EFT methods for the post-Newtonian framework Effective field theory methods in the post-Newtonian framework for the 2-body problem in General Relativity Riccardo Sturani Universit` a di Urbino/INFN (Italy) IHES, November 8 th , 2012
EFT methods for the post-Newtonian framework Effective Field Theory methods Outline 1 Effective Field Theory methods Introduction An Example of EFT at work 2 Binary conservative dynamics and the PN approximation EFT applied to 2-body systems Algorithm for computing PN-Hamiltonian dynamics 3 The dissipative sector Treating time-dependent problems
EFT methods for the post-Newtonian framework Effective Field Theory methods Introduction EFT principles: known fundamental theory • Fundamental theory known: effects of short distance physics r s (heavy d.o.f. Λ ) on large distance physics r ≫ r s (light modes ω ≪ Λ ) � D Φ( x ) e iS [ φ, Φ] exp ( iS eff [ φ ]) = � � d d x O i ( x ) S eff = c i i Wilson Coefficients local operators of φ ( x ) c i ( µ = Λ) ∼ Λ ∆ − d mass dim. ∆ : Decoupling Renormalize existing coefficients and generates new ones Dependence of large scale theory on small scale r given by simple power counting rule
EFT methods for the post-Newtonian framework Effective Field Theory methods Introduction EFT principles: unknown fundamental theory • Fundamental theory unknown: large scale effective Lagrangean can be expanded in terms of local operators ↓ write down the most general set of operators consistent with long scale system symmetries with unknown coefficients. ↓ Example: finite size effects in gravitational coupling of isolated bodies
EFT methods for the post-Newtonian framework Effective Field Theory methods An Example of EFT at work EFT for isolated compact object Fundamental Effective Fundamental List generic operators coupled gravitational fields to particle world-line Fundamental coupling to Diffeomorphism invariance particle world line � � S pp − fun = − m i dτ i Integrating out � � � x µ ˙ x ν + S pp − eff = − m dτ + c R dτR + c V dτR µν ˙ � dτ ( R µνρσ ) 2 + . . . c 2 (for a spherical body)
EFT methods for the post-Newtonian framework Effective Field Theory methods An Example of EFT at work EFT for isolated compact object Fundamental Effective Fundamental List generic operators coupled gravitational fields to particle world-line Fundamental coupling to Diffeomorphism invariance particle world line � � S pp − fun = − m i dτ i Integrating out � � � x µ ˙ x ν + S pp − eff = − m dτ + c R dτR + c V dτR µν ˙ PPPP ✟✟✟ ✏✏✏✏ ❍❍❍ � dτ ( R µνρσ ) 2 + . . . c 2 (for a spherical body)
EFT methods for the post-Newtonian framework Effective Field Theory methods An Example of EFT at work EFT applications Cosmology Generic gravity Lagrangean invariant under spatial rotations (time-dependent space diffeomorphisms) Short vs. Large inflaton fluctuation vs. Hubble scale of the background See P. Cheung et al. 2007 Hydrodynamics Derivative expansions: Short vs. Large Field time derivative vs. mean free time Field space derivatives vs. mean free length See Dubovsky et al. 2011
EFT methods for the post-Newtonian framework Binary conservative dynamics and the PN approximation Outline 1 Effective Field Theory methods Introduction An Example of EFT at work 2 Binary conservative dynamics and the PN approximation EFT applied to 2-body systems Algorithm for computing PN-Hamiltonian dynamics 3 The dissipative sector Treating time-dependent problems
EFT methods for the post-Newtonian framework Binary conservative dynamics and the PN approximation EFT applied to 2-body systems Different scales in EFT Very short distance � r s negligible up to 5PN (effacement principle) Short distance: potential gravitons k µ ∼ ( v/r, 1 /r ) Long distance: GW’s k µ ∼ ( v/r, v/r ) coupled to point particles with moments Goldberger and Rothstein PRD ’04
EFT methods for the post-Newtonian framework Binary conservative dynamics and the PN approximation EFT applied to 2-body systems Matching example Cross section for graviton scattering by a single black hole: s ω 4 + . . . r 10 s ω 8 . . . σ fund − BH = r 2 s f ( r s ω ) ∼ . . . r 6 Effective contribution to the amplitude: G N c 2 ω 4 C 2 ⇒ c 2 ∝ r 5 σ EFT − BH ∼ . . . + r s G N c 2 ω 4 + . . . + G 2 2 ω 8 = N c 2 s G N Goldberger and Rothstein PRD ’04
EFT methods for the post-Newtonian framework Binary conservative dynamics and the PN approximation EFT applied to 2-body systems Physical vs. gauge degrees of freedom h µν includes 1 4 gauge degrees of freedom 2 2 physical, radiative degrees of freedom 3 4 physical, non-radiative degrees of freedom 1&3 propagate with “the speed of thought” (Eddington ’22) After fixing the diffeomorphism invariance: � − 2Φ � Ξ i h µν = h TT Ξ i + θδ ij ij ∂ i Ξ i = h TT ij δ ij = ∂ i h ij = 0 : 6 degrees of freedom left, 4 eaten by gauge fixing Einstein eq’s: ∇ 2 Φ = ∇ 2 Ξ i = ∇ 2 Θ = 0 � h TT = 0 ij
EFT methods for the post-Newtonian framework Binary conservative dynamics and the PN approximation EFT applied to 2-body systems Conservative dynamics � exp [ iS eff ( x a )] = D h ( x ) exp [ iS EH ( h ) + iS pp ( h, x a )] � � � h 00 / 2 + v i h 0 i + v i v j h ij / 2 + G N h 2 � � S pp = − G N m dt 00 . . . � � ( ∂ i h ) 2 − ( ∂ t h ) 2 + G N h ( ∂h ) 2 + . . . � d 4 x � S EH = Power counting to integrate out potential gravitons h-M Vertex: ∼ dt d 3 k √ G N m k 2 (1 + k 2 Propagator: δ ( t ) δ (3) ( k ) 1 k 2 + . . . ) 0 dt d d k e ik ( x 1 ( t ) − x 2 ( t )) /k 2 � In k → 1 /r , k 0 → ∂ t ∼ kv ∼ v/r
EFT methods for the post-Newtonian framework Binary conservative dynamics and the PN approximation EFT applied to 2-body systems The 1PN potential Lv 2 Scaling: L Using virial theorem v 2 ∼ G N M/r v v2 v − Gm 1 m 2 � 1 − G N m 1 + 3 1 ) − 7 2 v 1 v 2 − 1 � 2( v 2 V = 2 v 1 ˆ rv 2 ˆ r + 2 r 2 r 1 ↔ 2
EFT methods for the post-Newtonian framework Binary conservative dynamics and the PN approximation EFT applied to 2-body systems Finite size effects enters at 5PN C 2 Lv 10 Re-derivation of the “Effacement principle” (Damour ’92)
EFT methods for the post-Newtonian framework Binary conservative dynamics and the PN approximation EFT applied to 2-body systems Quantum corrections are irrelevant m 1 m 2 � dt � G 2 ≃ � v 3 S eff ∼ N r 3 vs. leading order: m 1 m 2 � S eff dt G N ≃ L r See Donoghue PRD 1994
EFT methods for the post-Newtonian framework Binary conservative dynamics and the PN approximation EFT applied to 2-body systems What’s new to EFT in gravity? Systematic use of Feynman diagram with manifest power counting rule, enabling the construction of automatized algorithms Effective 2-body action is produced without the need to solve for the metric (however as in traditional ADM calculations) recast old problems in a field theory language: integrals in momentum space “look” easier to compute
EFT methods for the post-Newtonian framework Binary conservative dynamics and the PN approximation Algorithm for computing PN-Hamiltonian dynamics The 3PN computation automatized Topologies Graphs v and time derivative-insertions � 1 d d k d d k 1 A = G N m i v i k 2 ( k − k 1 ) 2 . . . Amplitudes Analytic integral in a database Evaluation S. Foffa & RS PRD 2011 original result Damour, Jaranowski and Sch¨ afer PRD 2001
EFT methods for the post-Newtonian framework Binary conservative dynamics and the PN approximation Algorithm for computing PN-Hamiltonian dynamics Feynman diagrams at 3PN order G N v 6 G 2 N v 4
EFT methods for the post-Newtonian framework Binary conservative dynamics and the PN approximation Algorithm for computing PN-Hamiltonian dynamics Feynman diagrams at 3PN order: G 3 N v 2
EFT methods for the post-Newtonian framework Binary conservative dynamics and the PN approximation Algorithm for computing PN-Hamiltonian dynamics Feynman diagrams at 3PN order: G 4 N Final result matches previous derivation of 3PN Hamiltonian see eq. (174) of Blanchet’s Living Review on Relativity
EFT methods for the post-Newtonian framework Binary conservative dynamics and the PN approximation Algorithm for computing PN-Hamiltonian dynamics The 4 PN status 3 graphs @ G N v 8 order 23 @ G 2 N v 6 202 @ G 3 N v 4 307 @ G 4 N v 2 50 @ G 5 N See also Jaranowski and Sch¨ afer PRD12
EFT methods for the post-Newtonian framework The dissipative sector Outline 1 Effective Field Theory methods Introduction An Example of EFT at work 2 Binary conservative dynamics and the PN approximation EFT applied to 2-body systems Algorithm for computing PN-Hamiltonian dynamics 3 The dissipative sector Treating time-dependent problems
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