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Gravitational Waves and Binary Systems (2) - Thibault Damour

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Waves on the lake: the astrophysics behind gravitational waves
May 28 – June 1, 2018

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Gravitational Waves and Binary Systems (2) - Thibault Damour

  1. 1. GRAVITATIONAL WAVES and BINARY SYSTEMS (lecture 2) Thibault Damour Institut des Hautes Etudes Scientifiques Waves on the lake: the astrophysics behind gravitational waves Lake Como School of Advanced Studies - May 28 - June 1, 2018, Como, Italy
  2. 2. Basics of Gravitational Waves ds2 = c2 dt2 + ( ij + hij)dxi dxj In linearized GR (Einstein 1916,1918): ⇤hµ⌫ + @µH⌫ + @⌫Hµ = 16⇡G(Tµ⌫ 1 D 2 ⌘µ⌫T) Hµ = 1 2 @µh @⌫ hµ⌫ @⌫ Tµ⌫ = 0 Massless, two helicity states s=±2, i.e. two Transverse-Traceless (TT) tensor polarizations propagating at v=c € hij = h+(xi x j − yi y j ) + h×(xi y j + yi x j ) € gij = δij + hij Joseph Weber (1919- 2000) L L = 1 2 hijni nj
  3. 3. BASICS OF BLACK HOLES 3 r = 0 SINGULARITÉ r = 2M HORIZON ÉCLAIR LUMINEUX ÉMIS DEPUIS LE CENTRE ÉTOILE EN EFFRONDREMENT temps espace 1916 Schwarzschild (non rotating) Black Hole (BH) Horizon: cylindrical-like regular null hyper-surface whose sectional area is nearly constant, and actually slowly increasing (Christodoulou ’70, Christodoulou-Ruffini ’71, Hawking ’71) ds2 = (1 2GM c2r )dt2 + dr2 1 2GM c2r + r2 (d✓2 + sin2 ✓d'2 ) Schwarzschild radius (singularity ?): r_S= 2GM/c^2 1963 Kerr Rotating BH: M, S 1939 Oppenheimer-Snyder « continued collapse » 1965 Penrose
  4. 4. 4 Rµ⌫ = 0 ds2 = gµ⌫(x ) dxµ dx⌫ = 0
  5. 5. Pioneering the GWs from coalescing compact binaries Freeman Dyson 1963, using Einstein 1918 + Landau-Lifshitz 1951 (+ Peters ’64) first vision of an intense GW flash from coalescing binary NS 2 4 6 8 10 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 -0.6 -0.4 -0.2 0.2 0.4 Challenge: describe the intense flash of GWs emitted by the last orbits and the merger of a binary BH, when v~c and r~GM/c^2
  6. 6. 6 Perturbative theory of motion Perturbative theory of gravitational radiation Motion of two BHs BH QNMs Mathematical Relativity BH QNMs Resummation EOB EOB[NR] Numerical Relativity IMRPhenomD=Phen[EOB+NR] ROM(EOB[NR]) PN (nonresummed) M < 4M
  7. 7. Long History of the GR Problem of Motion Einstein 1912 : geodesic principle Einstein 1913-1916 post-Minkowskian Einstein, Droste : post-Newtonian Weakly self-gravitating bodies: Einstein-Grossmann ’13, 1916 post-Newtonian: Droste, Lorentz, Einstein (visiting Leiden), De Sitter ; Lorentz-Droste ‘17, Chazy ‘28, Levi-Civita ’37 …., Eddington’ 21, …, Lichnerowicz ‘39, Fock ‘39, Papapetrou ‘51, … Dixon ‘64, Bailey-Israël ‘75, Ehlers-Rudolph ‘77….
  8. 8. Challenge: Motion of Strongly Self-gravitating Bodies (NS, BH) 8 Multi-chart approach to motion of strong-self-gravity bodies, and matched asymptotic expansions [EIH ’38], Manasse ’63, Demianski- Grishchuk ’74, D’Eath’75, Kates ’80, Damour ’82 Useful even for weakly self-gravitating bodies, i.e.“relativistic celestial mechanics”, Brumberg-Kopeikin ’89, Damour-Soffel-Xu ‘91-94 Combine two expansions in two charts: gµ⌫(x) = ⌘µ⌫ + Gh(1) µ⌫ (x) + G2 h(2) µ⌫ (x) + · · · G↵ (x) = G (0) ↵ (x) + G (1) ↵ (x) + · · ·
  9. 9. 9 Love numbers for compact objects and effacing property up to 5PN (Damour, 82)
  10. 10. Practical Techniques for Computing the Motion of Compact Bodies (NS or BH) 9 Skeletonization : point-masses (Mathisson ’31) delta-functions in GR : Infeld ’54, Infeld-Plebanski ’60 justified by Matched Asymptotic Expansions ( « Effacing Principle » Damour ’83) QFT’s analytic (Riesz ’49) or dimensional regularization (Bollini-Giambiagi ’72, t’Hooft-Veltman ’72) imported in GR (Damour ’80, Damour-Jaranowski-Schäfer ’01, …) Feynman-like diagrams and « Effective Field Theory » techniques Bertotti-Plebanski ’60, Damour-Esposito-Farèse ’96, Goldberger-Rothstein ’06, Porto ‘06, Gilmore-Ross’ 08, Levi ’10, Foffa-Sturani ’11 ‘13, Levi-Steinhoff ‘14, ‘15
  11. 11. 11 Fokker Action in Electrodynamics (1929) Stot[xµ a, Aµ] = X a Z madsa + X a Z eadxµ aAµ(xa) Z dD x 1 16⇡ Fµ⌫Fµ⌫ « Integrate out » the field A_mu in the total (particle+field) action One-photon-exchange diagram The effective action S_eff(x_a) was heavily used in the (second) Wheeler- Feynman paper (1949) together with similar diagrams to those used by Fokker + Sgf
  12. 12. g = ⌘ + h S(h, T) = Z ✓ 1 2 h⇤h + @@hhh + ... + (h + hh + ...)T ◆ ⇤h = T + ... ! h = G T + ... Sred(T) = 1 2 T G T + V3(G T, G T, G T) + ... Fokker-type Action in Gravity and its Diagrammatic Expansion Needs gauge-fixed* action and time-symmetric Green function G. *E.g. Arnowitt-Deser-Misner Hamiltonian formalism or harmonic coordinates. Perturbatively solving (in dimension D=4 - eps) Einstein’s equations to get the equations of motion and the action for the conservative dynamics Damour-Esposito-Farese ‘96 O(G)= Newtonian + (v/c)^n corrections O(G^2)=1PN =1 loop O(G^3)=2PN =2 loop O(G^5)=4PN =4 loop
  13. 13. Beyond 1-loop order needs to use PN-expanded Green function for explicit computations. This transforms spacetime diagrams (between two worldlines) into (massless) two-point space diagrams (in three dimensions) ⇤ 1 = ( 1 c2 @2 t ) 1 = 1 + 1 c2 @2 t 2 + ... PN computation of the Fokker-type Gravity Action E.g. at 3PN, a 3-loop space diagram ~ G^4 m_1^3 m_2^2 (Damour-Jaranowski-Schaefer 2001) E.g. at 4PN, some 4-loop space diagrams ~ G^5 m_1^3 m_2^3 among 515 4PN-level diagrams, (Damour-Jaranowski-Schaefer ’14, Bernard et al ’16, Foffa et al ’17) PN expansion: in powers of 1/c^2: 1PN= (v/c)^2; 2PN= (v/c)^4, etc nPN=(v/c)^(2n)
  14. 14. 14 Alternative Computation of Effective Action Instead of classically «  integrating out » the field dofs Formal functional integral over the field (QED: Feynman ’50; …; GR: « Effective Field Theory » approach (Goldberger-Rothstein ’06, Porto ‘06, Gilmore-Ross’ 08, Levi ’10,Foffa-Sturani ’11 ‘13, Levi-Steinhoff ‘14, ’15; Foffa-Mastrolia-Sturani-Sturm’16, Damour-Jaranowski '17 ) Saddle-point estimation: However, the explicit computations are done differently: by means of Wick’s theorem, and p-space integrations Damour-Jaranowski ’17 corrected the first version of an EFT computation by Foffa-Mastrolia-Sturani-Sturm’16 of some of the static 4-loop contributions, found a way of analytically computing a 2-point 4-loop master integral previously only numerically computed (Lee-Mingulov ’15)
  15. 15. 15 Analytic Continuation
  16. 16. 16 Analytic Continuation and Dimensional Regularization Self-energy of classical point-particles = 4⇡Gm (3) , = Gm r Eself ⇠ Z d3 x(r )2 ⇠ Gm2 Z r2 dr 1 r4 = Gm2 Z drr 2 = 1 Eself ⇠ m (0) = Gm2 [ 1 r ]r=0 = 1 in dimension d=3 = 4⇡Gmkd (d) , = Gm rd 2 Eself ⇠ m (0) = Gm2  1 rd 2 r=0 = 0 Justified by Matched Asymptotic Expansions, at least when being able to use Marcel Riesz kernels (Damour ’82) in dimension d=3+ eps Eself ⇠ Z dd x(r )2 ⇠ Gm2 Z rd 1 dr 1 r2(d 1) ⇠ Gm2 Z drr (d 1) = 0
  17. 17. 17 Four-loop static contribution to the gravitational interaction potential of two point masses (Foffa-Mastrolia-Sturani-Sturm’16, Damour-Jaranowski ’17) All integrals computable by the x-space generalized Riesz integral (Jaranowski-Schaefer ’00) (num.) Lee-Mingulov’15, Damour-Jaranowski ‘17
  18. 18. 18 Arnowitt-Deser-Misner (ADM) Hamiltonian approach (Jaranowski-Schaefer’18) constraints ADMTT coordinate gauge Elliptic eqs for phi and V^i and Hyperbolic eqs for hTT,piTT
  19. 19. Post-Newtonian Equations of Motion [2-body, wo spins] 10 • 1PN (including v2 /c2) [Lorentz-Droste ’17], Einstein-Infeld-Hoffmann ’38 • 2PN (inc. v4 /c4) Ohta-Okamura-Kimura-Hiida ‘74, Damour-Deruelle ’81 Damour ’82, Schäfer ’85, Kopeikin ‘85 • 2.5 PN (inc. v5 /c5) Damour-Deruelle ‘81, Damour ‘82, Schäfer ’85, Kopeikin ‘85 • 3 PN (inc. v6 /c6) Jaranowski-Schäfer ‘98, Blanchet-Faye ‘00, Damour-Jaranowski-Schäfer ‘01, Itoh-Futamase ‘03, Blanchet-Damour-Esposito-Farèse’ 04, Foffa-Sturani ‘11 • 3.5 PN (inc. v7 /c7) Iyer-Will ’93, Jaranowski-Schäfer ‘97, Pati-Will ‘02, Königsdörffer-Faye-Schäfer ‘03, Nissanke-Blanchet ‘05, Itoh ‘09 • 4PN (inc. v8 /c8) Jaranowski-Schäfer ’13, Foffa-Sturani ’13,’16 Bini-Damour ’13, Damour-Jaranowski-Schäfer ’14, Bernard et al’16 New feature : non-locality in time Inclusion of spin-dependent effects: Barker-O’ Connell’75, Faye-Blanchet-Buonanno’06, Damour-Jaranowski-Schaefer’08, Porto-Rothstein ’06, Levi ’10, Steinhoff-Hergt-Schaefer ’10, Steinhoff’11, Levi-Steinhoff’15-18
  20. 20. 2-body Taylor-expanded N + 1PN + 2PN Hamiltonian 11
  21. 21. 2-body Taylor-expanded 3PN Hamiltonian [DJS 01] 12
  22. 22. 2-body Taylor-expanded 4PN Hamiltonian [DJS, 2014] 13
  23. 23. 23 Nonlocality in time: Tail-transported hereditary effects (Blanchet-Damour ’88) Hereditary (time-dissymetric) modification of the quadrupolar radiation-damping force, signalling a breakdown of a basic tenet of PN expansion at the 4PN level: (v/c)^8 fractional generates a time-symmetric nonlocal-in-time 4PN-level action (Damour-Jaranowski-Schaefer’14) which was uniquely matched to the local-zone metric via the Regge-Wheeler- Zerilli-Mano-Suzuki-Takasugi- based work of Bini-Damour’13
  24. 24. 25
  25. 25. Perturbative Theory of the Generation of Gravitational Radiation Einstein ’16, ’18 (+ Landau-Lifshitz 41, and Fock ’55) : h+, hx and quadrupole formula Relativistic, multipolar extensions of LO quadrupole radiation : Sachs-Bergmann ’58, Sachs ’61, Mathews ’62, Peters-Mathews ’63, Pirani '64 Campbell-Morgan ’71, Campbell et al ’75, nonlinear effects: Bonnor-Rotenberg ’66, Epstein-Wagoner-Will ’75-76 Thorne ’80, .., Will et al 00 MPM Formalism: Blanchet-Damour ’86, Damour-Iyer ’91, Blanchet ’95 ‘98 Combines multipole exp. , Post Minkowkian exp., analytic continuation, and PN matching
  26. 26. 26 Multipolar Expansions Using STF Tensors A convenient form for (2l+1)-dim irrep of SO(3): STF tensors multi-index notation (Blanchet-Damour ’86) bTL = Thi1i2···ili TLSL @L = @i1i2···il Multipolar expansions with STF tensors
  27. 27. 27 STF Multipolar Analysis of Linearized Gravity (Damour-Iyer ’91) linearized gravity generalizing the scalar field case (Blanchet-Damour’89)
  28. 28. 28 Radiative multipole moments
  29. 29. MULTIPOLAR POST-MINKOWSKIAN FORMALISM (BLANCHET-DAMOUR-IYER) 29 Decomposition of space-time in various overlapping regions: 1. near-zone: r << lambda : PN theory 2. exterior zone: r >> r_source: MPM expansion 3. far wave-zone: Bondi-type expansion followed by matching between the zones in exterior zone, iterative solution of Einstein’s vacuum field equations by means of a double expansion in non-linearity and in multipoles, with crucial use of analytic continuation (complex B) for dealing with formal UV divergences at r=0 g = ⌘ + Gh1 + G2 h2 + G3 h3 + ..., ⇤h1 = 0, ⇤h2 = @@h1h1, ⇤h3 = @@h1h1h1 + @@h1h2, h1 = X ` @i1i2...i` ✓ Mi1i2...i` (t r/c) r ◆ + @@....@ ✓ ✏j1j2kSkj3...j` (t r/c) r ◆ , h2 = FPB⇤ 1 ret ✓ r r0 ◆B @@h1h1 ! + ..., h3 = FPB⇤ 1 ret....
  30. 30. 30 Nonlinearities in harmonic coordinates (Blanchet-Damour’88,’89,’92) h↵ 2 = ⇤ 1 R (r 2 Q↵ ) + ⇤ 1 R (r 3 N↵ 3 + · · · ) + X `=0,1 @L(r 1 T↵ L ) M/r light-cone deviation Both cured by coord. transformation X↵ = x↵ + G⇠↵ + G2 ↵ + O(G3 )
  31. 31. 31 Hereditary (tail and memory) effects in GW reaction and generation (Blanchet-Damour ’88,’89,’92,….) ( gnear zone 00 )hereditary = Hereditary tail effect (mass x quadrupole) in near-zone Memory and tail effects in wave-zone (radiative) GW multipoles (Blanchet-Damour ’89,92, Christodoulou’91,….) dependence on infinite past depends on multipoles of energy flux tail-transported hereditary effect
  32. 32. 32 Link radiative multipoles <-> source variables (Blanchet-Damour ’89’92, Damour-Iyer’91, Blanchet ’95…) tail memory instant. tail-of-tail
  33. 33. 33 Explicit Source Quadrupole Moment at 3.5 PN for a binary system (Blanchet-Damour-Esposito-Farese-Iyer’05; Blanchet et al;Faye-Marsat-Blanchet-Iyer’12)
  34. 34. Perturbative computation of GW flux from binary system • lowest order : Einstein 1918 Peters-Mathews 63 • 1 + (v2 /c2) : Wagoner-Will 76 • … + (v3 /c3) : Blanchet-Damour 92, Wiseman 93 • … + (v4 /c4) : Blanchet-Damour-Iyer Will-Wiseman 95 • … + (v5 /c5) : Blanchet 96 • … + (v6 /c6) : Blanchet-Damour-Esposito-Farèse-Iyer 2004 • … + (v7 /c7) : Blanchet x = ⇣v c ⌘2 = ✓ G(m1 + m2)⌦ c3 ◆2 3 = ✓ ⇡G(m1 + m2)f c3 ◆2 3 ⌫ = m1m2 (m1 + m2)2

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