Higher Dimensions & Higher Orders in EFT for Gravitational Waves. Ofek Birnholtz. 14 July th Marcel Grossmann Meeting, La Sapienza, Roma

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1 Higher Dimensions & Higher Orders in EFT for Gravitational Waves a OB, Shahar Hadar & Barak Kol, Phys. Rev. D 88, (2013) OB & Shahar Hadar, Phys. Rev. D 89, (2014) OB, Shahar Hadar & Barak Kol, IJMPA 29, 24, 30 (2014) OB, IJMPA 30, 02, 20 (2015) OB & Shahar Hadar, Phys. Rev. D 91, (2015) 14th Marcel Grossmann Meeting, La Sapienza, Roma 14 July 2015

2 Outline 1 Motivation 2 Plan 3 Math 4 Results 5 Higher Order interactions 6 Summary & Future

3 Outline 1 Motivation 2 Plan 3 Math 4 Results 5 Higher Order interactions 6 Summary & Future

4 Gravitational Waves from a 2-body system 1 g R d d 1 x S = 16πG d 2 h µν T µν d d x, Issues: µ T µν = 0, g µν = η µν + h µν ( c = 1, linearized source ) Metric elds are numerous, mixed, of dierent behaviours Some degrees of freedom are pure gauge Number of d.o.f (real and gauge) change with dimension Radiation and system zones have dierent symmetries Problem part-conservative, part dissipative Solution senstive to new eects & increasing order

5 Outline 1 Motivation 2 Plan 3 Math 4 Results 5 Higher Order interactions 6 Summary & Future

6 Everything should be made as simple as possible, but not simpler - Albert Einstein Zone Separation (λ R v R) System Zone: stationarity, NRG elds Radiation Zone: (hyper-)spherical symmetry Rad. Z. : Dimensional Reduction + Gauge Invariance = Master-Fields h S, h V, h T Sys. Z. : Match Master Sources T S, T V, T T = Master Action & Master Equation for outgoing waves

7 Everything should be made as simple as possible, but not simpler - Albert Einstein Rad. Z. : Dimensional Reduction + Gauge Invariance = Master-Fields h S, h V, h T Sys. Z. : Match Master Sources T S, T V, T T = Master Action & Master Equation for outgoing waves

8 Everything should be made as simple as possible, but not simpler - Albert Einstein Rad. Z. : Dimensional Reduction + Gauge Invariance = Master-Fields h S, h V, h T Sys. Z. : Match Master Sources T S, T V, T T = Master Action & Master Equation for outgoing waves = Field Doubling for Radiation-Reaction Eective Action

9 Outline 1 Motivation 2 Plan 3 Math 4 Results 5 Higher Order interactions 6 Summary & Future

10 Finding the G.I. Master Fields Starting from h ttn L h trn L h t Ω n L +h tvℵ n L ℵΩ h αβ = h rrn L h r Ω n L +h rvℵ n L ℵΩ e iωt h S n L + h ΩΩ S ñ L +h ΩΩ Vℵ n L +h ℵ ΩΩ ℵℶ n L ℵℶΩΩ we nd the tensors Master elds h ℵℶ = r (l+2) h ℵℶ. Using 1 algebraic equation we nd the vector Master elds h Vℵ = l(l+ˆd+1) ( 4 r l ω 2 ĉs ) 1 ( 2htVℵ +iωh Vℵ r 2 r 2 Using 3 algebraic equations we nd the scalars Φ := h tt + 2iωht ω 2 h S, ] canonically transformed to Φ with F [Φ, Φ, Φ, Φ = ĉ sr ˆd 1 ) 2 ( Φ Φ + ΦΦ, and nally the scalar Master elds h S = ˆd(l+ˆd+1) ˆd+1 (l+ r 2 ) Φ 2l )

11 Master Action: S ɛ = 1 dr 2 with 1D Master Action & Master Equation Lω [ Nl,ˆd N l,ˆd = Γ(1 + ˆd/2) 2 l Γ(1 + α) = ˆd!! (2l + ˆd)!! R S l,ˆd = ˆd(l+ˆd+1)(l+ˆd) (l 1) l Master Equation: G d R ɛ l,ˆd 0 = δs δh Lω = N l,ˆd ɛ G d R ɛ r 2l+ˆd+1 l,ˆd r 2l+ˆd+1 h ɛ L h ɛ (h ɛ T ɛ + c.c.), α = l+ ˆd 2, M l,ˆd = π 2 2α+1 N l,ˆd Γ2 (α+1), R V l,ˆd = (ˆd+1)(l+ˆd+1)l 2(l 1)(l+ˆd) ( ], R T l,ˆd = 8(ˆd+1) ˆd 2 c s(cs ˆd) ) ω 2 + r 2 + 2l + ˆd + 1 r h ɛ Lω TLω ɛ r

12 Field doubling Double the eld and the source (Schwinger, Galley) : These reect directed propagation Interpretation: "pulling" mirror / radiation "sink" h ĥ, Q ˆQ = δq δx ˆx Doubled action: [ ] [ δs Ŝ h, ĥ; Q, ˆQ = δh ĥ + δs ] ˆQ δq [ ( ) ] Nl,ˆd = ĥ ɛ G d R ɛ r 2l+ˆd+1 ω 2 + r 2 + 2l+ˆd+1 r h ɛ Q ɛ r l,ˆd EOM found by varying w.r.t. ĥ

13 Feynman Rules L r ɛ = Gret(r, r) = i G ω 2l+ˆd Ml,ˆd Rɛ l,ˆd j α (ωr < ) h+ α (ωr > )δ LL r j α := Γ(α + 1)2 α jα(x) x α = p=0 h α := Γ(α + 1)2 α hα(x) x α ( ) p (2α)!! (2p)!!(2p + 2α)!! x2p = 1 x 2 2(2α + 2) + = Q ɛ Lω, = ˆQɛ Lω Ŝ = = ( ˆQ G Q ), F = δŝ δ^x

14 Outline 1 Motivation 2 Plan 3 Math 4 Results 5 Higher Order interactions 6 Summary & Future

15 Quadrupole Moments (LO, NLO) Q ij E Q ij M Q ijk E Q L3 S : Mass quadrupole Q ij S n A=1 : Current quadrupole Q L2 : Mass octupole = Qijk S = n A=1 M = Qij M = 2 n m A ( x i x j 1 D δij x 2 ) A=1 A [ ( m r ) J (i x j)] A [ m A x i x j x k 1 ( δ ij x k + δ ik x j + δ jk x i) ] x 2 D + 2 A

16 General Dimension - RR Eective Action Ŝ linear = h S + h M + h T

17 General Dimension - RR Eective Action Ŝ linear = h S + h M + h T = ( ) ( ) G d ( ) l+ˆd l+ˆd+1 ˆd l+ˆd dt ˆQ ˆd!!(2l+ˆd)!! L 2l+ˆd (E) t Q (E) l (l 1) l L ) (ˆd+1 l + ( ) ˆQL (M) t 2l+ˆd Q (M) L + # (T) 2 l+ˆd

18 General Dimension - RR Eective Action Ŝ linear = h S + h M + h T = Ŝ (d=4) LO+NLO = G d ( ) ( ) G d ( ) l+ˆd l+ˆd+1 ˆd l+ˆd dt ˆQ ˆd!!(2l+ˆd)!! L 2l+ˆd (E) t Q (E) l (l 1) l L ) (ˆd+1 l + ( ) ˆQL (M) t 2l+ˆd Q (M) L + # (T) 2 l+ˆd dt [ 1 ˆQ ij 5 E 5 t Qij E 4 ˆQ ij 45 M 5 t Qij M + 1 ] ˆQ ijk 189 E 7 t Qijk E

19 General d Gravitation RR Eective Action S LO e ˆd+1 ˆd(ˆd + 2)(ˆd + 3) = ( ) 2 G d dt ˆQL 2 S 2 ˆd!! d+1 t Q L 2 S (ˆd + 4)!! S NLO e [ = ( ) ˆd+1 ˆd(ˆd+2)(ˆd+3) ( ) 2 G d dt ˆQL 2 S 2ˆd!!(ˆd d+1 t δ 1 Q L 2 S +δ1 ˆQL 2 S d+1 t Q L 2 S + 4)!! ˆd(ˆd+4)(ˆd+3) ˆQL 3 S 6 ˆd!!(ˆd+6)!! d+3 t Q L 3 S ] + 2 ˆd (ˆd+3) ˆQL 2 M 3 ˆd!! d+1 t Q L 2 M. (ˆd+4)!!

20 Outline 1 Motivation 2 Plan 3 Math 4 Results 5 Higher Order interactions 6 Summary & Future

21 In System Zone: Quadrupole Moments (LO, NLO) δ 1 Q ij E : Mass quadrupole (+1PN corrected including rst system zone nonlinear eect: gravitating potential energy, Gm Am B ) rˆd δ 1 Q L2 S = n A=1 m A ˆd+2 2 ˆd v2 A G d m B x L2 A x B A A x B ˆd 2(ˆd+1) ( ) ˆd(ˆd+2) t x A v A x L2 A (ˆd +6ˆd+4) 2 ( ) + 2ˆd(ˆd+2)(ˆd+6) 2 t r 2 A x L2 A

22 In Radiation Zone: Background Interactions h S Scalar / Cosmological Constant Λ h S Scalar / Total Mass Curvature (+(1 + ˆd 2 )PN) h V Vector / Total Mass Curvature (+(2 + ˆd 2 )PN)

23 Spin Eects co-rotation maximal-spin Spin-Orbit 1 + 2/ˆd 1/2 + 1/ˆd Spin-Spin 4/ˆd 2/ˆd

24 Outline 1 Motivation 2 Plan 3 Math 4 Results 5 Higher Order interactions 6 Summary & Future

25 GR radiation-reaction in general dimension Leading order and Next-to-leading order in any dimension Joint analytical Action formulation of radiation & reaction Economization of traditional computations Nonlinear eects - seperately for Sys. and Rad. zones Beginnings of +1.5PN, +2PN in any dimension Ready platform for other eects

26 Plans for the future The 2-body problem for systems of compact objects: Complete 4PN, 4.5PN, 5PN radiative eects Use EFT to manage Spin and tidal eects Study scattering trajectories in 4d and higher Numerical recipes from EFT Parameter determination from GW signals Other high-d systems Explore hyper-relativistic EM problem (Cerenkov) Explore ultra-relativistic systems Develope Analytical basis for Eective One Body

27 Thank you for your attention Questions?

28 Image credits K. Thorne (Caltech) & T. Carnahan (NASA GSFC) J. Tenniel in L. Carroll, Through the Looking-Glass, and What Alice Found There" (1871) Flickr's Flood G, (creative commons license)