POST-NEWTONIAN METHODS AND APPLICATIONS Luc Blanchet Gravitation et Cosmologie ( G R ε C O ) Institut d’Astrophysique de Paris 10 juin 2010 Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 1 / 43
ASTROPHYSICAL MOTIVATION Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 2 / 43
Ground-based laser interferometric detectors LIGO GEO LIGO/VIRGO/GEO observe the GWs in the high-frequency band 10 Hz � f � 10 3 Hz VIRGO Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 3 / 43
Space-based laser interferometric detector LISA LISA will observe the GWs in the low-frequency band 10 − 4 Hz � f � 10 − 1 Hz Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 4 / 43
The inspiral and merger of compact binaries Neutron stars spiral and coalesce Black holes spiral and coalesce Neutron star ( M = 1 . 4 M ⊙ ) events will be detected by ground-based 1 detectors LIGO/VIRGO/GEO Stellar size black hole ( 5 M ⊙ � M � 20 M ⊙ ) events will also be detected by 2 ground-based detectors Supermassive black hole ( 10 5 M ⊙ � M � 10 8 M ⊙ ) events will be detected 3 by the space-based detector LISA Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 5 / 43
Supermassive black-hole coalescences as detected by LISA When two galaxies collide their central supermassive black holes may form a bound binary system which will spiral and coalesce. LISA will be able to detect the gravitational waves emitted by such enormous events anywhere in the Universe Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 6 / 43
Extreme mass ratio inspirals (EMRI) for LISA A neutron star or stellar-size black hole follows a highly relativistic orbit around a supermassive black hole. Testing general relativity in the strong field regime and verifying the nature of the central object (is it a Kerr black hole?) are important goals of LISA. Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 7 / 43
The binary pulsar PSR 1913+16 The pulsar PSR 1913+16 is a rapidly rotating neutron star emitting radio waves like a lighthouse toward the Earth. This pulsar moves on a (quasi-)Keplerian close orbit around an unseen companion, probably another neutron star Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 8 / 43
The orbital decay of binary pulsar [Taylor & Weisberg 1989] Prediction from general relativity � 5 / 3 1 + 73 24 e 2 + 37 � 2 πG M 96 e 4 P = − 192 π µ ˙ ≈ − 2 . 4 10 − 12 s / s 5 c 5 (1 − e 2 ) 7 / 2 M P Newtonian energy balance argument [Peters & Mathews 1963] 2.5PN gravitational radiation reaction effect [Damour & Deruelle 1982] Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 9 / 43
GRAVITATIONAL WAVE TEMPLATES FOR BINARY INSPIRAL Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 10 / 43
Methods to compute gravitational-wave templates log 10 ( r 12 / m ) 4 Post-Newtonian Theory Post-Newtonian 3 & Theory Perturbation Theory 2 Numerical Perturbation 1 Relativity Theory 0 log 10 ( m 2 / m 1 ) 0 1 2 3 4 Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 11 / 43
Methods to compute gravitational-wave templates m 2 r 12 m 1 log 10 ( r 12 / m ) 4 Post-Newtonian Theory Post-Newtonian 3 & Theory Perturbation Theory 2 Numerical Perturbation 1 Relativity Theory 0 log 10 ( m 2 / m 1 ) 0 1 2 3 4 Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 11 / 43
Methods to compute gravitational-wave templates log 10 ( r 12 / m ) 4 Post-Newtonian Theory m 2 Post-Newtonian 3 & Theory Perturbation Theory 2 m 1 Numerical Perturbation 1 Relativity Theory 0 log 10 ( m 2 / m 1 ) 0 1 2 3 4 Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 11 / 43
Methods to compute gravitational-wave templates log 10 ( r 12 / m ) 4 Post-Newtonian Theory Post-Newtonian 3 & Theory Perturbation Theory 2 Numerical Perturbation 1 Relativity Theory 0 log 10 ( m 2 / m 1 ) 0 1 2 3 4 Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 11 / 43
Methods to compute gravitational-wave templates Effective-one-body (EOB) log 10 ( r 12 / m ) 4 Post-Newtonian Theory Post-Newtonian 3 & Theory Perturbation Theory 2 Numerical Perturbation 1 Relativity Theory 0 log 10 ( m 2 / m 1 ) 0 1 2 3 4 Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 11 / 43
PN templates for inspiralling compact binaries observer i The orbital phase φ ( t ) should be monitored in m 2 LIGO/VIRGO detectors with precision orbital plane φ m 1 δφ ∼ π ascending node d i r � GMω � − 5 / 3 � � e φ ( t ) = φ 0 − 1 1 +1PN + 1 . 5PN + · · · + 3PN c + · · · c 3 c 2 c 3 c 6 32 η � �� � � �� � needs to be computed with high PN precision result of the quadrupole formalism (sufficient for the binary pulsar) Detailed data analysis (using the sensitivity noise curve of LIGO/VIRGO detectors) show that the required precision is at least 2PN for detection and 3PN for parameter estimation Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 12 / 43
Equations of motion of compact binaries v 1 The equations of motion are written in y 1 Newtonian-like form (with t = x 0 /c playing the role of Newton’s “absolute time”) r 12 y 2 v 2 very difficult term to compute � 1 � �� � � d v 1 1 + 1 + 1 1 1 1 dt = A N c 2 A 1PN c 4 A 2PN c 5 A 2 . 5PN c 6 A 3PN c 7 A 3 . 5PN + + + + O 1 1 1 1 1 c 8 � �� � � �� � radiation reaction radiation reaction 1PN [Lorentz & Droste 1917; Einstein, Infeld & Hoffmann 1938] 2PN [Damour & Deruelle 1981, 1982] 2.5PN [Damour 1983; LB, Faye & Ponsot 1998] 3PN [Jaranowski & Sch¨ afer 1999; LB & Faye 2000, 2001; Itoh & Futamase 2003] 3.5PN [Pati & Will 2002; Nissanke & LB 2005] Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 13 / 43
Two equivalent PN wave generation formalisms The field equations are integrated in the exterior of an extended PN source by means of a multipolar expansion BD multipole moments [LB & Damour 1989; LB 1995, 1998] � M µν d 3 x x L τ µν ( x , t ) L ( t ) = Finite Part B =0 WW multipole moments [Will & Wiseman 1996] � W µν d 3 x x L τ µν ( x , t ) L ( t ) = M These formalisms solved the long-standing problem of divergencies in the PN expansion for general extended sources Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 14 / 43
Tails are an important part of the GW signal Tail of GW Tails are produced by backscatter of GWs on the curvature induced by the matter source’s total mass M They appear at 1.5PN order beyond the “Newtonian” approximation given by the Einstein quadrupole formula Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 15 / 43
The compact binary inspiral waveform Current precision of the PN inspiral waveform is 3.5PN [LB, Damour, Iyer, Will & Wiseman 1995; LB, Faye, Iyer & Siddhartha 2008] The PN waveform is now matched to the numerical merger waveform [Pretorius 2005, Baker et al 2006, Campanelli et al 2006] Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 16 / 43
GRAVITATIONAL SELF-FORCE THEORY Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 17 / 43
General problem of the self-force A particle is moving on a background u µ u µ = f µ = 0 m 2 space-time Its own stress-energy tensor modifies the background gravitational field f µ Because of the “back-reaction” the motion m 1 of the particle deviates from a background geodesic hence the appearance of a self force The self acceleration of the particle is proportional to its mass � m 1 � u µ D¯ = f µ = O d τ m 2 The gravitational self force includes both dissipative (radiation reaction) and conservative effects. Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 18 / 43
Self-force in perturbation theory The space-time metric g µν is decomposed as a background metric ¯ g µν plus h µν = linearized parturbation of the background space-time The field equation in an harmonic gauge reads � h µν + 2 R µ ν ρ σ h ρσ = − 16 π T µν µ x µ u The retarded solution is � u ρ ¯ u σ + O ( m 2 h µν ( x ) = 4 m 1 µν ρσ ( x, z ) ¯ 1 ) G ret Γ particle's trajectory Γ z µ Luc Blanchet ( G R ε C O ) Post-Newtonian methods and applications S´ eminaire IHES 19 / 43
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