General relativistic dynamics of binary black holes
Transcript of General relativistic dynamics of binary black holes
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]