Effective field theory methods in the post-newtonian framework for the 2-body problem in General Relativity

Transcription

1 Effective field theory methods in the post-newtonian framework for the 2-body problem in General Relativity Riccardo Sturani Università di Urbino/INFN (Italy) IHES, November 8 th, 2012

2 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

3 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) exp (is eff [φ]) = DΦ(x)e is[φ,φ] S eff = c i d d xo i (x) 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

4 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

5 Effective Field Theory methods An Example of EFT at work EFT for isolated compact object Fundamental Fundamental gravitational fields Fundamental coupling to particle world line S pp fun = i Effective m i List generic operators coupled to particle world-line Diffeomorphism invariance dτ Integrating out S pp eff = m dτ + c R dτr + c V c 2 dτ (R µνρσ ) dτr µν ẋ µ ẋ ν + (for a spherical body)

6 Effective Field Theory methods An Example of EFT at work EFT for isolated compact object Fundamental Fundamental gravitational fields Fundamental coupling to particle world line S pp fun = i Effective m i List generic operators coupled to particle world-line Diffeomorphism invariance dτ Integrating out S pp eff = m dτ + c R dτr + c V dτr µν ẋ µ ẋ ν + c 2 dτ (R µνρσ ) (for a spherical body)

7 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 Hydrodynamics Derivative expansions: Short vs. Large Field time derivative vs. mean free time Field space derivatives vs. mean free length See P. Cheung et al See Dubovsky et al. 2011

8 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

9 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

10 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: σ fund BH = r 2 sf(r s ω)... r 6 sω r 10 s ω 8... Effective contribution to the amplitude: G N c 2 ω 4 C 2 σ EF T BH... + r s G N c 2 ω G 2 Nc 2 2ω 8 = c 2 r5 s G N Goldberger and Rothstein PRD 04

11 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 Ξi ) h µν = Ξ i h T T ij + θδ ij i Ξ i = h T ij T = 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 T ij T = 0

12 Binary conservative dynamics and the PN approximation EFT applied to 2-body systems Conservative dynamics exp [is eff (x a )] = S pp = G N m S EH = d 4 x Dh(x) exp [is EH (h) + is pp (h, x a )] ( dt h 00 /2 + v i h 0i + v i v j h ij /2 + ) G N h ] [ ( i h) 2 ( t h) 2 + G N h( h) Power counting to integrate out potential gravitons h-m Vertex: dt d 3 k G N m Propagator: δ(t)δ (3) (k) 1 k 2 (1 + k2 0 k ) In dt d d k e ik(x 1(t) x 2 (t)) /k 2 k 1/r, k 0 t kv v/r

13 Binary conservative dynamics and the PN approximation EFT applied to 2-body systems The 1PN potential Scaling: L Lv 2 Using virial theorem v 2 G N M/r v V = Gm 1m 2 2r 1 2 v v 2 [ 1 G Nm r 2 (v2 1) 7 2 v 1v v 1ˆrv 2ˆr ] +

14 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)

15 Binary conservative dynamics and the PN approximation EFT applied to 2-body systems Quantum corrections are irrelevant S eff dt G 2 N m 1 m 2 r 3 v 3 vs. leading order: m 1 m 2 S eff dt G N L r See Donoghue PRD 1994

16 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

17 Binary conservative dynamics and the PN approximation Algorithm for computing PN-Hamiltonian dynamics The 3PN computation automatized Topologies Graphs Amplitudes Evaluation v and time derivative-insertions A = G N m iv i d d k d d k 1 1 k 2 (k k 1) 2... Analytic integral in a database S. Foffa & RS PRD 2011 original result Damour, Jaranowski and Schäfer PRD 2001

18 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 v4

19 Binary conservative dynamics and the PN approximation Algorithm for computing PN-Hamiltonian dynamics Feynman diagrams at 3PN order: G 3 N v2

20 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

21 Binary conservative dynamics and the PN approximation Algorithm for computing PN-Hamiltonian dynamics The 4 PN status 3 G N v 8 order G 2 N v6 G 3 N v4 G 4 N v2 G 5 N See also Jaranowski and Schäfer PRD12

22 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

23 The dissipative sector General EFT approach: Decompose binary motion into central wordline + moments describing internal dynamics S = m dτ + dτc (I) n I abi 1...i n i1... in E ab n + dτc (J) n J abi 1...i n i1... in B ab n E µν C µανβ ẋ α ẋ β B µν = ɛ µρσα C ρσ νβẋα ẋ β in GR GN S d 4 x h µν T µν 2 Take the multipole expansion: 1 h µν (t, x) = n! xi1 x in i1... in h µν (t, 0) n=0 = compute moment of the Energy Momentum Tensor, decomposed into SO(3) irreducible representations

24 The dissipative sector The dissipative sector Coupling gravitational waves in all possible ways to the composite systems e.g. Leading order S EF T diss = m m P l h 00 [ m P l ɛ ijk L k j h 0i 1 a NLO m a va 2 G Nm 1 m 2 r 2m P l Leading radiative coupling: T ij h ij t 2 ( T00 x i x j) h ij a ] m a x i x j R 0 i0j h 00 2m P l

25 The dissipative sector Radiation reaction Integrating out the radiation field only the sources are left: Radiation emitted and absorbed Q ij Q kl Using Feynman propagators Effective action modified S real = G N dt Q ij (t) d5 Q ij (t) 10 dt 5 Real part should modifies e.o.m. giving Burke-Thorne potential: (RR) ẍ ai (t) = 2G N 5 x aj(t)q (5) ij (t)

26 The dissipative sector Treating time-dependent problems In the standard Lagrangian formalism e iw [J+Q] = Dh GW e i h 2 GW + 2 h 3 GW +h GW (J+ Q) e i (J+Q) F (J+Q) +.. h GW (x) = δw? δj(x) = J=0 d 4 x D F (x x )J(x ) F (x) = θ(t) + (x) + θ( t) (x) R,A (x) = θ(±t) [ + (x) + (x)] F = i 2 ( R + A ) = h GW ( R + A )J A-causal! Lagrangean formalism looks not suitable to describe dissipative systems (Galley a Tiglio PRD 09)

27 The dissipative sector Treating time-dependent problems Solution: Doubling of the degrees of freedom e iw [J i+q i ] = Dh 1 Dh 2 e i[s(h 1) S(h 2 )+ (J 1 +Q 1 )h 1 (J 2 +Q 2 )h 2] Specific causal structure: W [J 1, J 2 ] = i ( ) ( ) GF G (J 1, J 2 ) J1 2 G + G D J 2 = i ( ) ( 0 ig (J +, J ) R J+ 2 ig A G H J ) with J ± J 1 ± J 2 and e.o.m. s: h GW = W J J =Q =0 G R Q Q + =Q Keldish 65, Chou et al. 80

28 The dissipative sector Treating time-dependent problems The radiation-reaction diagram Radiation emitted and absorbed S eff dtq ij Q (5) +ij Radiation emitted, scattered and absorbed Qij M Qkl (t,0) (t,0) is eff G 2 NM (t,x) dt Q (2) ij (t) dt Q (2) +ij (t ) dt d 3 x 2 t G R (t t, x)g R (t t, x) 1 r

29 The dissipative sector Treating time-dependent problems Conservative part of the self force At leading order the rad-reaction affects e.o.m. at 2.5PN order Burke-Thorne potential (SF ) ẍ ai (t) = 2G N 5 x aj(t)q (5) ij (t) 8 t [ 5 G2 NMx aj dt Q (7) (t t ] ) ij (t ) log T + relative 1.5PN tail correction Conservative part associated with tail integral (SF ) ẍ ai (t) = 8G2 N M ( x aj (t)q (6) r ) ij 5 (t) log λ Gravitational radiation emitted, scattered, and absorbed. L. Blanchet and T. Damour PRD 88 L. Blanchet, S.L. Detweiler, A. Le Tiec, B. F. Whiting PRD 10

30 The dissipative sector Treating time-dependent problems Observables associated with GW emission Emission rate F = ia h (k)= dγ h (k) = 1 T Q + Q M... d 3 k (2π) 3 2k A h(k) 2 k dγ h (k) = G ( N d 3 5 Q ij ) 2 (t) dt G ( N d 3 45 J ij ) 2 (t) dt The optical theorem: Q Im Q = Qij Q kl M with Feynman propagators time averaged flux

31 The dissipative sector Treating time-dependent problems Logs and renormalization A h Quad = i G N k 2 ɛ ij(k)i ij( k ) 4 A 2 [ (k) A h Quad = (G N M k ) ( )] 1 k2 + ln 105 d 4 µ True ultraviolet divergence and log appearance renormalization I ij( k ) = Z( k, µ)i R ij( k, µ) Z( k, µ) = (GN M k )2 1 d 4 leading to a classical RG equation µ dir ij dµ = 214 ( ) (GN M k )2 Iij( k, R µ) Iij( k, R µ 105 (Gmk)2 µ) = Iij( k, R µ 0) µ 0 ( ) Prediction of leading logs at higher order, e.g. #(G N M k ) 4 log 2 k 2 µ 2 Goldberger and Ross PRD 10

32 Conclusions Conclusions EFT is a powerful and flexible tool, applicable to problems exhibiting clear scale separation PN computations within EFT are equivalent to computations performed with tradition method: predictions for the same physical observables must give same values PN computations within EFT methods provide a healthy competition with traditional methods Merit of EFT methods in gravity: ricycle huge knowledge accumulated in theoretical particle physics

33 Phenomenology Spare slides Binary inspiral phenomenology

34 Phenomenology GW detection Inspiral Virial relation: P (v) de dt v (G N Mπf GW ) 1/3 ν = m 1 m 2 (m 1 + m 2 ) 2 E(v) = 1 2 νmv2 ( 1 + #(ν)v 2 + #(ν)v ) = 32 5G N v 10 ( 1 + #(ν)v 2 + #(ν)v ) E(v)(P(v)) known up to 3(3.5)PN 1 2π φ(t ) = 1 T v(t ) ω(v)de/dv ω(t)dt = dv 2π P (v) (1 + #(ν)v #(ν)v ) dv v 6

35 Phenomenology GW detection ( ) 10Hz 5/3 ( N cycles M f min Sensitivity Mc 5/3 Ncycles Mc 5/6 f Max M 1, M c (m 1 m 2 ) 3/5 (m 1 + m 2 ) 2/5 M c ) 5/3 Important to know the phase at O(1) when taking correlation of detector s output and model waveform

36 Phenomenology Detector sensitivity ] -1/2 Spectrum[Hz NS-NS M 60-60M AdvDet f GW [Hz]

37 Phenomenology Observational rate estimates LIGO/Virgo Advanced Observatories will detect Best case: NS-NS 10 M BH-BH Distance (Mpc) 300Mpc 1GPc Rates MWEG 1 Myear N = π ( DH 2.26M pc ) 3 MWEG r NS NS 400yr 1 r BH BH 10 3 yr 1 I. Mandell et al. PRD 2010

38 Phenomenology Fundamental gravity tests: Graviton self-interactions Conservative dynamics β 3 Emission V β 3 G 2 N m 1m 2 (m 1 + m 2 ) r 2 β 3 L pp h ij β 3 (νmx i ẍ j ) Example of tagging of fundamental physics effects

39 Phenomenology Bound on self-interaction triple vertex At present the binary pulsars give best constraint on non-conservative effect from β 3 P obs P β3 = P GR (1 + cβ 3 ) c 3.21 Given that P 1 0.1% = β 3 = (4.0 ± 6.4) 10 4 GR Conservative effect of β 3 already constrained by Lunar Laser Ranging, 1PN β 3 = β P P N < Cannella et al. 09

40 Phenomenology Bayesian analysis of GR vs. modgr Searching for waveforms whose phase is modified at any PPN waveforms φ(t) = φ N (t) [1 + φ 1 (t)(1 + δ 1 ) + φ 1.5 (1 + δφ 1.5 ) +φ 2 (1 + δφ 2 )] and injecting fake signals with φ inj (t) = φ GR (t) + φ N (t)δφ A (t) Li et al. 2011

41 Phenomenology Baysian analysis of GR vs. modgr O i = P (H i d) = P (H i ) P (d H i) P (d) O mgr GR = O mgr O GR P (d H mgr) P (d H GR ) (1 catalog = 15 sources)