general relativistic dynamics of binary black holes
TRANSCRIPT
General relativistic dynamics
of binary black holes
Alexandre Le Tiec
Laboratoire Univers et TheoriesObservatoire de Paris / CNRS
Outline
1 Gravitational wave source modelling
2 Post-Newtonian approximation
3 Black hole perturbation theory
4 Effective one-body model
5 Comparisons
Outline
1 Gravitational wave source modelling
2 Post-Newtonian approximation
3 Black hole perturbation theory
4 Effective one-body model
5 Comparisons
Need for 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 n(t) can be filteredand h(t) recovered by matched filtering −→ template waveforms
(Credit: L. Lindblom)
Need for 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 n(t) can be filteredand h(t) recovered by matched filtering −→ template waveforms
(Credit: L. Lindblom)
Modelling coalescing 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 & selfforce
(com
pact
ness
)
mass ratio
−1
Modelling coalescing 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 & selfforce
(com
pact
ness
)
mass ratio
−1m
1
m2
r v2
c2∼ Gm
c2r 1
Modelling coalescing 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 & selfforce
(com
pact
ness
)
mass ratio
−1 m1
m2
q ≡ m1
m2 1
Modelling coalescing 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 & selfforce
(com
pact
ness
)
mass ratio
−1
m1
m2
Modelling coalescing 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 & selfforce
m2
M = m1 + m
2
EOB
m1
Outline
1 Gravitational wave source modelling
2 Post-Newtonian approximation
3 Black hole perturbation theory
4 Effective one-body model
5 Comparisons
Small parameter
ε ∼ v212
c2∼ Gm
r12c2 1 m
2
m1
r12
v2
v1
Example
g00(t, x) = −1 +2Gm1
r1c2︸ ︷︷ ︸Newtonian
+4Gm2v
22
r2c4︸ ︷︷ ︸1PN term
+ · · ·+ (1↔ 2)
Notation
nPN order refers to effects O(c−2n) with respect to “Newtonian” solution
Small parameter
ε ∼ v212
c2∼ Gm
r12c2 1 m
2
m1
x
r1
r2
r12
v2
v1
Example
g00(t, x) = −1 +2Gm1
r1c2︸ ︷︷ ︸Newtonian
+4Gm2v
22
r2c4︸ ︷︷ ︸1PN term
+ · · ·+ (1↔ 2)
Notation
nPN order refers to effects O(c−2n) with respect to “Newtonian” solution
Small parameter
ε ∼ v212
c2∼ Gm
r12c2 1 m
2
m1
x
r1
r2
r12
v2
v1
Example
g00(t, x) = −1 +2Gm1
r1c2︸ ︷︷ ︸Newtonian
+4Gm2v
22
r2c4︸ ︷︷ ︸1PN term
+ · · ·+ (1↔ 2)
Notation
nPN order refers to effects O(c−2n) with respect to “Newtonian” solution
Metric potential
hαβ ≡ √−ggαβ − ηαβ
Einstein field equations
Gαβ = 8πTαβ
⇐⇒∂αh
αβ = 0
⇔ ∂αταβ = 0 ⇔ ∇αTαβ = 0
hαβ = 16π |g |Tαβ + Λαβ[h]︸ ︷︷ ︸nonlinearities∂h∂h + ···
≡ 16π ταβ
Weak-field approximation
|hαβ| 1
=⇒ perturbative nonlinear treatment
Metric potential
hαβ ≡ √−ggαβ − ηαβ
Einstein field equations
Gαβ = 8πTαβ
⇐⇒∂αh
αβ = 0
⇔ ∂αταβ = 0 ⇔ ∇αTαβ = 0
hαβ = 16π |g |Tαβ + Λαβ[h]︸ ︷︷ ︸nonlinearities∂h∂h + ···
≡ 16π ταβ
Weak-field approximation
|hαβ| 1
=⇒ perturbative nonlinear treatment
Metric potential
hαβ ≡ √−ggαβ − ηαβ
Einstein field equations
Gαβ = 8πTαβ ⇐⇒∂αh
αβ = 0
⇔ ∂αταβ = 0 ⇔ ∇αTαβ = 0
hαβ = 16π |g |Tαβ + Λαβ[h]︸ ︷︷ ︸nonlinearities∂h∂h + ···
≡ 16π ταβ
Weak-field approximation
|hαβ| 1
=⇒ perturbative nonlinear treatment
Metric potential
hαβ ≡ √−ggαβ − ηαβ
Einstein field equations
Gαβ = 8πTαβ ⇐⇒∂αh
αβ = 0
⇔ ∂αταβ = 0 ⇔ ∇αTαβ = 0
hαβ = 16π |g |Tαβ + Λαβ[h]︸ ︷︷ ︸nonlinearities∂h∂h + ···
≡ 16π ταβ
Weak-field approximation
|hαβ| 1
=⇒ perturbative nonlinear treatment
Metric potential
hαβ ≡ √−ggαβ − ηαβ
Einstein field equations
Gαβ = 8πTαβ ⇐⇒∂αh
αβ = 0 ⇔ ∂αταβ = 0
⇔ ∇αTαβ = 0
hαβ = 16π |g |Tαβ + Λαβ[h]︸ ︷︷ ︸nonlinearities∂h∂h + ···
≡ 16π ταβ
Weak-field approximation
|hαβ| 1
=⇒ perturbative nonlinear treatment
Metric potential
hαβ ≡ √−ggαβ − ηαβ
Einstein field equations
Gαβ = 8πTαβ ⇐⇒∂αh
αβ = 0 ⇔ ∂αταβ = 0 ⇔ ∇αTαβ = 0
hαβ = 16π |g |Tαβ + Λαβ[h]︸ ︷︷ ︸nonlinearities∂h∂h + ···
≡ 16π ταβ
Weak-field approximation
|hαβ| 1
=⇒ perturbative nonlinear treatment
Metric potential
hαβ ≡ √−ggαβ − ηαβ
Einstein field equations
Gαβ = 8πTαβ ⇐⇒∂αh
αβ = 0 ⇔ ∂αταβ = 0 ⇔ ∇αTαβ = 0
hαβ = 16π |g |Tαβ + Λαβ[h]︸ ︷︷ ︸nonlinearities∂h∂h + ···
≡ 16π ταβ
Weak-field approximation
|hαβ| 1
=⇒ perturbative nonlinear treatment
Metric potential
hαβ ≡ √−ggαβ − ηαβ
Einstein field equations
Gαβ = 8πTαβ ⇐⇒∂αh
αβ = 0 ⇔ ∂αταβ = 0 ⇔ ∇αTαβ = 0
hαβ = 16π |g |Tαβ + Λαβ[h]︸ ︷︷ ︸nonlinearities∂h∂h + ···
≡ 16π ταβ
Weak-field approximation
|hαβ| 1 =⇒ perturbative nonlinear treatment
Flat space retarded propagator
hαβ(t, ~x) = −4
∫R3
d3y
|~x − ~y | ταβ(t − |~x − ~y |/c , ~y)
source
(ct,x)
past lightcone
Post-Newtonian equations of motion
m1
m2
r
v2
v1
GW
GW
n
dv1
dt= −Gm2
r2n +
A1PN
c2+A2PN
c4︸ ︷︷ ︸conservative terms
+A2.5PN
c5︸ ︷︷ ︸rad. reac.
+A3PN
c6︸ ︷︷ ︸cons. term
+A3.5PN
c7︸ ︷︷ ︸rad. reac.
+ · · ·
Conservative post-Newtonian dynamics
dv
dt= −Gm
r2n
+A1PN
c2+A2PN
c4+A3PN
c6+
A4PN
c8︸ ︷︷ ︸non local
+ · · ·
0PN Newton (1687)
1PN Lorentz & Droste (1917); Einstein et al. (1938)
2PN Ohta et al. (1973); Damour & Deruelle (1982)
3PN Jaranowski & Schafer (1999); Blanchet & Faye (2001)
Itoh & Futamase (2003); Foffa & Sturani (2011)
4PN Jaranowski & Schafer (2013); Damour et al. (2014)
Bernard, Blanchet, Bohe, Faye & Marsat (2016)
Poincare group symmetries −→ 10 conserved quantities
Conservative post-Newtonian dynamics
dv
dt= −Gm
r2n +
A1PN
c2
+A2PN
c4+A3PN
c6+
A4PN
c8︸ ︷︷ ︸non local
+ · · ·
0PN Newton (1687)
1PN Lorentz & Droste (1917); Einstein et al. (1938)
2PN Ohta et al. (1973); Damour & Deruelle (1982)
3PN Jaranowski & Schafer (1999); Blanchet & Faye (2001)
Itoh & Futamase (2003); Foffa & Sturani (2011)
4PN Jaranowski & Schafer (2013); Damour et al. (2014)
Bernard, Blanchet, Bohe, Faye & Marsat (2016)
Poincare group symmetries −→ 10 conserved quantities
Conservative post-Newtonian dynamics
dv
dt= −Gm
r2n +
A1PN
c2+A2PN
c4
+A3PN
c6+
A4PN
c8︸ ︷︷ ︸non local
+ · · ·
0PN Newton (1687)
1PN Lorentz & Droste (1917); Einstein et al. (1938)
2PN Ohta et al. (1973); Damour & Deruelle (1982)
3PN Jaranowski & Schafer (1999); Blanchet & Faye (2001)
Itoh & Futamase (2003); Foffa & Sturani (2011)
4PN Jaranowski & Schafer (2013); Damour et al. (2014)
Bernard, Blanchet, Bohe, Faye & Marsat (2016)
Poincare group symmetries −→ 10 conserved quantities
Conservative post-Newtonian dynamics
dv
dt= −Gm
r2n +
A1PN
c2+A2PN
c4+A3PN
c6
+A4PN
c8︸ ︷︷ ︸non local
+ · · ·
0PN Newton (1687)
1PN Lorentz & Droste (1917); Einstein et al. (1938)
2PN Ohta et al. (1973); Damour & Deruelle (1982)
3PN Jaranowski & Schafer (1999); Blanchet & Faye (2001)
Itoh & Futamase (2003); Foffa & Sturani (2011)
4PN Jaranowski & Schafer (2013); Damour et al. (2014)
Bernard, Blanchet, Bohe, Faye & Marsat (2016)
Poincare group symmetries −→ 10 conserved quantities
Conservative post-Newtonian dynamics
dv
dt= −Gm
r2n +
A1PN
c2+A2PN
c4+A3PN
c6+
A4PN
c8︸ ︷︷ ︸non local
+ · · ·
0PN Newton (1687)
1PN Lorentz & Droste (1917); Einstein et al. (1938)
2PN Ohta et al. (1973); Damour & Deruelle (1982)
3PN Jaranowski & Schafer (1999); Blanchet & Faye (2001)
Itoh & Futamase (2003); Foffa & Sturani (2011)
4PN Jaranowski & Schafer (2013); Damour et al. (2014)
Bernard, Blanchet, Bohe, Faye & Marsat (2016)
Poincare group symmetries −→ 10 conserved quantities
Conservative post-Newtonian dynamics
dv
dt= −Gm
r2n +
A1PN
c2+A2PN
c4+A3PN
c6+
A4PN
c8︸ ︷︷ ︸non local
+ · · ·
0PN Newton (1687)
1PN Lorentz & Droste (1917); Einstein et al. (1938)
2PN Ohta et al. (1973); Damour & Deruelle (1982)
3PN Jaranowski & Schafer (1999); Blanchet & Faye (2001)
Itoh & Futamase (2003); Foffa & Sturani (2011)
4PN Jaranowski & Schafer (2013); Damour et al. (2014)
Bernard, Blanchet, Bohe, Faye & Marsat (2016)
Poincare group symmetries −→ 10 conserved quantities
Gravitational-wave tails
• Gravitational waves arescattered by the bkgdcurvature generated bythe source’s mass M
space
time
space
source
• Starting at 4PN order, the near-zone metric depends onthe entire past history of the source [Blanchet & Damour 88]
δg tail00 (x, t) = −8G 2M
5c10x ix j
∫ t
−∞dt ′M
(7)ij (t ′) ln
(c(t − t ′)
2r
)
Phasing for inspiralling compact binaries
• Conservative orbital dynamics → 4PN binding energy
E (ω) = −µ2
(mω)2/3︸ ︷︷ ︸Newtonian
binding energy
(1 + · · ·
)︸ ︷︷ ︸4PN relative
correction
• Wave generation formalism → 3.5PN GW energy flux
F(ω) =32
5
µ2
m2(mω)5︸ ︷︷ ︸
Einstein’squad. formula
(1 + · · ·
)︸ ︷︷ ︸
3.5PN relativecorrection
• Energy balance → 3.5PN orbital phase and GW phase
dE
dt= −F
=⇒ dω
dt= −F(ω)
E ′(ω)=⇒ φ(t) =
∫ t
ω(t ′)dt ′
Phasing for inspiralling compact binaries
• Conservative orbital dynamics → 4PN binding energy
E (ω) = −µ2
(mω)2/3︸ ︷︷ ︸Newtonian
binding energy
(1 + · · ·
)︸ ︷︷ ︸4PN relative
correction
• Wave generation formalism → 3.5PN GW energy flux
F(ω) =32
5
µ2
m2(mω)5︸ ︷︷ ︸
Einstein’squad. formula
(1 + · · ·
)︸ ︷︷ ︸
3.5PN relativecorrection
• Energy balance → 3.5PN orbital phase and GW phase
dE
dt= −F
=⇒ dω
dt= −F(ω)
E ′(ω)=⇒ φ(t) =
∫ t
ω(t ′)dt ′
Phasing for inspiralling compact binaries
• Conservative orbital dynamics → 4PN binding energy
E (ω) = −µ2
(mω)2/3︸ ︷︷ ︸Newtonian
binding energy
(1 + · · ·
)︸ ︷︷ ︸4PN relative
correction
• Wave generation formalism → 3.5PN GW energy flux
F(ω) =32
5
µ2
m2(mω)5︸ ︷︷ ︸
Einstein’squad. formula
(1 + · · ·
)︸ ︷︷ ︸
3.5PN relativecorrection
• Energy balance → 3.5PN orbital phase and GW phase
dE
dt= −F
=⇒ dω
dt= −F(ω)
E ′(ω)=⇒ φ(t) =
∫ t
ω(t ′)dt ′
Phasing for inspiralling compact binaries
• Conservative orbital dynamics → 4PN binding energy
E (ω) = −µ2
(mω)2/3︸ ︷︷ ︸Newtonian
binding energy
(1 + · · ·
)︸ ︷︷ ︸4PN relative
correction
• Wave generation formalism → 3.5PN GW energy flux
F(ω) =32
5
µ2
m2(mω)5︸ ︷︷ ︸
Einstein’squad. formula
(1 + · · ·
)︸ ︷︷ ︸
3.5PN relativecorrection
• Energy balance → 3.5PN orbital phase and GW phase
dE
dt= −F =⇒ dω
dt= −F(ω)
E ′(ω)
=⇒ φ(t) =
∫ t
ω(t ′)dt ′
Phasing for inspiralling compact binaries
• Conservative orbital dynamics → 4PN binding energy
E (ω) = −µ2
(mω)2/3︸ ︷︷ ︸Newtonian
binding energy
(1 + · · ·
)︸ ︷︷ ︸4PN relative
correction
• Wave generation formalism → 3.5PN GW energy flux
F(ω) =32
5
µ2
m2(mω)5︸ ︷︷ ︸
Einstein’squad. formula
(1 + · · ·
)︸ ︷︷ ︸
3.5PN relativecorrection
• Energy balance → 3.5PN orbital phase and GW phase
dE
dt= −F =⇒ dω
dt= −F(ω)
E ′(ω)=⇒ φ(t) =
∫ t
ω(t ′) dt ′
PN vs NR waveformsEqual masses and no spins
0 1000 2000 3000 4000
-0.0004
0
0.0004
NRTaylorT1 3.5/2.5
3200 3400 3600 3800
-0.004
0
0.004
(t-r*)/M
Re(ψ22 )
0 1000 2000 3000 4000
-0.0004
0
0.0004
NRTaylorT4 3.5/3.0
3500 3600 3700 3800 3900
-0.005
0
0.005
(t-r*)/M
Re(ψ22 )
[PRD 76 (2007) 124038]
PN vs NR waveformsUnequal masses and no spins
-0.2
-0.1
0
0.1
0.2
H22
Re[H22, NR
]
Re[H22, PN
]
-2000 -1500 -1000 -500 0 500 1000 1500 2000
(t-Rex
) / M
-0.06
-0.04
-0.02
0
0.02
0.04
H33
Re[H33, NR
]
Re[H33, PN
]
q = 4
[CQG 28 (2011) 134004]
PN vs NR waveforms
-100 -50 0
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
-1 0-31 -30-101 -100
10 100
10-23
10-22
|h~2
2(f
)| f
1/2
[H
z−1
/2]
q=6Buchmanet al. ’12
(t − tpeak) / 1000M
Re(
DL h
22
/ M
) q=7 new longsimulation
f [Hz]
350 GW cycles!
[PRL 115 (2015) 031102]
Further reading
Review articles
• Gravitational radiation from post-Newtonian sources. . .L. Blanchet, Living Rev. Rel. 17, 2 (2014)
• Post-Newtonian methods: Analytic results on the binary problemG. Schafer, in Mass and motion in general relativityEdited by L. Blanchet et al., Springer (2011)
• The post-Newtonian approximation for relativistic compact binariesT. Futamase and Y. Itoh, Living Rev. Rel. 10, 2 (2007)
Topical books
• Gravity: Newtonian, post-Newtonian, relativisticE. Poisson and C. M. Will, Cambridge University Press (2015)
• Gravitational waves: Theory and experimentsM. Maggiore, Oxford University Press (2007)
Outline
1 Gravitational wave source modelling
2 Post-Newtonian approximation
3 Black hole perturbation theory
4 Effective one-body model
5 Comparisons
Extreme mass ratio inspirals
• LISA sensitive to MBH ∼ 105 − 107M → q ∼ 10−7 − 10−4
• Torb ∝ MBH ∼ hr and Tinsp ∝ MBH/q ∼ yrs
(Credit: L. Barack)
Large to extreme mass ratios
numerical relativity
postNewtonian theory
log10
(m2 /m
1)
0 1 2 3
0
1
2
3
4
4
log10
(r /m)
perturbation theory & selfforce
(com
pact
ness
)
mass ratio
−1 m1
m2
q ≡ m1
m2 1
Black hole perturbation theory
Metric perturbation
hαβ ≡ gαβ︸︷︷︸exact
− gαβ︸︷︷︸bkgd
= O(q)
Lorenz gauge condition
∇αhαβ = 0
Einstein field equations
g hαβ + 2Rµ να β hµν = −16πTαβ
Linear equation but involved Green’s function
Black hole perturbation theory
Metric perturbation
hαβ ≡ gαβ︸︷︷︸exact
− gαβ︸︷︷︸bkgd
= O(q)
Lorenz gauge condition
∇αhαβ = 0
Einstein field equations
g hαβ + 2Rµ να β hµν = −16πTαβ
Linear equation but involved Green’s function
Black hole perturbation theory
Metric perturbation
hαβ ≡ gαβ︸︷︷︸exact
− gαβ︸︷︷︸bkgd
= O(q)
Lorenz gauge condition
∇αhαβ = 0
Einstein field equations
g hαβ + 2Rµ να β hµν = −16πTαβ
Linear equation but involved Green’s function
Black hole perturbation theory
Metric perturbation
hαβ ≡ gαβ︸︷︷︸exact
− gαβ︸︷︷︸bkgd
= O(q)
Lorenz gauge condition
∇αhαβ = 0
Einstein field equations
g hαβ + 2Rµ να β hµν = −16πTαβ
Linear equation but involved Green’s function
Gravitational self-force
• Dissipative component ←→ gravitational waves
• Conservative component ←→ secular effects
(Credit: A. Pound)
Gravitational self-force
Spacetime metric
gαβ = gαβ
+ hαβ
Small parameter
q ≡ m1
m2 1
Gravitational self-force
uα ≡ uβ∇βuα = f α[h]
m1→ 0
m2
u
Gravitational self-force
Spacetime metric
gαβ = gαβ
+ hαβ
Small parameter
q ≡ m1
m2 1
Gravitational self-force
uα ≡ uβ∇βuα = f α[h]
m1
m2
u
Gravitational self-force
Spacetime metric
gαβ = gαβ + hαβ
Small parameter
q ≡ m1
m2 1
Gravitational self-force
uα ≡ uβ∇βuα = f α[h]
m1
m2
u
h
Gravitational self-force
Spacetime metric
gαβ = gαβ + hαβ
Small parameter
q ≡ m1
m2 1
Gravitational self-force
uα ≡ uβ∇βuα = f α[h]
m1
m2
f
u
Gravitational self-force
Spacetime metric
gαβ = gαβ + hαβ
Small parameter
q ≡ m1
m2 1
Gravitational self-force
uα ≡ uβ∇βuα = f α[h]
m1
m2
f
u u
f
Equation of motion
Metric perturbation
hαβ = hdirectαβ + htail
αβ
MiSaTaQuWa equation
uα =(gαβ + uαuβ
)︸ ︷︷ ︸projector⊥ uα
(12∇βhtail
λσ −∇λhtailβσ
)uλuσ︸ ︷︷ ︸
“force”
≡ f α[htail]
Beware: the self-force is gauge-dependant
(Credit: A. Pound)
Equation of motion
Metric perturbation
hαβ = hdirectαβ + htail
αβ
MiSaTaQuWa equation
uα =(gαβ + uαuβ
)︸ ︷︷ ︸projector⊥ uα
(12∇βhtail
λσ −∇λhtailβσ
)uλuσ︸ ︷︷ ︸
“force”
≡ f α[htail]
Beware: the self-force is gauge-dependant
(Credit: A. Pound)
Equation of motion
Metric perturbation
hαβ = hdirectαβ + htail
αβ
MiSaTaQuWa equation
uα =(gαβ + uαuβ
)︸ ︷︷ ︸projector⊥ uα
(12∇βhtail
λσ −∇λhtailβσ
)uλuσ︸ ︷︷ ︸
“force”
≡ f α[htail]
Beware: the self-force is gauge-dependant
(Credit: A. Pound)
Equation of motion
Metric perturbation
hαβ = hdirectαβ + htail
αβ
MiSaTaQuWa equation
uα =(gαβ + uαuβ
)︸ ︷︷ ︸projector⊥ uα
(12∇βhtail
λσ −∇λhtailβσ
)uλuσ︸ ︷︷ ︸
“force”
≡ f α[htail]
Beware: the self-force is gauge-dependant
(Credit: A. Pound)
Generalized equivalence principle
singular/self field
hS ∼ m/r
hS ∼ −16πT
f α[hS ] = 0
regular/residual field
hR ∼ htail + local terms
hR ∼ 0
uα = f α[hR ]
Self-acc. motion in gαβ ⇐⇒ Geodesic motion in gαβ + hRαβ
(Credit: A. Pound)
Generalized equivalence principle
singular/self field
hS ∼ m/r
hS ∼ −16πT
f α[hS ] = 0
regular/residual field
hR ∼ htail + local terms
hR ∼ 0
uα = f α[hR ]
Self-acc. motion in gαβ ⇐⇒ Geodesic motion in gαβ + hRαβ
(Credit: A. Pound)
Generalized equivalence principle
singular/self field
hS ∼ m/r
hS ∼ −16πT
f α[hS ] = 0
regular/residual field
hR ∼ htail + local terms
hR ∼ 0
uα = f α[hR ]
Self-acc. motion in gαβ ⇐⇒ Geodesic motion in gαβ + hRαβ
(Credit: A. Pound)
Generalized equivalence principle
singular/self field
hS ∼ m/r
hS ∼ −16πT
f α[hS ] = 0
regular/residual field
hR ∼ htail + local terms
hR ∼ 0
uα = f α[hR ]
Self-acc. motion in gαβ ⇐⇒ Geodesic motion in gαβ + hRαβ
(Credit: A. Pound)
Further reading
Review articles
• Motion of small objects in curved spacetimesA. Pound, in Equations of Motion in Relativistic GravityEdited by D. Puetzfeld et al., Springer (2015)
• The motion of point particles in curved spacetimeE. Poisson, A. Pound and I. Vega, Living Rev. Rel. 14, 7 (2011)
• Gravitational self force in extreme mass-ratio inspiralsL. Barack , Class. Quant. Grav. 26, 213001 (2009)
• Analytic black hole perturbation approach to gravitational radiationM. Sasaki and H. Tagoshi, Living Rev. Rel. 6, 5 (2003)
Outline
1 Gravitational wave source modelling
2 Post-Newtonian approximation
3 Black hole perturbation theory
4 Effective one-body model
5 Comparisons
1m
2m
1m 2m
g
realE Eeff
realJ Nreal effJ effN
Real description
g eff
Effective description
Eeff(J,N) = f (Ereal(J,N))(Credit: Buonanno & Sathyaprakash 2015)
m2
M = m1+ m
2
EOB
m1
HH
real
eff
• Motivated by the exact solution in the Newtonian limit
• By construction, the EOB model:
Recovers the known PN dynamics as c−1 → 0 Recovers the geodesic dynamics when q → 0
• Idea extended to spinning binaries and to tidal effects
m2
M = m1+ m
2
EOB
m1
HH
real
eff
• Motivated by the exact solution in the Newtonian limit
• By construction, the EOB model:
Recovers the known PN dynamics as c−1 → 0 Recovers the geodesic dynamics when q → 0
• Idea extended to spinning binaries and to tidal effects
m2
M = m1+ m
2
EOB
m1
HH
real
eff
• Motivated by the exact solution in the Newtonian limit
• By construction, the EOB model:
Recovers the known PN dynamics as c−1 → 0
Recovers the geodesic dynamics when q → 0
• Idea extended to spinning binaries and to tidal effects
m2
M = m1+ m
2
EOB
m1
HH
real
eff
• Motivated by the exact solution in the Newtonian limit
• By construction, the EOB model:
Recovers the known PN dynamics as c−1 → 0 Recovers the geodesic dynamics when q → 0
• Idea extended to spinning binaries and to tidal effects
m2
M = m1+ m
2
EOB
m1
HH
real
eff
• Motivated by the exact solution in the Newtonian limit
• By construction, the EOB model:
Recovers the known PN dynamics as c−1 → 0 Recovers the geodesic dynamics when q → 0
• Idea extended to spinning binaries and to tidal effects
EOB Hamiltonian dynamics
EOB Hamiltonian
HEOBreal = M
√1 + 2ν
(Heff
µ− 1
), ν ≡ µ
M∈ [0, 1/4]
Effective Hamiltonian
Heff = µ
√√√√g efftt (r)
(1 +
p2φ
r2+
p2r
g effrr (r)
+ · · ·)
Hamilton’s equations
r =∂HEOB
real
∂pr, pr = −∂H
EOBreal
∂r+ Fr , · · ·
EOB Hamiltonian dynamics
EOB Hamiltonian
HEOBreal = M
√1 + 2ν
(Heff
µ− 1
), ν ≡ µ
M∈ [0, 1/4]
Effective Hamiltonian
Heff = µ
√√√√g efftt (r)
(1 +
p2φ
r2+
p2r
g effrr (r)
+ · · ·)
Hamilton’s equations
r =∂HEOB
real
∂pr, pr = −∂H
EOBreal
∂r+ Fr , · · ·
EOB Hamiltonian dynamics
EOB Hamiltonian
HEOBreal = M
√1 + 2ν
(Heff
µ− 1
), ν ≡ µ
M∈ [0, 1/4]
Effective Hamiltonian
Heff = µ
√√√√g efftt (r)
(1 +
p2φ
r2+
p2r
g effrr (r)
+ · · ·)
Hamilton’s equations
r =∂HEOB
real
∂pr, pr = −∂H
EOBreal
∂r+ Fr , · · ·
EOB effective metric
Effective metric
ds2eff = −g eff
tt (r ; ν)dt2 + g effrr (r ; ν)dr2 + r2 dΩ2
Effective potentials
g efftt = 1− 2M
r︸ ︷︷ ︸Schwarzschild
+ ν
[2
(M
r
)3
+
(94
3− 41
32π2
)(M
r
)4
+ · · ·]
︸ ︷︷ ︸finite mass-ratio “deformation”
Pade resummation
Motivation: improve convergence of PN series in strong-field regime
EOB effective metric
Effective metric
ds2eff = −g eff
tt (r ; ν)dt2 + g effrr (r ; ν)dr2 + r2 dΩ2
Effective potentials
g efftt = 1− 2M
r︸ ︷︷ ︸Schwarzschild
+ ν
[2
(M
r
)3
+
(94
3− 41
32π2
)(M
r
)4
+ · · ·]
︸ ︷︷ ︸finite mass-ratio “deformation”
Pade resummation
Motivation: improve convergence of PN series in strong-field regime
EOB effective metric
Effective metric
ds2eff = −g eff
tt (r ; ν)dt2 + g effrr (r ; ν)dr2 + r2 dΩ2
Effective potentials
g efftt = 1− 2M
r︸ ︷︷ ︸Schwarzschild
+ ν
[2
(M
r
)3
+
(94
3− 41
32π2
)(M
r
)4
+ · · ·]
︸ ︷︷ ︸finite mass-ratio “deformation”
Pade resummation
Motivation: improve convergence of PN series in strong-field regime
EOB waveform prediction
-0.2-0.1
00.10.20.3
grav
itatio
nal w
avef
orm inspiral-plunge
merger-ringdown
-250 -200 -150 -100 -50 0 50c3t/GM
0.10.20.30.40.5
freq
uenc
y: G
Mω
/c3 2 x orbital freq
inspiral-plunge GW freqmerger-ringdown GW freq
(Credit: Buonanno & Sathyaprakash 2015)
EOB vs NR waveformsEqual masses and no spins
3800 3820 3840 3860 3880 3900 3920 3940 3960 3980 4000
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
t
Merger time
[PRD 79 (2009) 081503]
EOB vs NR waveformsEqual masses and aligned spins
0 1000 2000 3000 4000 5000 6000(t − r
*) / M
-0.4
-0.2
0.0
0.2
0.4R
e(R
/M h
22)
6000 6100 6200 6300 6400 6500(t − r
*) / M
-0.4
-0.2
0.0
0.2
0.4
Re(
R/M
h22
)NREOB
(q, χ1, χ
2) = (1, +0.98, +0.98)
[PRD 89 (2014) 061502]
EOB vs NR waveformsUnequal masses and a precessing spin
(q, χ1, χ2) = (5,+0.5, 0), ι = π/3
0 1000 2000 3000 4000 5000 6000 7000
(t− r∗)/M
−0.05
0.00
0.05
0.10
(DL/M
)h
+
precessing NRprecessing EOBNR
6900 7000 7100 7200 7300 7400 7500
(t− r∗)/M
[gr-qc/1607.05661]
Recent developments
• Extension of EOB model to spinning binaries[Barausse & Buonanno (2011), Nagar (2011), Damour & Nagar (2014)]
• Addition of tidal interactions for neutrons stars[Damour & Nagar (2010), Bini et al. (2012), Hinderer et al. (2016)]
• Various calibrations to numerical relativity simulations[Damour & Nagar (2014), Pan et al. (2014), Taracchini et al. (2014)]
• Calibration of EOB potentials by comparison to self-force[Barack et al. (2011), Le Tiec (2015), Akcay & van de Meent (2016)]
Recent developments
• Extension of EOB model to spinning binaries[Barausse & Buonanno (2011), Nagar (2011), Damour & Nagar (2014)]
• Addition of tidal interactions for neutrons stars[Damour & Nagar (2010), Bini et al. (2012), Hinderer et al. (2016)]
• Various calibrations to numerical relativity simulations[Damour & Nagar (2014), Pan et al. (2014), Taracchini et al. (2014)]
• Calibration of EOB potentials by comparison to self-force[Barack et al. (2011), Le Tiec (2015), Akcay & van de Meent (2016)]
Recent developments
• Extension of EOB model to spinning binaries[Barausse & Buonanno (2011), Nagar (2011), Damour & Nagar (2014)]
• Addition of tidal interactions for neutrons stars[Damour & Nagar (2010), Bini et al. (2012), Hinderer et al. (2016)]
• Various calibrations to numerical relativity simulations[Damour & Nagar (2014), Pan et al. (2014), Taracchini et al. (2014)]
• Calibration of EOB potentials by comparison to self-force[Barack et al. (2011), Le Tiec (2015), Akcay & van de Meent (2016)]
Recent developments
• Extension of EOB model to spinning binaries[Barausse & Buonanno (2011), Nagar (2011), Damour & Nagar (2014)]
• Addition of tidal interactions for neutrons stars[Damour & Nagar (2010), Bini et al. (2012), Hinderer et al. (2016)]
• Various calibrations to numerical relativity simulations[Damour & Nagar (2014), Pan et al. (2014), Taracchini et al. (2014)]
• Calibration of EOB potentials by comparison to self-force[Barack et al. (2011), Le Tiec (2015), Akcay & van de Meent (2016)]
Further reading
Review articles
• Sources of gravitational waves: Theory and observationsA. Buonanno and B. S. Sathyaprakash, in General relativity andgravitation: A centennial perspectiveEdited by A. Ashtekar et al., Cambridge University Press (2015)
• The general relativistic two body problem and the EOB formalismT. Damour, in General relativity, cosmology and astrophysicsEdited by J. Bicak and T. Ledvinka, Springer (2014)
• The effective one-body description of the two-body problemT. Damour and A. Nagar, in Mass and motion in general relativityEdited by L. Blanchet et al., Springer (2011)
Outline
1 Gravitational wave source modelling
2 Post-Newtonian approximation
3 Black hole perturbation theory
4 Effective one-body model
5 Comparisons
numerical relativity
postNewtonian theory
log10
(m2 /m
1)
0 1 2 3
0
1
2
3
4
4
log10
(r /m)
perturbation theory & selfforce
(com
pact
ness
)
mass ratio
−1
numerical relativity
postNewtonian theory
log10
(m2 /m
1)
0 1 2 3
0
1
2
3
4
4
log10
(r /m)
perturbation theory & selfforce
(com
pact
ness
)
mass ratio
−1 ?
??
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 future null infinity
• Conservative effects on the orbital dynamics
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 future null infinity
• Conservative effects on the orbital dynamics
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 future 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 3Szilagyi et al. 2015 NR/PN/EOB waveformOssokine et al. 2015 NR/PN waveform 3
Paper Year Methods Observable Orbit 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 3Akcay et al. 2015 BHP/PN averaged redshift eccentricShah & Pound 2015 BHP/PN precession angle 3Zimmerman et al. 2016 NR/PN surface gravityAkcay et al. 2016 BHP/PN precession angle eccentric
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
Periastron advance in black hole binaries
• Generic eccentric orbit parametrized bythe two invariant frequencies
Ωr =2π
P, Ωϕ =
1
P
∫ P
0ϕ(t) dt
• Periastron advance per radial period
K ≡ Ωϕ
Ωr= 1 +
∆Φ
2π
• In the circular orbit limit e → 0, therelation K (Ωϕ) is coordinate-invariant
m2
m1
t=0 t=P
Periastron advance vs orbital frequency
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
[PRL 107 (2011) 141101]
Periastron advance vs orbital frequency
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
[PRL 107 (2011) 141101]