post-newtonian methods and applications · post-newtonian methods and applications luc blanchet...

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

http://www.iap.fr

Ground-based laser interferometric detectors

LIGO GEO

LIGO/VIRGO/GEO observe the GWs inthe high-frequency band

10 Hz . f . 103 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 = 1.4M�) events will be detected by ground-baseddetectors LIGO/VIRGO/GEO

2 Stellar size black hole (5M� .M . 20M�) events will also be detected byground-based detectors

3 Supermassive black hole (105M� .M . 108M�) events will be detectedby the space-based detector LISA


Supermassive black-hole coalescences as detected by LISA

When two galaxies collide their central supermassive black holes may form abound binary system which will spiral and coalesce. LISA will be able to detect thegravitational waves emitted by such enormous events anywhere in the Universe


GRAVITATIONAL WAVE TEMPLATES


Methods to compute gravitational-wave templates

Numerical Relativity

Perturbation Theory

PostNewtonian Theory

log10

(m2/m

1)

log10

(r12

/m)

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Perturbation Theory


log10

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Perturbation Theory


log10

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/m)

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m2




Perturbation Theory


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/m)

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The inspiral (or chirp) of compact binary systemsThe most interesting known source of gravitational waves for theLIGO/VIRGO detectors and a very important one for LISAThe dynamics of these systems is driven by gravitational radiation reactioneffects or equivalently by the loss of energy by gravitational radiationTheoretical waveforms (templates) for detection and analysis of the signalsshould be very accurate in terms of a post-Newtonian expansionPost-Newtonian waveforms for the inspiral should be completed by numericalcalculations of the merger and ringdown phases


PN templates for inspiralling compact binaries

m1

2m

observer

direction of ascending node

ascending node

orbital plane

i

The orbital phase φ(t) should be monitored inLIGO/VIRGO detectors with precision

δφ ∼ π

φ(t) = φ0−1

32ν

(GMω

c3

)−5/3

︸︷︷︸result of the quadrupole formalism

(sufficient for the binary pulsar)

{1 +

1PNc2

+1.5PNc3

+ · · ·+ 3PNc6

+ · · ·︸︷︷︸needs to be computed with high PN precision

}

Detailed data analysis (using the sensitivity noise curve of LIGO/VIRGOdetectors) show that the required precision is at least 2PN for detection and 3PNfor parameter estimation


Matching the PN inspiral to numerical merger waveforms

The numerical merger waveform is matched with high accuracy to the PN inspiralwaveform. Current precision of the PN waveform is

3.5PN order in phase [LB, Faye, Iyer & Joguet 2002; LB, Damour, Esposito-Farese & 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 Taylorexpanded PN series is very good.Numerical point is from [Gourgoulhon,

Grandclement & Bonazzola 2001]

There is an excellent agreement with thestandard 3PN prediction. Numericalpoints 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 160341PN +59.3 +4411.5PN −51.4+16.0κ1 χ1 + 16.0κ2 χ2 −211+65.7κ1 χ1 + 65.7κ2 χ22PN +4.1−3.3κ1 κ2 χ1 χ2 + 1.1 ξ χ1 χ2 +9.9−8.0κ1 κ2 χ1 χ2 + 2.8 ξ χ1 χ22.5PN −7.1+5.5κ1 χ1 + 5.5κ2 χ2 −11.7+9.0κ1 χ1 + 9.0κ2 χ23PN +2.2 +2.63.5PN −0.8 −0.9

Table: PN cycles accumulated in the bandwidth of LISA

(106 + 106)M� (105 + 105)M�

Newtonian 2267 95701PN +134 +3231.5PN −92.4+28.8κ1 χ1 + 28.8κ2 χ2 −170+53κ1 χ1 + 53κ2 χ22PN +6.0−4.8κ1 κ2 χ1 χ2 + 1.7 ξ χ1 χ2 +8.7−7.1κ1 κ2 χ1 χ2 + 2.4 ξ χ1 χ22.5PN −9.0+6.9κ1 χ1 + 6.9κ2 χ2 −11.0+8.5κ1 χ1 + 8.5κ2 χ23PN +2.3 +2.53.5PN −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 backgroundspace-time

Its own stress-energy tensor modifies thebackground gravitational field

Because of the “back-reaction” the motionof the particle deviates from a backgroundgeodesic hence the appearance of a self force

m1

m2

f

u u

f

The self acceleration of the particle is proportional to its mass

Duµ

dτ= fµ = O

(m1

m2

)The gravitational self force includes both dissipative (radiation reaction) andconservative effects.


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

�hµν + 2Rµ νρ σ h

ρσ = −16π Tµν

particle's trajectory

x

z

u

The retarded solution is

hµν(x) = 4m1

∫ΓGret

µνρσ(x, z) uρ uσ +O(m2

1)


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

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

GS

=12

[Gret

+ Gadv−H

]where H is homogeneous solution of the wave equation

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

GR

(x, z) = Gret

(x, z)−GS

(x, z) =12

[Gret− G

adv+H

]


Causal behaviour of the radiative Green function

The radiative Green function is

1 causal in the sense that

GR

(x) = limx→z

GR

(x, z)

depends only on the particle’s past history

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

x

z

G (x,z)ret

x

z

G (x,z)adv

zx

G (x,z)S

zx

G (x,z)R


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

The metric perturbation is decomposed as

hµν = hSµν︸︷︷︸

particular solution

+ hRµν︸︷︷︸

homogeneous solution

The particular solution hµνS (symmetric in a time reversal) diverges on theparticle’s location but the homogeneous solution hµνR is perfectly regular

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

gSFµν ≡ gµν + h

Rµν

The divergence on the particle’s trajectory due to GS can be renormalized in aredefinition of the particle’s mass


POST-NEWTONIAN VERSUS SELF-FORCE PREDICTIONS


Common regime of validity of SF and PN


Perturbation Theory


log10

(m2/m

1)

log10

(r12

/m)

0 1 2 3

0

1

2

3


&Perturbation

Theory

4

4

m1

m2

r12


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

Both the PN and SF approaches use a self-field regularization for point particlesfollowed 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 onceregular and causal

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

Light cylinder

Particle's trajectory

k k k

u

Black hole

1

1


Choice of a gauge-invariant observable [Detweiler 2008]

1 For exactly circular orbits the geometry admits ahelical Killing vector (HKV) with

kµ∂µ = ∂t + ω ∂ϕ (asymptotically)

2 The four-velocity of the particle is necessarilytangent to the HKV hence

uµ1 = uT1 kµ1

3 The relation uT1 (ω) is well-defined in both PN andSF approaches and is gauge-invariant

space

time u

ut

m2

m1


Post-Newtonian calculation

In a coordinate system such that kµ∂µ = ∂t + ω ∂ϕ everywhere this invariantquantity reduces to the zero component of the particle’s four-velocity,

ut1 =(− Reg1 [gµν ]︸︷︷︸

regularized metric

vµ1 vν1

c2

)−1/2

v1

y1

y2

r12

v2

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


Dimensional regularization [’t Hooft & Veltman 1972]

1 Perform the PN iteration in d spatial dimensions

P (x) = ∆−1F (x) ≡ − k

4π

∫ddx′

F (x′)|x′ − x|d−2

2 Obtain value when x→ y1 using analytic dimensional continuation

P (y1) = − k

4π

∫ddx

F (x)rd−21

3 Expand when ε ≡ d− 3→ 0 to obtain

P (y1) =1εP−1(y1)︸︷︷︸pole part

+ P0(y1)︸︷︷︸finite part

+O (ε)

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 ≡(GMωc3

)3/2as

uT = 1 + CN x+ C1PN x2 + C2PN x

3 + C3PN x4 +O(x5)

The coefficients depend on mass ratios η ≡ m1m2/M2 and ∆ ≡ (m1 −m2)/M .

C3PN =2835256

+2835256

∆−[

218348− 41

64π2

]η −

[12199384

− 4164π2

]∆ η

+[

17201576

− 41192

π2

]η2 +

795128

∆ η2 − 2827864

η3 +25

1728∆ η3 +

3510368

η4

We find that the poles ∝ ε−1 cancel out


3PN prediction for the self-force

With q ≡ m1/m2 � 1 and y ≡(Gm2ωc3

)3/2uT = uTSchw + q uTSF︸︷︷︸

self-force

+ q2 uTPSF︸︷︷︸post-self-force

+O(q3)

where uTSF = −y − 2y2 − 5y3 +(−121

3+

4132π2

)y4 +O(y5)

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

C3PN = −1213

+4132π2 = −27.6879 · · ·

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

CSF3PN = −27.677± 0.005


Comparison between PN and SF predictions

0

0.1

0.2

0.3

0.4

0.5

6 8 10 12 14 16 18 20

ut S

F

y-1

N1PN2PN3PN

Exact

↑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

v2

m1

m2

smaller mass

Vrecoil

motioncenter−of−mass

v1

larger mass momentumejection

In the Newtonian approximation [with f(η) ≡ η2√

1− 4η]

Vrecoil = 20 km/s(

6Mr

)4f(η)fmax

= 1500 km/s(

2Mr

)4f(η)fmax

[Fitchett 1983]


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

(dP i

dt

)GW

= −464105

f(η)x11/2

[1 +

1PN︷︸︸︷(−452

87− 1139

522η

)x+

tail︷︸︸︷30958

π x3/2

+(−71345

22968+

367612088

η +14710168904

η2

)x2︸︷︷︸

2PN

]λi

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

dP irecoil

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 ageodesic 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 iplunge = L

∫ horizon

ISCO

(1

M ω

dP i

dt

)dy√

E2 − (1− 2y)(1 + L2 y2)

M

plunging geodesicof Schwarzschild

ISCO

E and L are the constant energy andangular momentum of the Schwarzschildplunging orbit


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


Comparison with numerical works

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

Koppitz, van Meter & Miller 2006]

Kick at the maximum is 175 km/sFinal kick is 105 km/s

0 100 200 300t (M

ADM)

0

50

100

150

200

250

300

v (k

m/s

)

h1h2h3

0 100 200 300 400 500t (M

ADM)

0

50

100

150

200

250

300

v (k

m/s

)

r0 = 6.0 M

r0 = 7.0 M

r0 = 8.0 M

Mass ratio η = 0.19 [Gonzalez, Sperhake,

Bruegmann, Hannam & Husa 2006]

Kick at the maximum is 250 km/sFinal kick is 160 km/s


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

1 Start with the 2PN-accurate metricof two point-masses

g2PN00 = −1 +

2Gm1

c2r1+

2Gm2

c2r2+ ...

2 Expand it formally in CL form i.e.

r12

r→ 0

3 Identify the perturbation from theSchwarzschild BH

g2PNµν = gSchw

µν + hµν

βx

y

z

m1

m2

n

n12

θ

φ

rr

2

r1

r12

n1

n2

x

y1

y2


Numerical evolution of the perturbation

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

hµν = h(e)µν︸︷︷︸

polar modes

+ h(o)µν︸︷︷︸

axial modes

2 Starting from these PN conditions the Regge-Wheeler and Zerilli masterfunctions are evolved numerically(

∂2

∂t2− ∂2

∂r2∗

+ V(e,o)`

)Ψ(e,o)`,m = 0

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

dPxdt

+ idPydt

= − 18π

∑`,m

[i a`,m Ψ(e)

`,m˙Ψ(o)`,m+1

+b`,m(

Ψ(e)`,m

˙Ψ(e)`+1,m+1 + Ψ(o)

`,m˙Ψ(o)`+1,m+1

)]Luc Blanchet (GRεCO) Post-Newtonian methods and applications Chevaleret 2009 36 / 38

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

0.1 0.15 0.2 0.25η

0

50

100

150

200

250

Kic

k V

eloc

ity (

km/s

)

Sopuerta et al.Baker et al.CampanelliDamour & GopakumarHerrmann et al.González et al. 07González et al. 09BQWThis work


The unreasonable effectiveness of the PN approximation1

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

1 Post-Newtonian theory has proved to be the appropriate tool to describe theinspiral phase of compact binaries up to the ICO.

2 The 3.5PN templates should be sufficient for detection and analysis ofneutron star binary inspirals in LIGO/VIRGO

3 For massive BH binaries the PN templates should be matched to the resultsof numerical relativity for the merger and ringdown phases

4 The PN approximation is now tested against different approaches such as theSF and performs very well. This provides a test of the self-field regularizationscheme for point particles

5 A combination of semi-analytic approximations based on PN theory gives thecorrect result for the recoil (essentially generated in the strong field regime)


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