Modelling gravitational waves with post-Newtonian theory Laura - - PowerPoint PPT Presentation

Modelling gravitational waves with post-Newtonian theory Laura Bernard

(LUTH - Observatoire de Paris) 23-27 September Kavli - RISE Summer School on Gravitational Waves

Outline

• 1. Introduction
• 2. The post-Newtonian expansion
• 3. The wave generation formalism
• 4. Methodology and example
• 5. Concluding remarks

1

Introduction

Compacity vs mass ratio

[Le Tiec 2014]

2

Compacity vs mass ratio

[Le Tiec 2014]

3

Compacity vs mass ratio

[Le Tiec 2014]

4

Compacity vs mass ratio

[Le Tiec 2014]

5

Compacity vs mass ratio

[Le Tiec 2014]

6

Analytical approximation method

In the past Solar system precession of Mercury’s orbit, light deflection Compact objects binary pulsars : when the mutual gravitational field is weak Gravitational waves compact binaries, . . . Cosmological observations

7

Analytical approximation method

In the past Solar system precession of Mercury’s orbit, light deflection Compact objects binary pulsars : when the mutual gravitational field is weak Gravitational waves compact binaries, . . . Cosmological observations Post-Newtonian theory

• approximation solution to GR
• applies to weak gravitational fields and slow motion
• relies on perturbative techniques

⊲ we call nPN order the O 1 c2n

• correction w.r.t. the Newtonian order

7

Problems to solve

Motion what are the conservative equations of motion of the source including non-linear effects ? Propagation what is the propagation of gravitational waves from the source to the detector including non-linear effects ? Generation what is the gravitational radiation field generated in a detector far from the source ? Reaction what are the dissipative radiation forces inside the source as a reaction to the emission of gravitational waves ?

8

The post-Newtonian expansion

Hypotheses

Post-Newtonian source Isolated, compact support, smooth T µν, slowly moving, weakly stressed and weakly gravitating

Exterior zone Near zone Buffer zone 9

Hypotheses

Post-Newtonian source Isolated, compact support, smooth T µν, slowly moving, weakly stressed and weakly gravitating

Exterior zone Near zone Buffer zone

ǫ ≡ v2

12

c2 ∼ Gm r12c2 ≪ 1

9

Hypotheses

Post-Newtonian source Isolated, compact support, smooth T µν, slowly moving, weakly stressed and weakly gravitating

Exterior zone Near zone Buffer zone

ǫ ≡ v2

12

c2 ∼ Gm r12c2 ≪ 1

9

Hypotheses

Post-Newtonian source Isolated, compact support, smooth T µν, slowly moving, weakly stressed and weakly gravitating

Exterior zone Near zone Buffer zone

ǫ ≡ v2

12

c2 ∼ Gm r12c2 ≪ 1 Boundary condition at infinity

• no incoming radiation at past null infinity

lim

r→∞ t+ r

c =cst

d dr + d cdt r hαβ = 0

9

Hypotheses

Post-Newtonian source Isolated, compact support, smooth T µν, slowly moving, weakly stressed and weakly gravitating

Exterior zone Near zone Buffer zone

ǫ ≡ v2

12

c2 ∼ Gm r12c2 ≪ 1 Boundary condition at infinity

• no incoming radiation at past null infinity
• in practice : stationary source in the past

∂ ∂t

• hαβ(x, t)
• = 0

when t < −T

9

Approximation

Tidal moment sourced by the companion body E(A)

ij

= − [∂ijUext]A

10

Approximation

Tidal moment sourced by the companion body E(A)

ij

= − [∂ijUext]A Induced quadrupole moment Q(A)

ij

= −λ(A)

2

E(A)

ij

with λ(A)

2

= 2 k(A)

2

R5

A

3 G the tidal Love number of body A

10

Approximation

Tidal moment sourced by the companion body E(A)

ij

= − [∂ijUext]A Induced quadrupole moment Q(A)

ij

= −λ(A)

2

E(A)

ij

with λ(A)

2

= 2 k(A)

2

R5

A

3 G the tidal Love number of body A

Effacement of the internal structure for a compact object Ftidal FN ∼ RA r12 5 k(A)

2

∝

GmA RAc2 ∼1

GmA r12 c2 5 = O v c 10

10

Approximation

Tidal moment sourced by the companion body E(A)

ij

= − [∂ijUext]A Induced quadrupole moment Q(A)

ij

= −λ(A)

2

E(A)

ij

with λ(A)

2

= 2 k(A)

2

R5

A

3 G the tidal Love number of body A

Effacement of the internal structure for a compact object Ftidal FN ∼ RA r12 5 k(A)

2

∝

GmA RAc2 ∼1

GmA r12 c2 5 = O v c 10 = ⇒ Point-particle approximation valid up to 5PN

• explicitely tested at 2PN [Mitchell & Will 2007]

10

Rewriting the Einstein equation

Gµν = 8πG c4 T µν We define the gothic metric hµν ≡ √−ggµν − ηµν rewrite the field equations hµν

flat d’Alembertian ηρσ∂ρ∂σ

=

matter fields

• 16πG

c4 |g|T µν + Λµν[h, ∂h, ∂2h]

• non-linearities : Λ∼h∂2h+∂h∂h+h∂h∂h+···

and impose the harmonic gauge condition ∂ν hµν = 0

11

Rewriting the Einstein equation

Gµν = 8πG c4 T µν We define the gothic metric hµν ≡ √−ggµν − ηµν rewrite the field equations hµν

flat d’Alembertian ηρσ∂ρ∂σ

=

matter fields

• 16πG

c4 |g|T µν + Λµν[h, ∂h, ∂2h]

• non-linearities : Λ∼h∂2h+∂h∂h+h∂h∂h+···

and impose the harmonic gauge condition ∂ν hµν = 0 ⊲ This is a well-posed wave equation in flat spacetime (Choquet-Bruhet, 1956)

11

Rewriting the Einstein equation

Gµν = 8πG c4 T µν We define the gothic metric hµν ≡ √−ggµν − ηµν rewrite the field equations hµν

flat d’Alembertian ηρσ∂ρ∂σ

=

matter fields

• 16πG

c4 |g|T µν + Λµν[h, ∂h, ∂2h]

• non-linearities : Λ∼h∂2h+∂h∂h+h∂h∂h+···

and impose the harmonic gauge condition ∂ν hµν = 0 ⊲ The field equations contains the matter conservation equations ∂νhµν = 0 ⇐ ⇒ ∇νT µν = 0

11

Rewriting the Einstein equation

Gµν = 8πG c4 T µν We define the gothic metric hµν ≡ √−ggµν − ηµν rewrite the field equations hµν

flat d’Alembertian ηρσ∂ρ∂σ

=

matter fields

• 16πG

c4 |g|T µν + Λµν[h, ∂h, ∂2h]

• non-linearities : Λ∼h∂2h+∂h∂h+h∂h∂h+···

and impose the harmonic gauge condition ∂ν hµν = 0 ⊲ Λ contains all non-linearities c4 16πG Λµν =

Landau-Lifshitz pseudo-tensor

• tµν

LL

+ tµν

H

• ∼h∂2h+∂h∂h

11

The relaxed Einstein field equations

   hµν = 16πG c4 τ µν ∂ν hµν = 0 with τ µν ≡ |g| T µν +

c4 16πG (tµν LL + tµν H ) the stress-energy pseudo-tensor

⊲ This is an exact formulation of Einstein equations ⊲ The wave equation determines hµν for a certain distribution of matter. ⊲ The matter is governed by the conservation equation ∂ντ µν = 0 ⇐ ⇒ ∂ν hµν = 0

12

The relaxed Einstein field equations

   hµν = 16πG c4 τ µν ∂ν hµν = 0 with τ µν ≡ |g| T µν +

c4 16πG (tµν LL + tµν H ) the stress-energy pseudo-tensor

⊲ This is an exact formulation of Einstein equations ⊲ The wave equation determines hµν for a certain distribution of matter. ⊲ The matter is governed by the conservation equation ∂ντ µν = 0 ⇐ ⇒ ∂ν hµν = 0 The harmonic gauge condition ∂ν hµν = 0

• either we impose it in the field equations tµν

H |harmonic = 0

• or we solve the eqs. without enforcing it ⇒ relaxed Einstein field equations

12

Solving the wave equation

The retarded solution hµν(x, t) = 16πG c4

• −1

retτ µν

(x, t)

• Flat-space retarded propagator
• −1

retτ

• (x, t) ≡ − 1

4π

• R3

d3x′ |x − x′| τ

• x′, t − |x − x′|

c

• ⊲ This is an integral over the past light cone of the point (x, t)

source

13

The different zones

Near zone

near zone r ≤ R

14

The different zones

Near zone Exterior zone

near zone r ≤ R exterior zone r > a

14

The different zones

Near zone Exterior zone Buffer zone

near zone r ≤ R buffer zone a < r ≤ R exterior zone r > a

14

The different zones

Near zone Exterior zone Buffer zone Wave zone

near zone r ≤ R buffer zone a < r ≤ R exterior zone r > a wave zone r ≫ λ

14

Problems

In the near zone : PN expansion τ

• x′, t − |x−x′|

c

• |x − x′|

= τ (x′, t) |x − x′| − ˙ τ (x′, t) c + |x − x′| 2c2 ¨ τ

• x′, t
• + · · ·

⊲ generates an expansion in powers of 1

c

• each term is instantaneous

⊲ how to include information about the boundary conditions at infinity ?

• a

λGW ∼ v c ≪ 1 = ⇒ expansion ill-behaved when r ≫ λGW

15

Problems

In the near zone : PN expansion τ

• x′, t − |x−x′|

c

• |x − x′|

= τ (x′, t) |x − x′| − ˙ τ (x′, t) c + |x − x′| 2c2 ¨ τ

• x′, t
• + · · ·

⊲ generates an expansion in powers of 1

c

• each term is instantaneous

⊲ how to include information about the boundary conditions at infinity ?

• a

λGW ∼ v c ≪ 1 = ⇒ expansion ill-behaved when r ≫ λGW In the wave zone : multipolar expansion τ

• x′, t − |x−x′|

c

• |x − x′|

= τ

• x′, t − r

c

• r

− x′j∂j

• τ
• x′, t − r

c

• r
• + · · ·

⊲ expression ill-behaved when r − → 0

15

The different zones

Near zone Exterior zone Buffer zone Wave zone

near zone r ≤ R buffer zone a < r ≤ R exterior zone r > a wave zone r ≫ λ

16

The different zones

Near zone Exterior zone Buffer zone Wave zone

near zone r ≤ R buffer zone a < r ≤ R exterior zone r > a wave zone r ≫ λ

16

The different zones

Near zone Exterior zone Buffer zone Wave zone

near zone r ≤ R buffer zone a < r ≤ R exterior zone r > a wave zone r ≫ λ

16

The multipolar post-Newtonian formalism

In the near zone : post-Newtonian expansion, nPN = O

• 1

c2n

• ¯

hµν =

∞

• m=2

1 cm ¯ hµν

m

with ¯ hµν

m = 16πG ¯

τ µν

m

Exterior zone Near zone Buffer zone

In the exterior zone : multipolar post-Minkowskian expansion M(h)αβ =

∞

• n=1

Gnhαβ

(n)

with hαβ

(n) = Λαβ n

• h(1), . . . , h(n−1)
• Matching in the buffer zone :

M(h) = M ¯ h

• everywhere

radiative moments ← −

• exp. in 1/R source moments

− →

matching source 17

The wave generation formalism

In the exterior region

The vacuum field equations

• hµν

ext = Λµν

∂ν hµν

ext = 0 18

In the exterior region

The vacuum field equations

• hµν

ext = Λµν

∂ν hµν

ext = 0

The post-Minkowskian expansion

• Nonlinear PM expansion

hµν

ext = G hµν (1) + G2 hµν (2) + G3 hµν (3) + · · ·

• Hierarchy of PM equations

   hµν

(n) = Λµν (n)

• h(1), . . . , h(n−1)
• ∂ν hµν

(n) = 0 18

Linearized solution : h(1)

hµν

(1) = 0

General multipolar solution hµν

(1) (x, t) = +∞

• l=0

∂L

• Rµν

L

• t − r

c

• r
• No-incoming radiation =

⇒ function only of u ≡ t − r

c 19

Linearized solution : h(1)

hµν

(1) = 0

General multipolar solution hµν

(1) (x, t) = +∞

• l=0

∂L

• Rµν

L

• t − r

c

• r
• No-incoming radiation =

⇒ function only of u ≡ t − r

c

Gauge considerations

10

• symmetric trace-free

− 4

• gauge conditions ∂νhµν=0

= 6

• independent multipole moments

19

Linearized solution : h(1)

hµν

(1) = 0

General multipolar solution hµν

(1) (x, t) = +∞

• l=0

∂L

• Rµν

L

• t − r

c

• r
• No-incoming radiation =

⇒ function only of u ≡ t − r

c

Gauge considerations

10

• symmetric trace-free

− 4

• gauge conditions ∂νhµν=0

= 6

• independent multipole moments

IL (u) : mass-moment of order l JL (u) : current-moment of order l WL, XL, YL, ZL : gauge-moments of order l

19

Multipole moments and conservation laws

Conservation laws ∂ν hµν

(1) = 0

= ⇒ 10 conserved quantities M ≡ I = cst : total mass Xi ≡ Ii

T = cst : center-of-mass position

Pi ≡ ˙ Ii = cst : linear momentum Si ≡ Ji = cst : angular momentum ⊲ Exact total quantities for the source ⊲ includes the GWs emitted by the source

20

Multipole moments and conservation laws

Conservation laws ∂ν hµν

(1) = 0

= ⇒ 10 conserved quantities M ≡ I = cst : total mass Xi ≡ Ii

T = cst : center-of-mass position

Pi ≡ ˙ Ii = cst : linear momentum Si ≡ Ji = cst : angular momentum ⊲ Exact total quantities for the source ⊲ includes the GWs emitted by the source Source multipole moments

• IL, JL : encode all information from the source at linear order
• WL, XL, YL, ZL : enter at non-linear order

20

Non-linear vacuum solution

hµν

(n) = Λµν (n)

• h(1), . . . , h(n−1)
• Most general multipolar solution

gext

µν (x; IL, JL, WL, XL, YL, ZL) 21

Non-linear vacuum solution

hµν

(n) = Λµν (n)

• h(1), . . . , h(n−1)
• Most general multipolar solution

gext

µν (x; IL, JL, WL, XL, YL, ZL)

⊲ Coordinate transformation x′µ = xµ + Gϕµ (x; WL, XL, YL, ZL)) ⊲ Canonical metric gcan

µν (x; ML, SL)

⊲ Canonical moments ML = IL + O(G)

nonlinar functional of

mass type SL = JL +

IL,JL,WL,...,ZL

O(G) current type

21

Asymptotic behaviour of the solution

remove the logarithmic dependence at J + = ⇒ coordinate transformation (t, x) − → (T, X) Asymptotic waveform

HT T

ij

= 4G c2R Pijkl(N)

+∞

• l=2

1 cll!

• NL−2UklL−2 (U) + 1

c NaL−2εab(kVl)bL−2 (U)

• +O

1 R2

• with U ≡ T − R

c , asymptotically a null coordinate at J +

Radiative multipole moments UL

• T − R

c

• = M (l)

L + O(G) : mass-type moment of order l

VL

• T − R

c

• = S(l)

L + O(G) : current-type moment of order l 22

Asymptotic behaviour of the solution

remove the logarithmic dependence at J + = ⇒ coordinate transformation (t, x) − → (T, X) Asymptotic waveform

HT T

ij

= 4G c2R Pijkl(N)

+∞

• l=2

1 cll!

• NL−2UklL−2 (U) + 1

c NaL−2εab(kVl)bL−2 (U)

• +O

1 R2

• with U ≡ T − R

c , asymptotically a null coordinate at J +

Radiative multipole moments UL

• T − R

c

• = M (l)

L + O(G) : mass-type moment of order l

VL

• T − R

c

• = S(l)

L + O(G) : current-type moment of order l

To summarize IL, JL, WL, XL, YL, ZL

• source

− → ML, SL

• canonical

− → UL, VL

radiative 22

The buffer zone

Near zone Exterior zone Buffer zone Wave zone

M(h) ∼ M

• h
• when

a ≤ r ≤ R

23

Theory of matched asymptotic expansions

The matching equation M(h)

PN expansion of each multipolar coeff.

≡ M

• h
• multipole expansion of each PN coeff.

everywhere

• This is a formal equality between two formal asymptotic series

⊲ Common structure between the two expansions M(h) ∼

• ˆ

nL rp (ln r)q F(t) ∼ M

• h
• ⊲ General expression of the multipole moments

IL

• ¯

τ αβ JL

• ¯

τ αβ

24

General expression of the multipole moments

The exterior field M(h) M(hµν) = −1

ret M (Λµν) + homogeneous retarded solution

• +∞
• l=0

∂L

• F µν

L

• t − r

c

• r
• with

F µν

L (t) =

• d3x ˆ

xL 1

−1

dz δl(z) τ µν x − zr c

• ⊲ gives the source multipole moments ML, SL at any PN order

25

General expression of the multipole moments

The exterior field M(h) M(hµν) = −1

ret M (Λµν) + homogeneous retarded solution

• +∞
• l=0

∂L

• F µν

L

• t − r

c

• r
• with

F µν

L (t) =

• d3x ˆ

xL 1

−1

dz δl(z) τ µν x − zr c

• ⊲ gives the source multipole moments ML, SL at any PN order

The inner field h ¯ hµν = 16πG c4 −1

ret ¯

τ µν −

homogeneous antisymmetric solution

• 4G

c4

+∞

• l=0

∂L

• Rµν

L

• t − r

c

• − Rµν

L

• t + r

c

• 2r
• with

Rµν

L (t) =

• d3x ˆ

xL +∞

1

dz γl(z) M (τ µν)

• x − zr

c

• ⊲ solves the radiation-reaction problem

25

Radiation-reaction problem

source

t

mass - quadrupole interaction

¯ hµν = 16πG c4 −1

ret ¯

τ µν − 4G c4

+∞

• l=0

∂L

• Rµν

L

• t − r

c

• − Rµν

L

• t + r

c

• 2r
• 26

Radiation-reaction problem

source

t

mass - quadrupole interaction

¯ hµν = 16πG c4 −1

ret ¯

τ µν − 4G c4

+∞

• l=0

∂L

• Rµν

L

• t − r

c

• − Rµν

L

• t + r

c

• 2r
• standard linear radiation-reaction effect −

→ starts at 2.5PN 16πG c4 −1

ret ¯

τ µν

• rad reac

= −4G c4

+∞

• l=0

∂L

• F µν

L

• t − r

c

• − F µν

L

• t + r

c

• 2r
• 26

Radiation-reaction problem

source

t

mass - quadrupole interaction

¯ hµν = 16πG c4 −1

ret ¯

τ µν − 4G c4

+∞

• l=0

∂L

• Rµν

L

• t − r

c

• − Rµν

L

• t + r

c

• 2r
• standard linear radiation-reaction effect −

→ starts at 2.5PN tails non-linear radiation-reaction effect − → starts at 4PN

26

Problem of divergences

Near zone Exterior zone Buffer zone Wave zone

IR divergences PN solution diverges when r − → +∞ UV divergences multipolar solution diverges when r − → 0

27

Problem of divergences

Near zone Exterior zone Buffer zone Wave zone

IR divergences PN solution diverges when r − → +∞ UV divergences multipolar solution diverges when r − → 0 ⊲ Solution Dimensional regularization

27

Historical finite part regularization

Regularization of the mPM solution

M(hµν) = FP −1

ret M (Λµν) + +∞

• l=0

∂L

• F µν

L

• t − r

c

• r
• with

F µν

L (t) = FP

• d3x ˆ

xL 1

−1

dz δl(z) τ µν x − zr c

• 28

Historical finite part regularization

Regularization of the mPM solution

M(hµν) = FP −1

ret M (Λµν) + +∞

• l=0

∂L

• F µν

L

• t − r

c

• r
• with

F µν

L (t) = FP

• d3x ˆ

xL 1

−1

dz δl(z) τ µν x − zr c

• 1. add a regulator rB for B ∈ C

uµν(B) ≡ −1

ret rBM (Λµν) 28

Historical finite part regularization

Regularization of the mPM solution

M(hµν) = FP −1

ret M (Λµν) + +∞

• l=0

∂L

• F µν

L

• t − r

c

• r
• with

F µν

L (t) = FP

• d3x ˆ

xL 1

−1

dz δl(z) τ µν x − zr c

• 1. add a regulator rB for B ∈ C

uµν(B) ≡ −1

ret rBM (Λµν)

• 2. perform a Laurent expansion when B → 0

uµν(B) =

+∞

• j=jmin

uµν

(j) Bj

s.t.      uµν

(j) = 0 for j ≤ −1

uµν

(j) = (ln r)j

j! Λµν for j ≥ 0

28

Historical finite part regularization

Regularization of the mPM solution

M(hµν) = FP −1

ret M (Λµν) + +∞

• l=0

∂L

• F µν

L

• t − r

c

• r
• with

F µν

L (t) = FP

• d3x ˆ

xL 1

−1

dz δl(z) τ µν x − zr c

• 1. add a regulator rB for B ∈ C

uµν(B) ≡ −1

ret rBM (Λµν)

• 2. perform a Laurent expansion when B → 0

uµν(B) =

+∞

• j=jmin

uµν

(j) Bj

s.t.      uµν

(j) = 0 for j ≤ −1

uµν

(j) = (ln r)j

j! Λµν for j ≥ 0

• 3. define the finite part as being the zeroth coefficient

uµν = FP −1

ret rBM (Λµν) ≡ uµν (0) 28

Self-field divergences

Point-particle approximation U (x, t) = Gm1 |x − y1(t)| + Gm2 |x − y2(t)|

• U (y1(t), t) = ?

− → self-field divergence for each particle ⊲ self-field regularization to remove the infinite self-field of each particle ⊲ each particle follows the geodesic motion of a regularized metric dP i

1

dt = F i

1 29

Treatment of divergences

Dimensional regularization

• work in d ∈ C spatial dimensions with distributional function

∆U = −4πG2(d − 2) d − 1 ρ = ⇒ U = 2(d − 2)k d − 1

• Gm1

|x − y1|d−2 + Gm2 |x − y2|d−2

• calculations for large negative R(d) to kill self-terms
• analytical continuation for all d ∈ C except some integer values

⊲ same procedure for all divergences

30

Methodology and example

Methodology

• 1. derive the nPN equations of motion
• 2. compute the conserved (or not) quantities
• 3. determine the gravitational energy and angular momentum fluxes
• 4. compute the required multipole moments
• 5. determine the gravitational waveform from the energy-balance equation
• 6. calculate the polarisation modes

31

Metric parametrization

Perturbed metric decomposition

hµν = ˜ gµν − ηµν and h = (h00ii ≡ h00 + hii, h0i, hij)

                 h00ii = −4V c2 − 8V 2 c4 + O 1 c6

• h0i = −4V i

c3 − 8 c5

• Ri + V V i

+ O 1 c7

• hij = − 4

c4

• Wij − 1

2δijW

• + O

1 c6

• Poissson-type equations

∆Wij = −4πG (σij − δijσkk)

• matter source

− ∂iV ∂jV

• non-compact support

32

Conservative equations of motion

dv1 dt = −Geff m2 r2

12

n12 + A1PN c2

• conservative terms

+ A2PN c4

cons.

+ A2.5PN c5

• rad. reac.

+ A3PN c6

cons.

+ A3.5PN c7

• rad. reac.

+ Ainst

4PN

c8

cons, local

+ Atail

4PN

c8

cons, nonloc.

Tails Stail ∝ G2m 5c8 dtdt′ |t − t′|I(3)

ij (t)I(3) ij (t′)

Conserved quantities energy E angular momentum J linear momentum P center of mass Q

33

Einstein quadrupolar formula

Qij(t) =

• source

d3xρ(x, t)(xixj − 1 3δijx2) Gravitational wave field HT T

ij

= 2G c4RPijkl(N)

• ¨

Qkl

• T − R

C

• + O

1 c

• + O

1 R2

• Energy balance equation

dE dt = −F with F ≡ dE dt GW = G c5 ... Qij ... Qij + O 1 c2

• Keplerian orbit period decay dP

dt , eccentricity de dt evolution amplitude a(t) ∝ (tc − t)1/4, phase φ(t) ∝ (tc − t)5/8

34

Concluding remarks

Other topics

Comparison PN - PM to post-Minkowskian Lagrangian (valid at any PN order) PN - SF to gravitational self-force using gauge-invariant quantities PN - NR to numerical relativity results redshift variable : z1 = 1 ut

1

=

• − (gµν)1

vµ

1 vν 1

c2

35

Other topics

Comparison PN - PM to post-Minkowskian Lagrangian (valid at any PN order) PN - SF to gravitational self-force using gauge-invariant quantities PN - NR to numerical relativity results redshift variable : z1 = 1 ut

1

=

• − (gµν)1

vµ

1 vν 1

c2 Additional effects Tidal effects starts at 5PN Eccentricity quasi-Keplerian representation of the orbit (n, K, ar, er, eφ, et) Spins spin-spin and spin-orbit coupling

35

Spins

dSA dt = ΩA×SA+O 1 c2

• H [x, p, S] = Horb [x, p]+1

2ΩA·SA

36

Spins

dSA dt = ΩA×SA+O 1 c2

• H [x, p, S] = Horb [x, p]+1

2ΩA·SA SO spin - orbit SS quadratic in spin : S1 − S1, S1 − S2 S· · · S cubic and higher orders in spin

36

State of the art

no spin S - O S - S tides eccentricity conservative dynamics 4PN 3.5PN 4PN 7PN 4PN radiation reaction 4.5PN* 4PN 4.5PN 6PN 3PN energy flux 4PN 4PN 2PN 2PN* 3PN waveform amplitude 3PN* 3PN 2PN 6PN 3PN waveform phase 3.5PN 4PN 2PN 6PN 3PN

37

State of the art

no spin S - O S - S tides eccentricity conservative dynamics 4PN 3.5PN 4PN 7PN 4PN radiation reaction 4.5PN* 4PN 4.5PN 6PN 3PN energy flux 4PN 4PN 2PN 2PN* 3PN waveform amplitude 3PN* 3PN 2PN 6PN 3PN waveform phase 3.5PN 4PN 2PN 6PN 3PN Methods

• harmonic coordinates
• ADM Hamiltonian formalism
• effective field theory
• surface integrals

37

State of the art

no spin S - O S - S tides eccentricity conservative dynamics 4PN 3.5PN 4PN 7PN 4PN radiation reaction 4.5PN* 4PN 4.5PN 6PN 3PN energy flux 4PN 4PN 2PN 2PN* 3PN waveform amplitude 3PN* 3PN 2PN 6PN 3PN waveform phase 3.5PN 4PN 2PN 6PN 3PN Methods

• multipolar post-Minkowskian
• DIRE (direct integration of relaxed field equations)
• effective field theory

37

Full waveform templates

Phenomenological waveforms ⊲ indirect matching between NR and PN waveforms Effective one-body waveforms ⊲ mapping the two-body dynamics to an effective one-body dynamics

[Buonanno & Sathyaprakash 2015]

Both models can now describe :

• spinning binaries
• tidal effects for neutron stars
• gravitational self-force information for EMRIs

38

Last remarks

⊲ the convergence of the PN and PM series is not known ⊲ complicated and heavy calculations ⊲ for alternative theories of gravity

• model compact objects
• more parameters −

→ find a good perturbative schemes

• more polarization modes

39

Further readings

Books

• Gravity : Newtonian, post-Newtonian and relativistic, E. Poisson & C.

Will, Cambridge University Press (2015)

• Gravitational waves : Theory and experiments, M. Maggiore, Oxford

University Press (2007) Review articles

• Gravitational radiation from post-Newtonian sources and inspiralling

compact binaries, L. Blanchet, Living Rev. Rel. (2014)

• Hamiltonian formulation of general relativity and post-Newtonian dynamics
• f compact binaries, G. Sch¨

afer and P. Jaranowski, Living Rev. Rel. (2018)

• The effective field theorist’s approach to gravitational dynamics, R. Porto,
• Phys. Rept. (2016)

40