVirgo,KAGRA, eLISA,DECIGO, ET,...
( (
Kinosaki Camp — February 24, 2014 Alexandre Le Tiec