Post-Newtonian methods and applications - Philippe G. LeFloch

POST-NEWTONIAN METHODS AND APPLICATIONS 

Luc Blanchet 

Gravitation et Cosmologie (GRεCO) 

Institut d’Astrophysique de Paris 

4 novembre 2009 

Luc Blanchet (GRεCO) Post-Newtonian methods and applications Chevaleret 2009 1 / 38

Ground-based laser interferometric detectors 

LIGO GEO 

LIGO/VIRGO/GEO observe the GWs in 

the high-frequency band 

10 Hz f 10 3 Hz 

VIRGO 


Space-based laser interferometric detector 

LISA 

LISA will observe the GWs in the low-frequency band 

10 −4 Hz f 10 −1 Hz 


The inspiral and merger of compact binaries 

Neutron stars spiral and coalesce Black holes spiral and coalesce 

1 Neutron star (M = 1.4 M⊙) events will be detected by ground-based 

detectors LIGO/VIRGO/GEO 

2 Stellar size black hole (5 M⊙ M 20 M⊙) events will also be detected by 

ground-based detectors 

3 Supermassive black hole (10 5 M⊙ M 10 8 M⊙) events will be detected 

by the space-based detector LISA 


Supermassive black-hole coalescences as detected by LISA 

When two galaxies collide their central supermassive black holes may form a 

bound binary system which will spiral and coalesce. LISA will be able to detect the 

gravitational waves emitted by such enormous events anywhere in the Universe 


GRAVITATIONAL WAVE TEMPLATES 


Methods to compute gravitational-wave templates 

4 

3 

2 

1 

0 

log 10 (r 12 /m) 

PostNewtonian 

Theory 

Numerical 

Relativity 

0 1 2 3 


Theory 

& 

Perturbation 

Theory 

Perturbation 

Theory 

4 

log 10 (m 2 /m 1 ) 



4 

3 

2 

1 

0 

log 10 (r 12 /m) 


Theory 

Numerical 

Relativity 

m 2 

r 12 

0 1 2 3 

m 1 


Theory 

& 

Perturbation 

Theory 

Perturbation 

Theory 

4 

log 10 (m 2 /m 1 ) 



4 

3 

2 

1 

0 

log 10 (r 12 /m) 


Theory 

Numerical 

Relativity 

0 1 2 3 


Theory 

& 

Perturbation 

Theory 

Perturbation 

Theory 

4 

m 1 

log 10 (m 2 /m 1 ) 

Luc Blanchet (GRεCO) Post-Newtonian methods and applications Chevaleret 2009 7 / 38 

m 2


4 

3 

2 

1 

0 

log 10 (r 12 /m) 


Theory 

Numerical 

Relativity 

0 1 2 3 


Theory 

& 

Perturbation 

Theory 

Perturbation 

Theory 

4 

log 10 (m 2 /m 1 ) 


The inspiral (or chirp) of compact binary systems 

The most interesting known source of gravitational waves for the 

LIGO/VIRGO detectors and a very important one for LISA 

The dynamics of these systems is driven by gravitational radiation reaction 

effects or equivalently by the loss of energy by gravitational radiation 

Theoretical waveforms (templates) for detection and analysis of the signals 

should be very accurate in terms of a post-Newtonian expansion 

Post-Newtonian waveforms for the inspiral should be completed by numerical 

calculations of the merger and ringdown phases 


PN templates for inspiralling compact binaries 

d 

i 

r 

e 

c 

t 

i 

o 

n 

a 

s 

c 

e 

n 

d 

i 

n 

g 

n 

o 

d 

e 

m 2 

ascending node 

 

i 

observer 

m 1 

orbital plane 

The orbital phase φ(t) should be monitored in 

LIGO/VIRGO detectors with precision 

δφ ∼ π 

φ(t) = φ0 − 

o 

f 

1 

 

GMω 

32ν c3 −5/3 1 + 

 

result of the quadrupole formalism 

(sufficient for the binary pulsar) 

1PN 1.5PN 

+ 

c2 c3 

3PN 

+ · · · + + · · · 

c6 

needs to be computed with high PN precision 

Detailed data analysis (using the sensitivity noise curve of LIGO/VIRGO 

detectors) show that the required precision is at least 2PN for detection and 3PN 

for parameter estimation 


Matching the PN inspiral to numerical merger waveforms 

The numerical merger waveform is matched with high accuracy to the PN inspiral 

waveform. Current precision of the PN waveform is 

3.5PN order in phase [LB, Faye, Iyer & Joguet 2002; LB, Damour, Esposito-Farèse & Iyer 2004] 

3PN order in amplitude [LB, Faye, Iyer & Siddhartha 2008] 


PN prediction for the ICO of binary black holes 

The convergence of the standard Taylor 

expanded PN series is very good. 

Numerical point is from [Gourgoulhon, 

Grandclément & Bonazzola 2001] 

There is an excellent agreement with the 

standard 3PN prediction. Numerical 

points are from [Caudill, Cook, Grigsby & Pfeiffer 

2006] 


PN contributions to the number of GW cycles 

Table: PN cycles accumulated in the bandwidth of LIGO/VIRGO 

(10 + 10)M⊙ 

(1.4 + 1.4)M⊙ 

Newtonian 601 16034 

1PN +59.3 +441 

1.5PN −51.4+16.0 κ1 χ1 + 16.0 κ2 χ2 −211+65.7 κ1 χ1 + 65.7 κ2 χ2 

2PN +4.1−3.3 κ1 κ2 χ1 χ2 + 1.1 ξ χ1 χ2 +9.9−8.0 κ1 κ2 χ1 χ2 + 2.8 ξ χ1 χ2 

2.5PN 

3PN 

−7.1+5.5 κ1 χ1 + 5.5 κ2 χ2 

+2.2 

−11.7+9.0 κ1 χ1 + 9.0 κ2 χ2 

+2.6 

3.5PN −0.8 −0.9 

Table: PN cycles accumulated in the bandwidth of LISA 

(10 6 + 10 6 )M⊙ 

(10 5 + 10 5 )M⊙ 

Newtonian 2267 9570 

1PN +134 +323 

1.5PN −92.4+28.8 κ1 χ1 + 28.8 κ2 χ2 −170+53 κ1 χ1 + 53 κ2 χ2 

2PN +6.0−4.8 κ1 κ2 χ1 χ2 + 1.7 ξ χ1 χ2 +8.7−7.1 κ1 κ2 χ1 χ2 + 2.4 ξ χ1 χ2 

2.5PN −9.0+6.9 κ1 χ1 + 6.9 κ2 χ2 −11.0+8.5 κ1 χ1 + 8.5 κ2 χ2 

3PN +2.3 +2.5 

3.5PN −0.9 −0.9 

Spin effects are indicated in green and tail effects are in red 


GRAVITATIONAL SELF-FORCE THEORY 


General problem of the self-force 

A particle is moving on a background 

space-time 

Its own stress-energy tensor modifies the 

background gravitational field 

Because of the “back-reaction” the motion 

of the particle deviates from a background 

geodesic hence the appearance of a self force 

u 

u 

f 

The self acceleration of the particle is proportional to its mass 

Dū µ 

dτ = f µ 

m1 

= O 

The gravitational self force includes both dissipative (radiation reaction) and 

conservative effects. 


m2 

m 1 

f 

m 2

Self-force in perturbation theory 

The space-time metric gµν is decomposed as a background metric ¯gµν plus 

hµν = linearized parturbation of the background space-time 

The field equation in an harmonic gauge reads 

u 

 

particle's trajectory 

 

z 

h µν µ ν 

+ 2R ρ σ h ρσ = −16π T µν 

x 

The retarded solution is 

h µν 

µν 

(x) = 4m1 G ρσ(x, z) ū 

ret 

ρ ū σ + O(m 2 1) 


Γ

Green function responsible for the self-force [Detweiler & Whiting 2003] 

The symmetric (S) Green function is defined by the prescription 

G = 

S 

1 

 

G 

2 ret + G 

−H 

adv 

where H is homogeneous solution of the wave equation 

1 GS is symmetric under a time reversal hence corresponds to stationary waves 

at infinity and does not produce a reaction force on the particle 

2 It has the same divergent behavior as Gret on the particle’s worldline 

3 It is non zero only when x and z are related by a space-like interval 

The radiative (R) Green function responsible for the self force is then 

G(x, z) = G(x, z) − G(x, z) = 

R 

ret S 

1 

 

G 

2 ret − G 

+H 

adv 


Causal behaviour of the radiative Green function 

The radiative Green function is 

1 causal in the sense that 

G(x) = lim G(x, z) 

R x→z R 

depends only on the particle’s past history 

2 smooth (i.e. C ∞ ) on the particle’s trajectory 

z 

G (x,z) 

ret 

z 

x 

G (x,z) 

S 

x 

z 

G (x,z) 

adv 

z 

G (x,z) 

R 


x 

x

Computation of the self-force [Mino, Sasaki & Tanaka 1997; Quinn & Wald 1997] 

The metric perturbation is decomposed as 

hµν = h µν 

S 

 

+ h µν 

R 

 

particular solution homogeneous solution 

The particular solution h µν 

S (symmetric in a time reversal) diverges on the 

particle’s location but the homogeneous solution h µν 

R is perfectly regular 

The self-force f µ is computed from the geodesic motion with respect to 

g SF 

µν ≡ ¯gµν + h R µν 

The divergence on the particle’s trajectory due to GS can be renormalized in a 

redefinition of the particle’s mass 


POST-NEWTONIAN VERSUS SELF-FORCE PREDICTIONS 


Common regime of validity of SF and PN 

4 

3 

2 

1 

0 

log 10 (r 12 /m) 


Theory 

Numerical 

Relativity 

0 1 2 3 


Theory 

& 

Perturbation 

Theory 

Perturbation 

Theory 

4 

log 10 (m 2 /m 1 ) 


m 2 

r 12 

m 1

Motivation for this work [LB, Detweiler, Le Tiec & Whiting 2009] 

Both the PN and SF approaches use a self-field regularization for point particles 

followed by a renormalization. However, the prescription are very different 

1 SF theory is based on a prescription for the Green function GR that is at once 

regular and causal 

2 PN theory uses dimensional regularization and it was shown that subtle issues 

appear at the 3PN order due to the appearance of poles ∝ (d − 3) −1 

How can we make a meaningful comparison? 

To find a gauge-invariant observable computable in both formalisms 


Circular orbits admit a helical Killing vector 

Black hole 

u 

1 

Particle's trajectory 

Light cylinder 

 

k 1 

k k 


Choice of a gauge-invariant observable [Detweiler 2008] 

1 For exactly circular orbits the geometry admits a 

helical Killing vector (HKV) with 

k µ ∂µ = ∂t + ω ∂ϕ (asymptotically) 

2 The four-velocity of the particle is necessarily 

tangent to the HKV hence 

u µ 

1 = uT1 k µ 

1 

3 The relation u T 1 (ω) is well-defined in both PN and 

SF approaches and is gauge-invariant 


time 

m 2 

u 

 

u t 

space 

m 1

Post-Newtonian calculation 

In a coordinate system such that k µ ∂µ = ∂t + ω ∂ϕ everywhere this invariant 

quantity reduces to the zero component of the particle’s four-velocity, 

u t 1 = 

 

− Reg 1 [gµν] 

 

regularized metric 

v µ 

1 vν 1 

c2 −1/2 One needs a self-field regularization 

Hadamard regularization will yield an ambiguity at 3PN order 

Dimensional regularization will be free of any ambiguity at 3PN order 


y 1 

v 1 

r 12 

v 2 

y 2

Dimensional regularization [’t Hooft & Veltman 1972] 

1 Perform the PN iteration in d spatial dimensions 

P (x) = ∆ −1 F (x) ≡ − k 

 

d 

4π 

d x ′ F (x ′ ) 

|x ′ − x| d−2 

2 Obtain value when x → y1 using analytic dimensional continuation 

P (y1) = − k 

 

4π 

3 Expand when ε ≡ d − 3 → 0 to obtain 

d d x 

F (x) 

r d−2 

1 

P (y1) = 1 

ε P−1(y1) + P0(y1) +O (ε) 

 

 

finite part 

pole part 

4 Compute the finite part by means of Hadamard’s partie finie 


3PN result for the gauge-invariant observable 

The result is expressed in terms of the orbital frequency through x ≡ GMω 

c3 3/2 as 

u T = 1 + CN x + C1PN x 2 + C2PN x 3 + C3PN x 4 + O(x 5 ) 

The coefficients depend on mass ratios η ≡ m1m2/M 2 and ∆ ≡ (m1 − m2)/M. 

C3PN = 2835 

 

2835 2183 41 

+ ∆ − − 

256 256 48 64 π2 

 

12199 41 

η − − 

384 64 π2 

 

∆ η 

 

17201 41 

+ − 

576 192 π2 

 

η 2 + 795 

128 ∆ η2 − 2827 

864 η3 + 25 

1728 ∆ η3 + 35 

10368 η4 

We find that the poles ∝ ε −1 cancel out 


3PN prediction for the self-force 

With q ≡ m1/m2 ≪ 1 and y ≡ Gm2ω 

c3 3/2 

u T = u T Schw + q u T SF + q 

 

self-force 

2 u T PSF 

 

+O(q 

post-self-force 

3 ) 

where u T SF = −y − 2y 2 − 5y 3 

+ − 121 41 

+ 

3 32 π2 

 

y 4 + O(y 5 ) 

The 3PN self-force coefficient is [LB, Detweiler, Le Tiec & Whiting 2009] 

C3PN = − 121 41 

+ 

3 32 π2 = −27.6879 · · · 

which agrees with the SF numerical calculation at the 2σ level 

C SF 

3PN = −27.677 ± 0.005 


Comparison between PN and SF predictions 

u t SF 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

N 

1PN 

2PN 

3PN 

Exact 

6 8 10 12 14 16 18 20 

y -1 

↑ 

ISCO 


GRAVITATIONAL RECOIL OF BINARY BLACK HOLES 


Gravitational recoil of BH binaries 

The linear momentum ejection is in the direction of the lighter mass’ velocity 

v 2 

momentum 

ejection 

smaller mass 

m2 center−of−mass 

motion 

Vrecoil 

v 

1 

m 

1 

larger mass 

In the Newtonian approximation [with f(η) ≡ η 2√ 1 − 4η] 

4 6M f(η) 

Vrecoil = 20 km/s 

r fmax 

4 2M f(η) 

= 1500 km/s 

r 

fmax 

[Fitchett 1983] 


The 2PN linear momentum [LB, Qusailah & Will 2005] 

dP i 

dt 

GW 

1PN 

= − 464 

 

f(η) x11/2 1 + − 

105 452 1139 

− 

87 522 η 

tail 

 

309 

x + π x3/2 

58 

 

+ − 71345 36761 147101 

+ η + 

22968 2088 68904 η2 

 

x 2 

 

ˆλ 

 

2PN 

i 

The recoil of the center-of-mass follows from integrating 

dP i recoil 

dt 

= − 

dP i 

dt 

GW 

We find a maximum recoil velocity of 22 km/s at the ISCO 


Estimating the recoil during the plunge 

1 The plunge is approximated by that of a test particle of mass µ moving on a 

geodesic of the Schwarzschild metric of a BH of mass M 

2 The 2PN linear momentum flux is integrated on that orbit (y ≡ M/r) 

∆V i 

plunge = L 

ISCO 

M 

horizon 

ISCO 

plunging geodesic 

of Schwarzschild 

1 

M ω 

dP i 

dt 

dy 

E 2 − (1 − 2y)(1 + L 2 y 2 ) 

E and L are the constant energy and 

angular momentum of the Schwarzschild 

plunging orbit 


Recoil up to coalescence at r = 2M [LB, Qusailah & Will 2005] 


Comparison with numerical works 

v (km/s) 

v (km/s) 

300 

250 

200 

150 

100 

50 

0 

0 100 200 300 

t (M ) 

ADM 

300 

250 

200 

150 

100 

50 

h1 

h2 

h3 

r 0 = 6.0 M 

r 0 = 7.0 M 

r 0 = 8.0 M 

0 

0 100 200 300 400 500 

t (M ) ADM 

Mass ratio η = 0.24 [Baker, Centrella, Choi, 

Koppitz, van Meter & Miller 2006] 

Kick at the maximum is 175 km/s 

Final kick is 105 km/s 

Mass ratio η = 0.19 [Gonzalez, Sperhake, 

Bruegmann, Hannam & Husa 2006] 

Kick at the maximum is 250 km/s 

Final kick is 160 km/s 


Close-limit expansion with PN initial conditions [Le Tiec & LB 2009] 

1 Start with the 2PN-accurate metric 

of two point-masses 

g 2PN 

00 

= −1 + 2Gm1 

c 2 r1 

+ 2Gm2 

c 2 r2 

2 Expand it formally in CL form i.e. 

r12 

r 

→ 0 

+ ... 

3 Identify the perturbation from the 

Schwarzschild BH 

g 2PN 

µν = g Schw 

µν + hµν 


x 

m 2 

y 2 

n 2 

r 2 

φ 

θ 

z 

β 

r 

n 

n 12 

x 

r 12 

r 1 

y 1 

n 1 

m 1 

y

Numerical evolution of the perturbation 

1 We recast the initial PN perturbation in Regge-Wheeler-Zerilli formalism 

hµν = h (e) 

µν 

 

+ h 

polar modes 

(o) 

µν 

 

axial modes 

2 Starting from these PN conditions the Regge-Wheeler and Zerilli master 

functions are evolved numerically 

∂ 2 

∂2 

− 

∂t2 ∂r2 + V 

∗ 

(e,o) 

ℓ 

 

Ψ (e,o) 

ℓ,m 

= 0 

3 The linear momentum flux is obtained in a standard way as 

dPx 

dt 

+ i dPy 

dt 

= − 1 

8π 

 

ℓ,m 

+bℓ,m 

 

i aℓ,m ˙ Ψ (e) 

ℓ,m ˙¯ Ψ (o) 

ℓ,m+1 

 

˙Ψ (e) 

ℓ,m ˙¯ (e) 

Ψ ℓ+1,m+1 + ˙ Ψ (o) 

ℓ,m ˙¯ 

 

(o) 

Ψ ℓ+1,m+1 


Final recoil velocity [Le Tiec, LB & Will 2009] 

Kick Velocity (km/s) 

250 

200 

150 

100 

50 

0 

Sopuerta et al. 

Baker et al. 

Campanelli 

Damour & Gopakumar 

Herrmann et al. 

González et al. 07 

González et al. 09 

BQW 

This work 

0.1 0.15 0.2 0.25 

η 


The unreasonable effectiveness of the PN approximation 1 

1 Clifford Will, adapting “The unreasonable effectiveness of mathematics in the natural sciences” 

1 Post-Newtonian theory has proved to be the appropriate tool to describe the 

inspiral phase of compact binaries up to the ICO. 

2 The 3.5PN templates should be sufficient for detection and analysis of 

neutron star binary inspirals in LIGO/VIRGO 

3 For massive BH binaries the PN templates should be matched to the results 

of numerical relativity for the merger and ringdown phases 

4 The PN approximation is now tested against different approaches such as the 

SF and performs very well. This provides a test of the self-field regularization 

scheme for point particles 

5 A combination of semi-analytic approximations based on PN theory gives the 

correct result for the recoil (essentially generated in the strong field regime)

Post-Newtonian methods and applications - Philippe G. LeFloch

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