Compact binary systems in scalar-tensor theories Laura Bernard (IST, - PowerPoint PPT Presentation

Compact binary systems in scalar-tensor theories Laura Bernard (IST, Lisbon) Gravity and Cosmology 2018 based on arXiv: 1803.10201 Laura BERNARD Compact binary systems in ST theories

The complete waveform [PRL 116, 241103 (2016)] Laura BERNARD Compact binary systems in ST theories

Post-Newtonian formalism Post-Newtonian source − → Slow moving, weakly-stressed compact source ǫ ≡ v 2 c 2 ∼ Gm 12 r 12 c 2 ≪ 1 1 � � post-Newtonian order : n PN = O ≡ O (2 n ) . c 2 n Laura BERNARD Compact binary systems in ST theories

Massless scalar-tensor theories ⊲ First introduced by Jordan, Fierz, Brans and Dicke more than 50 years ago, ⊲ Only one additional massless scalar field, minimally coupled to gravity. ⊲ It is the simplest , well motivated and most studied alternative theory of gravity, ⊲ Binary BHs gravitational radiation indistinguishable from GR (Hawking, 1972), ⊲ But strong deviations from GR are expected for neutron stars (scalarization). Laura BERNARD Compact binary systems in ST theories

Scalar-tensor theories The action c 3 � d 4 x √− g � φR − ω ( φ ) � g αβ ∂ α ∂ β φ S ST = + S m ( m , g αβ ) 16 πG φ • Metric g µν , • Scalar field φ and scalar function ω ( φ ) , • Matter fields m , minimally coupled to the physical metric, • No potential or mass for the scalar field. • No direct coupling between the matter and scalar fields, Laura BERNARD Compact binary systems in ST theories

Conformal vs physical frame Metric (Jordan) frame ⊲ Physical metric g αβ : Scalar field only coupled to the gravitational sector, ⊲ Frame for physical results and observations. Conformal (Einstein) frame ϕ = φ ˜ g µν = ϕ g µν , with φ 0 = φ ( ∞ ) = cst φ 0 • Scalar field only coupled to the matter sector. • Scalar field and metric decoupling = ⇒ BHs are the same as in GR. • Simpler to do calculations. Laura BERNARD Compact binary systems in ST theories

The matter action : Eardley’s approach In ST theories : violation of the Strong Equivalence Principle , Self-gravitating bodies : M A ( φ ) � A v β − g αβ v α � � d t M A ( φ ) c 2 A S m = − c 2 A � d ln M A ( φ ) ⊲ Sensitivities : s A = 0 , and all higher order derivatives, � d ln φ � • Neutron stars : s A ∼ 0 . 2 , • Black holes : s A = 1 / 2 , • related to the scalar charge α A ∝ 1 − 2 s A . Laura BERNARD Compact binary systems in ST theories

State of the art in ST theories • Equations of motion at 2 . 5 PN [Mirshekari & Will, 2013], • Tensor gravitational waveform to 2 PN [Lang, 2013], • Scalar waveform to 1 . 5 PN [Lang, 2014] : starts at − 0 . 5 PN , • Energy flux to 1 PN [Lang, 2014] : starts at − 1 PN , � ˜ � 3 ( s 2 − s 1 ) 2 = 4 mν 2 d E dipole Gαm 3 rc 3 d t r α (4 + 2 ω 0 ) Laura BERNARD Compact binary systems in ST theories

State of the art in ST theories • Equations of motion at 2 . 5 PN [Mirshekari & Will, 2013], • Tensor gravitational waveform to 2 PN [Lang, 2013], • Scalar waveform to 1 . 5 PN [Lang, 2014] : starts at − 0 . 5 PN , • Energy flux to 1 PN [Lang, 2014] : starts at − 1 PN , � ˜ � 3 ( s 2 − s 1 ) 2 = 4 mν 2 d E dipole Gαm 3 rc 3 d t r α (4 + 2 ω 0 ) What’s next • Flux and gravitational waveform at 2 PN : on-going (A. Heffernan, C. Will), ⊲ We need the EoM at 3 PN . Laura BERNARD Compact binary systems in ST theories

The multipolar post-Newtonian formalism • In the near zone : post-Newtonian expansion ∞ 1 h µν = ¯ c m ¯ � ¯ � h µν h µν τ µν m , with m = 16 πG ¯ m , m =2 ∞ 1 ¯ c m ¯ � ¯ � τ ( s ) ψ = ψ m , with ψ m = − 8 πG ¯ m m =2 Laura BERNARD Compact binary systems in ST theories

The multipolar post-Newtonian formalism • In the near zone : post-Newtonian expansion ∞ 1 h µν = ¯ c m ¯ � ¯ � h µν h µν τ µν m , with m = 16 πG ¯ m , m =2 ∞ 1 ¯ c m ¯ � ¯ � τ ( s ) ψ = ψ m , with ψ m = − 8 πG ¯ m m =2 • In the wave zone : multipolar expansion ∞ M ( h ) αβ = � G n h αβ � h αβ ( n ) = Λ αβ � � ( n ) , with h (1) , . . . , h ( n − 1) ; ψ , n n =1 ∞ � G n ψ ( n ) , � ψ ( n ) = Λ ( s ) � � M ( ψ ) = with ψ (1) , . . . , ψ ( n − 1) ; h , n n =1 Laura BERNARD Compact binary systems in ST theories

The multipolar post-Newtonian formalism • In the near zone : post-Newtonian expansion ∞ 1 h µν = ¯ c m ¯ � ¯ � h µν h µν τ µν m , with m = 16 πG ¯ m , m =2 ∞ 1 ¯ c m ¯ � ¯ � τ ( s ) ψ = ψ m , with ψ m = − 8 πG ¯ m m =2 • In the wave zone : multipolar expansion ∞ M ( h ) αβ = � G n h αβ � h αβ ( n ) = Λ αβ � � ( n ) , with h (1) , . . . , h ( n − 1) ; ψ , n n =1 ∞ � G n ψ ( n ) , � ψ ( n ) = Λ ( s ) � � M ( ψ ) = with ψ (1) , . . . , ψ ( n − 1) ; h , n n =1 • Buffer zone = ⇒ matching between the near zone and far zone solutions : � ¯ � M ( h ) = M h everywhere , � ¯ � M ( ψ ) = M ψ everywhere . Laura BERNARD Compact binary systems in ST theories

Fokker action What is the Fokker Lagrangian ? ⊲ Replace the gravitational degrees of freedom by their solution S Fokker [ y A , v A , . . . ] = S [ g sol ( y B , v B , . . . ) , φ sol ( y B , v B , . . . ) ; v A ] ⊲ Generalized Lagrangian : dependent on the accelerations. ⊲ Same dynamics as the original action. Laura BERNARD Compact binary systems in ST theories

Fokker action What is the Fokker Lagrangian ? ⊲ Replace the gravitational degrees of freedom by their solution S Fokker [ y A , v A , . . . ] = S [ g sol ( y B , v B , . . . ) , φ sol ( y B , v B , . . . ) ; v A ] ⊲ Generalized Lagrangian : dependent on the accelerations. ⊲ Same dynamics as the original action. Why a Fokker Lagrangian ? • The “ n + 2 ” method : we need to know the metric at only half the order we would have expected, O ( n + 2) instead of O (2 n ) . Laura BERNARD Compact binary systems in ST theories

Scalar-tensor theories The gravitational part g µν = √ ˜ g µν , • Conformal gothic metric ˜ g ˜ � � � � S ST = c 3 φ 0 − 1 g µρ − 1 d 4 x g λγ ∂ λ ˜ g µν ∂ γ ˜ g ρσ ˜ g µσ ˜ 2˜ g µν ˜ ˜ g ρσ 32 πG 2 � g σν ) − 3 + 2 ω g ρµ ∂ ρ ˜ g σν − ∂ ρ ˜ g ρµ ∂ σ ˜ g αβ ∂ α ϕ∂ β ϕ + ˜ g µν ( ∂ σ ˜ ˜ ϕ 2 → harmonic coordinates ∂ ν h µν = 0 Γ ν − g µν ˜ Γ µ ˜ ⊲ gauge-fixing term − 1 2 ˜ The matter part � � A v β v α � d t M A ( φ ) c 2 A S m = − − g αβ c 2 A • Solve flat space-time wave equations for the PN potentials. Laura BERNARD Compact binary systems in ST theories

Dimensional regularisation UV divergences • At the position of the particles ⊲ simple pole 1 /ε ⊲ vanishes through a redefinition of the trajectory of the particles : ok IR divergences • Divergence of the PN solution at infinity ⊲ simple pole 1 /ε ⊲ does not vanish through a redefinition of the trajectory of the particles ! ⊲ New effect in ST theories Laura BERNARD Compact binary systems in ST theories

A scalar tail effect • Non-local tail terms in the conservative dynamics at 3 PN : � τ � + ∞ L tail = 2 G 2 M � � − 1 � (3 + 2 ω 0 ) I (2) I (3) ( t ) d t ln ( t − τ ) i i 3 c 6 2 ε τ 0 0 ⊲ Exactly compensate the pole 1 /ε from the IR divergences. • New effect in ST theories Laura BERNARD Compact binary systems in ST theories

Result 3PN equations of motion d v 1 − G eff m 2 n 12 + A 1PN + A 1 . 5PN + A 2PN + A 2 . 5PN = r 2 c 2 c 3 c 4 c 5 d t 12 � �� � � �� � � �� � � �� � cons. terms rad. reac. rad. reac. conservative terms A inst A tail 3PN 3PN + + + · · · c 6 c 6 � �� � � �� � cons. & local cons. & nonlocal • Confirmation of the previous 2PN result by Mirshekari & Will (2013). • Renormalisation of the trajectories ⇐ ⇒ the poles disappear : ok • GR limit : ok • 2-black-hole limit : ok • Lorentz invariance : ok Laura BERNARD Compact binary systems in ST theories

Conclusion Equations of motion at 3 PN in scalar-tensor theories On-going calculations • Lorentz-Poincar´ e symmetry − → 10 conserved quantities • to be used in the scalar waveform and the scalar flux at 2 PN, Prospects ⊲ Incorporate the tidal effects for neutron stars − → start at 3PN. • Construct a full IMR waveform, • Comparison with numerical relativity or self-force results in ST theories. Laura BERNARD Compact binary systems in ST theories