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 � 1 � ⊲ we call n PN order the O correction w.r.t. the Newtonian order c 2 n 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 Near zone Exterior zone Buffer zone 9

Hypotheses Post-Newtonian source Isolated, compact support, smooth T µν , slowly moving, weakly stressed and weakly gravitating Near zone ǫ ≡ v 2 c 2 ∼ Gm Exterior zone 12 r 12 c 2 ≪ 1 Buffer zone 9

Hypotheses Post-Newtonian source Isolated, compact support, smooth T µν , slowly moving, weakly stressed and weakly gravitating Near zone ǫ ≡ v 2 c 2 ∼ Gm Exterior zone 12 r 12 c 2 ≪ 1 Buffer zone 9

Hypotheses Post-Newtonian source Isolated, compact support, smooth T µν , slowly moving, weakly stressed and weakly gravitating Near zone ǫ ≡ v 2 c 2 ∼ Gm Exterior zone 12 r 12 c 2 ≪ 1 Buffer zone Boundary condition at infinity • no incoming radiation at past null infinity � d � � d r + d r h αβ � lim = 0 c d t r →∞ t + r c =cst 9

Hypotheses Post-Newtonian source Isolated, compact support, smooth T µν , slowly moving, weakly stressed and weakly gravitating Near zone ǫ ≡ v 2 c 2 ∼ Gm Exterior zone 12 r 12 c 2 ≪ 1 Buffer zone Boundary condition at infinity • no incoming radiation at past null infinity • in practice : stationary source in the past � � ∂ h αβ ( x , t ) = 0 when t < −T ∂t 9

Approximation Tidal moment sourced by the companion body E ( A ) = − [ ∂ ij U ext ] A ij 10

Approximation Tidal moment sourced by the companion body E ( A ) = − [ ∂ ij U ext ] A ij Induced quadrupole moment Q ( A ) = − λ ( A ) E ( A ) 2 ij ij = 2 k ( A ) R 5 with λ ( A ) 2 A the tidal Love number of body A 2 3 G 10

Approximation Tidal moment sourced by the companion body E ( A ) = − [ ∂ ij U ext ] A ij Induced quadrupole moment Q ( A ) = − λ ( A ) E ( A ) 2 ij ij = 2 k ( A ) R 5 with λ ( A ) 2 A the tidal Love number of body A 2 3 G Effacement of the internal structure for a compact object � R A � 5 � Gm A � 5 � v � 10 F tidal k ( A ) ∼ ∝ = O 2 F N r 12 r 12 c 2 c � �� � Gm A R A c 2 ∼ 1 10

Approximation Tidal moment sourced by the companion body E ( A ) = − [ ∂ ij U ext ] A ij Induced quadrupole moment Q ( A ) = − λ ( A ) E ( A ) 2 ij ij = 2 k ( A ) R 5 with λ ( A ) 2 A the tidal Love number of body A 2 3 G Effacement of the internal structure for a compact object � R A � 5 � Gm A � 5 � v � 10 F tidal k ( A ) ∼ ∝ = O 2 F N r 12 r 12 c 2 c � �� � Gm A R A c 2 ∼ 1 = ⇒ Point-particle approximation valid up to 5PN • explicitely tested at 2PN [Mitchell & Will 2007] 10

Rewriting the Einstein equation G µν = 8 πG T µν c 4 We define the gothic metric h µν ≡ √− gg µν − η µν rewrite the field equations matter fields � �� � 16 πG | g | T µν + � h µν Λ µν [ h, ∂h, ∂ 2 h ] = c 4 � �� � � �� � flat d’Alembertian η ρσ ∂ ρ ∂ σ non-linearities : Λ ∼ h∂ 2 h + ∂h∂h + h∂h∂h + ··· and impose the harmonic gauge condition ∂ ν h µν = 0 11

Rewriting the Einstein equation G µν = 8 πG T µν c 4 We define the gothic metric h µν ≡ √− gg µν − η µν rewrite the field equations matter fields � �� � 16 πG | g | T µν + � h µν Λ µν [ h, ∂h, ∂ 2 h ] = c 4 � �� � � �� � flat d’Alembertian η ρσ ∂ ρ ∂ σ non-linearities : Λ ∼ h∂ 2 h + ∂h∂h + h∂h∂h + ··· and impose the harmonic gauge condition ∂ ν h µν = 0 ⊲ This is a well-posed wave equation in flat spacetime (Choquet-Bruhet, 1956) 11

Rewriting the Einstein equation G µν = 8 πG T µν c 4 We define the gothic metric h µν ≡ √− gg µν − η µν rewrite the field equations matter fields � �� � 16 πG | g | T µν + � h µν Λ µν [ h, ∂h, ∂ 2 h ] = c 4 � �� � � �� � flat d’Alembertian η ρσ ∂ ρ ∂ σ non-linearities : Λ ∼ h∂ 2 h + ∂h∂h + h∂h∂h + ··· and impose the harmonic gauge condition ∂ ν h µν = 0 ⊲ The field equations contains the matter conservation equations ∂ ν h µν = 0 ∇ ν T µν = 0 ⇐ ⇒ 11

Rewriting the Einstein equation G µν = 8 πG T µν c 4 We define the gothic metric h µν ≡ √− gg µν − η µν rewrite the field equations matter fields � �� � 16 πG | g | T µν + � h µν Λ µν [ h, ∂h, ∂ 2 h ] = c 4 � �� � � �� � flat d’Alembertian η ρσ ∂ ρ ∂ σ non-linearities : Λ ∼ h∂ 2 h + ∂h∂h + h∂h∂h + ··· and impose the harmonic gauge condition ∂ ν h µν = 0 ⊲ Λ contains all non-linearities Landau-Lifshitz pseudo-tensor c 4 ���� 16 πG Λ µν = t µν t µν + LL H ���� ∼ h∂ 2 h + ∂h∂h 11

The relaxed Einstein field equations  � h µν = 16 πG τ µν  c 4 ∂ ν h µν = 0  with τ µν ≡ | g | T µν + c 4 16 πG ( t µν LL + t µν H ) the stress-energy pseudo-tensor ⊲ This is an exact formulation of Einstein equations ⊲ The wave equation determines h µν for a certain distribution of matter. ⊲ The matter is governed by the conservation equation ∂ ν τ µν = 0 ⇐ ⇒ ∂ ν h µν = 0 12

The relaxed Einstein field equations  � h µν = 16 πG τ µν  c 4 ∂ ν h µν = 0  with τ µν ≡ | g | T µν + c 4 16 πG ( t µν LL + t µν H ) the stress-energy pseudo-tensor ⊲ This is an exact formulation of Einstein equations ⊲ The wave equation determines h µν for a certain distribution of matter. ⊲ The matter is governed by the conservation equation ∂ ν τ µν = 0 ⇐ ⇒ ∂ ν h µν = 0 The harmonic gauge condition ∂ ν h µν = 0 • either we impose it in the field equations t µν H | harmonic = 0 • or we solve the eqs. without enforcing it ⇒ relaxed Einstein field equations 12

Solving the wave equation The retarded solution h µν ( x , t ) = 16 πG � ret τ µν � � − 1 ( x , t ) c 4 • Flat-space retarded propagator � � � d 3 x ′ x ′ , t − | x − x ′ | ( x , t ) ≡ − 1 � � � − 1 ret τ | x − x ′ | τ 4 π c R 3 ⊲ 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 exterior zone r ≤ R r > a 14

The different zones Near zone Buffer zone Exterior zone near zone buffer zone exterior zone r ≤ R a < r ≤ R r > a 14

The different zones Near zone Buffer zone Exterior zone Wave zone near zone buffer zone wave zone exterior zone r ≤ R r ≫ λ a < r ≤ R r > a 14

Problems In the near zone : PN expansion � � x ′ , t − | x − x ′ | τ = τ ( x ′ , t ) τ ( x ′ , t ) + | x − x ′ | | x − x ′ | − ˙ c � � x ′ , t τ ¨ + · · · | x − x ′ | c 2 c 2 ⊲ generates an expansion in powers of 1 c • each term is instantaneous ⊲ how to include information about the boundary conditions at infinity ? λ GW ∼ v a • c ≪ 1 = ⇒ expansion ill-behaved when r ≫ λ GW 15