Equations of motion of compact binaries at the fourth post-Newtonian - PowerPoint PPT Presentation

Equations of motion of compact binaries at the fourth post-Newtonian order Laura BERNARD in collaboration with L.Blanchet, A. Boh´ e, G. Faye, S. Marsat Hot Topics in General Relativity and Gravitation 2015 10/08/2015 Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Outline Introduction The post-Newtonian Fokker action Results and consistency checks Conclusion Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Motivations A Global Network of Interferometers A Global Network of Interferometers LIGO Hanford 4 & 2 km GEO Hannover 600 m Kagra Japan 3 km LIGO South Indigo Virgo Cascina 3 km LIGO Livingston 4 km Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Coalescing compact binary systems NS-NS merger BH-BH merger Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Coalescing compact binary systems merger phase numerical relativity inspiralling phase post-Newtonian theory ringdown phase perturbation theory Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Principle of the Fokker action ⊲ Starting from the action S tot [ g µν , y B ( t ) , v B ( t )] = S grav [ g µν ] + S mat [( g µν ) B , y B ( t ) , v B ( t )] ⊲ we solve the Einstein equation δS tot δg µν = 0 → g µν [ y A ( t ) , v A ( t ) , · · · ] ⊲ and construct the Fokker action � � S Fokker [ y B ( t ) , v B ( t ) , · · · ] = S tot g µν ( y A ( t ) , v A ( t ) , · · · ) , y B ( t ) , v B ( t ) Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Principle of the Fokker action ⊲ Starting from the action S tot [ g µν , y B ( t ) , v B ( t )] = S grav [ g µν ] + S mat [( g µν ) B , y B ( t ) , v B ( t )] ⊲ we solve the Einstein equation δS tot δg µν = 0 → g µν [ y A ( t ) , v A ( t ) , · · · ] ⊲ and construct the Fokker action � � S Fokker [ y B ( t ) , v B ( t ) , · · · ] = S tot g µν ( y A ( t ) , v A ( t ) , · · · ) , y B ( t ) , v B ( t ) ⊲ The dynamics for the particles is unchanged � � δS Fokker δS tot � · δg µν + δS tot � � � = � � δy A δg µν δy A δy A � � g = g g = g � �� � = 0 � δS tot � � = � δy A � g = g Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Our Fokker action   � c 3 d 4 x √− g  g µν � � 1   Γ ρ µλ Γ λ νρ − Γ ρ µν Γ λ 2 g µν Γ µ Γ ν S grav = −  ,   ρλ 16 πG � �� � gauge fixing term � � � v µ A v ν m A c 2 S mat = − d t − ( g µν ) A A . c 2 A Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Our Fokker action   � c 3 d 4 x √− g  g µν � � 1   Γ ρ µλ Γ λ νρ − Γ ρ µν Γ λ 2 g µν Γ µ Γ ν S grav = −  ,   ρλ 16 πG � �� � gauge fixing term � � � v µ A v ν m A c 2 S mat = − d t − ( g µν ) A A . c 2 A Relaxed Einstein equations � h µν = 16 πG | g | T µν + Λ µν � � h, ∂h, ∂ 2 h c 4 � ◮ with h µν = | g | g µν − η µν the metric perturbation variable. ◮ We don’t impose the harmonicity condition ∂ ν h µν = 0. ◮ Λ µν encodes the non-linearities, with supplementary harmonicity terms containing H µ = ∂ ν h µν . Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Near zone / Wave zone h Multipole expansion Exterior zone Matching zone Near zone PN expansion Actual solution m r 1 m 2 ⊲ Near zone : Post-Newtonian expansion h = h , ⊲ Wave zone : Multipole expansion h = M ( h ), � � ⊲ Matching zone : h = M ( h ) = ⇒ M h = M ( h ). � r � r � � � B � � � B d 3 x d 3 x S g = FP d t L F + FP d t M ( L F ) r 0 r 0 B =0 B =0 Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Near zone / Wave zone h Multipole expansion Exterior zone Matching zone Near zone PN expansion Actual solution m r 1 m 2 ⊲ Near zone : Post-Newtonian expansion h = h , ⊲ Wave zone : Multipole expansion h = M ( h ), � � ⊲ Matching zone : h = M ( h ) = ⇒ M h = M ( h ) everywhere. � r � r � � � B � � � B d 3 x d 3 x S g = FP d t L F + FP d t M ( L F ) r 0 r 0 B =0 B =0 � �� � O (5 . 5 P N ) Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Post-Newtonian counting in a Fokker action Thanks to the property of the Fokker action, cancellations between gravitational and matter terms in the action occur. � 1 � ⊲ To get the Lagrangian at n PN i . e . O , we only need to c 2 n know the metric at roughly half the order we would have expected : � � 1 � h 00 ii , h 0 i , h ij � = O . c n +2 Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Post-Newtonian counting in a Fokker action Thanks to the property of the Fokker action, cancellations between gravitational and matter terms in the action occur. � 1 � ⊲ To get the Lagrangian at n PN i . e . O , we only need to c 2 n know the metric at roughly half the order we would have expected : � � 1 � h 00 ii , h 0 i , h ij � = O . c n +2 � 1 � c 6 , 1 c 5 , 1 � h 00 ii , h 0 i , h ij � ⊲ For 4 PN : = O c 6 Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Tail effects at 4PN ◮ At 4PN we have to insert some tail effects, + ∞ � A µν � L ( t − r/c ) − A µν ( − 1) l µν = h part − 2 G L ( t + r/c ) � µν h ∂ L c 4 l ! r l =0 ◮ When inserted into the Fokker action it gives in the following contribution � � S tail = G 2 ( m 1 + m 2 ) d t d t ′ | t − t ′ | I (3) ij ( t ) I (3) ij ( t ′ ) Pf 5 c 8 2 s 0 c ⊲ The two constant of integration are linked through s 0 = r 0 e − α . Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Different regularization schemes IR Singularity of the PN expansion at infinity : r 0 Tail effects : s 0 ⊲ The two constants of integration are linked through s 0 = r 0 e − α . ⊲ α will be determined by comparison with self-force results. Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Different regularization schemes IR Singularity of the PN expansion at infinity : r 0 Tail effects : s 0 ⊲ The two constants of integration are linked through s 0 = r 0 e − α . ⊲ α will be determined by comparison with self-force results. UV Singularity at the location of the point particles ⊲ Dimensional regularization, 1. We calculate the Lagrangian in d = 3 + ε dimensions. 2. We expand the results when ε → 0 : appearance of a pole 1 /ε . 3. We eliminate the pole through a redefinition of the variables. ⊲ The physical result should not depend on ε . Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

The equations of motion at 4PN The generalized Lagrangian L 4PN = Gm 1 m 2 + 1 1 + 1 2 m 1 v 2 2 m 2 v 2 2 + L 1pn + L 2pn + L 3pn r 12 + L 4pn [ y A ( t ) , v A ( t ) , a A ( t ) , ∂a A ( t ) , · · · ] The equations of motion 1 , 4PN = − Gm 2 a i n i 12 + a i 1 , 1pn + a i 1 , 2pn + a i 1 , 3pn + a i 1 , 4pn [ α ] r 2 12 ⊲ Previous results at 4PN were obtained with the Hamiltonian formalism (Jaranowski, Schaffer 2013 and Damour, Jaranowski, Schaffer 2014) and partially with EFT (Foffa, Sturani 2012). Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Binding energy for circular orbits ⊲ The constant α is determined by comparison of the binding energy for circular orbits with another method, such as self-force calculations: E ( x ; ν ) = − µc 2 x � � 3 � � − ν 2 � 4 + ν − 27 8 + 19 ν x 2 1 − x + 2 12 8 24 − 205 π 2 ν − 155 ν 2 − 35 ν 3 � � 34445 � � − 675 x 3 + 64 + 576 96 96 5184 � 9037 π 2 � � − 3969 1536 − 123671 + 448 + 128 + 15 (2 γ + ln(16 x )) ν 5760 � 3157 π 2 ν 2 + 301 ν 3 1728 + 77 ν 4 � � � − 198449 x 4 − 576 3456 31104 � 2 / 3 � G ( m 1 + m 2 )Ω m 1 m 2 with x = and ν = ( m 1 + m 2 ) 2 the symmetric c 3 mass ratio. Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Consistency checks We have checked that ⊲ the IR regularization is in agreement with the tail part : no r 0 , ⊲ the result does not depend on the UV regularization : no pole 1 /ε , ⊲ the equations of motion are manifestly Lorentz invariant , ⊲ in the test mass limit we recover the Schwarzschild geodesics , ⊲ we recover the conserved energy for circular orbits . Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Summary ◮ We obtained the equations of motion at 4PN from a Fokker Lagrangian, in harmonic coordinates. ◮ We recover all the physical results that we expected. Laura BERNARD EoM of compact binaries at 4PN 10/08/2015

Summary ◮ We obtained the equations of motion at 4PN from a Fokker Lagrangian, in harmonic coordinates. ◮ We recover all the physical results that we expected. ◮ We are now systematically computing the conserved quantities. ◮ The important goal is now to compute the gravitational radiation field at 4PN. Laura BERNARD EoM of compact binaries at 4PN 10/08/2015