State of the Art in PN Gravity Theory Mich` ele Levi Niels Bohr - PowerPoint PPT Presentation

State of the Art in PN Gravity Theory Mich` ele Levi Niels Bohr International Academy Niels Bohr Institute University of Copenhagen QCD meets Gravity 2019 IPAM, UCLA December 10, 2019

PN Gravity Theory State of the Art State of the Art in PN Gravity Theory Complete state-of-the-art of PN theory for compact binary dynamics ❍❍❍❍ n (N 0 )LO N (1) LO N 2 LO N 3 LO N 4 LO l ❍ S 0 1 0 3 0 25 S 1 2 7 32 S 2 2 2 18 S 3 4 S 4 3 Each entry at PN order n + l + Parity( l ) / 2 A measure for loop computational scale: number of (highest) n -loop graphs that enter at N n LO of up to the l th multipole moment S l . All (but top right one) are derived in the public “EFTofPNG” code: https://github.com/miche-levi/pncbc-eftofpng Mich` ele Levi (NBIA, NBI, U. of Copenhagen) State of the Art in PN Gravity Theory December 2019 1 / 22

PN Gravity Theory State of the Art Kaluza-Klein decomposition of field Reduction over time dimension ` a la Kaluza-Klein ds 2 = g µν dx µ dx ν ≡ e 2 φ ( dt − A i dx i ) 2 − e − 2 φ γ ij dx i dx j φ, A i , γ ij ≡ δ ij + σ ij , KK fields Newtonian potential scalar φ Gravitomagnetic vector A i Hierarchy in coupling to mass and to spin Advantageous, preferable e.g. over Lorentz covariant, ADM decompositions... Mich` ele Levi (NBIA, NBI, U. of Copenhagen) State of the Art in PN Gravity Theory December 2019 2 / 22

PN Gravity Theory State of the Art State of the Art in PN Gravity Theory Complete state-of-the-art of PN theory for compact binary dynamics ❍❍❍❍ n (N 0 )LO N (1) LO N 2 LO N 3 LO N 4 LO l ❍ S 0 1 0 3 0 25 S 1 2 7 32 S 2 2 2 18 S 3 4 S 4 3 Each entry at the PN order n + l + Parity( l ) / 2 A measure for loop computational scale: number of (highest) n -loop graphs that enter at N n LO of up to the l th multipole moment S l . All (but top right one) are derived in the public “EFTofPNG” code: https://github.com/miche-levi/pncbc-eftofpng Mich` ele Levi (NBIA, NBI, U. of Copenhagen) State of the Art in PN Gravity Theory December 2019 3 / 22

Tower of EFTs Setup and Strategy EFTs are Universal There is a Hierarchy of Scales 1 r s , scale of internal structure, r s ∼ m r s r ∼ v 2 2 r , orbital separation scale, r 3 λ , radiation wavelength scale, λ ∼ v v ≪ 1 → nPN ≡ v 2 n correction in Classical Gravity to Newtonian gravity Multistage strategy for EFTs of inspiraling binaries [Goldberger & Rothstein 2007] It’s a multiscale! 1 One-Particle EFT 2 EFT of a Composite Particle 3 Effective Theory of Dynamical Multipoles Mich` ele Levi (NBIA, NBI, U. of Copenhagen) State of the Art in PN Gravity Theory December 2019 4 / 22

Tower of EFTs Setup and Strategy One-Particle EFT 1st Stage Remove scale r S of isolated compact object In the full theory we only have a vacuum gravitational field: � 1 d 4 x √ gR [ g µν ] S [ g µν ] = − 16 π G Integrate out strong field modes g s µν , g µν ≡ g s µν + ¯ g µν via bottom-up approach: � � � ∞ 1 d 4 x √ ¯ g µν , y µ ( σ ) , e µ S eff [¯ A ( σ )] = − gR [¯ g µν ( x )] + C i ( r s ) d σ O i ( σ ) 16 π G i =1 � �� � ≡ S pp ( σ ) with Wilson coefficients The operators O i ( σ ) must respect the symmetries that pertain at low energies. � d 4 x √ ¯ 1 g µν , y µ ] = − S eff [¯ gR [¯ g µν ( x )] 16 π G � � � y β � 2 + · · · y α ˙ − md σ + c 5PN d σ R µανβ ˙ � �� � finite size effects Mich` ele Levi (NBIA, NBI, U. of Copenhagen) State of the Art in PN Gravity Theory December 2019 5 / 22

Tower of EFTs Setup and Strategy EFT of Composite Particle 2nd Stage Remove orbital scale r of binary, (first) via the top-down approach: + � g µν ≡ η µν + H µν ¯ h µν ���� ���� orbital radiation ∂ t H µν ∼ v ∂ i H µν ∼ 1 h µν ∼ v ∂ ρ � � r H µν , r H µν , h µν r � � � 1 d 4 x √ ¯ g µν , y µ 1 , y µ µ µ S eff ¯ 2 , e (1) A , e (2) = − gR [¯ g µν ] + S pp ( σ 1 ) + S pp ( σ 2 ) A 16 π G Integrate out orbital field modes - in this classical context - only tree level � � � � h µν , y µ , e µ g µν , y µ 1 , y µ µ µ iS eff(composite) D H µν e iS eff [ ¯ A ] 2 , e (1) A , e (2) ⇒ e ( Comp ) A ≡ Stop here for effective action strictly in conservative sector, that is WITHOUT any remaining (orbital scale) field modes Mich` ele Levi (NBIA, NBI, U. of Copenhagen) State of the Art in PN Gravity Theory December 2019 6 / 22

Tower of EFTs EFT of spinning particle Spinning Particle: DOFs Assume isolated object has no intrinsic permanent multipoles beyond mass (monopole) and spin (dipole) 1 Gravitational field Metric g µν ( x ) Tetrad field η ab ˜ µ ( x )˜ ν ( x ) = g µν ( x ) e a e b 2 Particle Coordinate y µ ( σ ) function of arbitrary affine parameter σ Particle worldline position does not in general coincide with object’s ‘center’ 3 Particle rotating DOFs Worldline tetrad, η AB e A µ ( σ ) e B ν ( σ ) = g µν De A ν ⇒ Angular velocity Ω µν ( σ ) ≡ e µ D σ + conjugate Spin S µν ( σ ) A ⇒ Lorentz matrices η AB Λ Aa ( σ )Λ B b ( σ ) = η ab + conjugate local spin S ab ( σ ) Mich` ele Levi (NBIA, NBI, U. of Copenhagen) State of the Art in PN Gravity Theory December 2019 7 / 22

Tower of EFTs EFT of spinning particle Spinning Particle: Symmetries 1 General coordinate invariance, and parity invariance 2 Worldline reparametrization invariance 3 Internal Lorentz invariance of local frame field 4 SO(3) invariance of ‘body-fixed’ spatial triad 5 Spin gauge invariance, that is invariance under choice of completion of ‘body-fixed’ spatial triad through timelike vector Mich` ele Levi (NBIA, NBI, U. of Copenhagen) State of the Art in PN Gravity Theory December 2019 8 / 22

Tower of EFTs EFT of spinning particle Spin as extra particle DOF Effective action of spinning particle [Hanson & Regge 1974, Bailey & Israel 1975] u µ ≡ dy µ / d σ , Ω µν ≡ e µ De A ν g µν , u µ , Ω µν ] D σ ⇒ L pp [¯ A ∂ L S µν ≡ − 2 ∂ Ω µν spin as further particle DOF – classical source � � � − p µ u µ − 1 2 S µν Ω µν + L SI [¯ g µν ( y µ ) , u µ , S µν ] ⇒ S pp ( σ ) = d σ p µ S µν p ν = 0 This form assumes covariant gauge, e.g. e µ √ [0] = p 2 , ∂ u µ = m u µ Linear momentum p µ ≡ − ∂ L u 2 + O ( RS 2 ) √ For EFT of spin – gauge of both rotational DOFs should be fixed at level of one-particle action Mich` ele Levi (NBIA, NBI, U. of Copenhagen) State of the Art in PN Gravity Theory December 2019 9 / 22

Tower of EFTs EFT of spinning particle Extra term in minimal coupling Introduce gauge freedom into tetrad by boosting its timelike component → entails transformed gauge of spin ˆ S µν , traditionally called “SSC” ⇒ Extra term in action appears! From minimal coupling ˆ S µρ p ρ 2 S µν Ω µν = 1 1 Dp µ Ω µν + ˆ S µν ˆ 2 p 2 D σ Extra term with covariant derivative of momentum, contributes to finite size effects, yet carries no Wilson coefficient As of LO with spin, to all orders in spin! Essentially Thomas precession Beyond minimal coupling we use the relation ˆ ˆ S µρ p ρ p ν S νρ p ρ p µ S µν = ˆ S µν − + p 2 p 2 Mich` ele Levi (NBIA, NBI, U. of Copenhagen) State of the Art in PN Gravity Theory December 2019 10 / 22

Tower of EFTs EFT of spinning particle LO non-minimal couplings to all orders in spin S µ ≡ ∗ S µν 2 ǫ αβγµ R αβ p ν S αβ ≡ 1 2 ǫ αβµν S µν ; E µν ≡ R µανβ u α u β , B µν ≡ 1 √ δν u γ u δ p 2 , ∗ New spin-induced Wilson coefficients: � ∞ ( − 1) n C ES 2 n E µ 1 µ 2 u 2 S µ 1 S µ 2 · · · S µ 2 n − 1 S µ 2 n L SI = m 2 n − 1 D µ 2 n · · · D µ 3 √ (2 n )! n =1 � ( − 1) n ∞ C BS 2 n +1 B µ 1 µ 2 u 2 S µ 1 S µ 2 · · · S µ 2 n − 1 S µ 2 n S µ 2 n +1 + D µ 2 n +1 · · · D µ 3 √ m 2 n (2 n + 1)! n =1 LO spin couplings up to 5PN order L ES 2 = − C ES 2 E µν u 2 S µ S ν , Quadrupole @2PN √ 2 m L BS 3 = − C BS 3 D λ B µν u 2 S µ S ν S λ , Octupole @3.5PN √ 6 m 2 L ES 4 = C ES 4 D λ D κ E µν S µ S ν S λ S κ , Hexadecapole @4PN √ 24 m 3 u 2 Mich` ele Levi (NBIA, NBI, U. of Copenhagen) State of the Art in PN Gravity Theory December 2019 11 / 22

Tower of EFTs Integrating out the orbital modes LO sectors beyond Newtonian Feynman graphs of non-spinning sector to 1PN order Newton One-loop diagram – absent from 1PN . with KK parametrization of field LO Feynman diagrams with spin – to quadratic-in-spin Spin-Orbit Spin-Spin Mich` ele Levi (NBIA, NBI, U. of Copenhagen) State of the Art in PN Gravity Theory December 2019 12 / 22

Tower of EFTs Integrating out the orbital modes LO cubic & quartic in spin Feynman diagrams of LO cubic in spin sector On left pair – quadrupole-dipole, on right – octupole-monopole Note analogy of each pair with LO spin-orbit Feynman diagrams of LO quartic in spin sector On left and right – quadrupole-quadrupole and hexadecapole-monopole Each is analogous to LO spin-squared In middle – octupole-dipole analogous to LO spin1-spin2 Mich` ele Levi (NBIA, NBI, U. of Copenhagen) State of the Art in PN Gravity Theory December 2019 13 / 22