Gravitational Waves from Inspiralling Compact Binaries: PN - PDF document

Gravitational Waves from Inspiralling Compact Binaries: PN Waveforms and Resummed Extensions Bala R Iyer Raman Research Institute Bangalore, India 30 March 2002 Most Recent Review: Luc Blanchet, Living Reviews in Relativity, gr-qc 0202016 Alessandra Buonanno, gr-qc 0203030

Phasing Formula � v LSO dv E ′ ( v ) t ( v F ) = t LSO + m F ( v ) , v F � v LSO dvv 3 E ′ ( v ) φ ( v F ) = φ LSO + 2 F ( v ) . v F Newtonian Waveform C M ( π M F ( t )) 2 / 3 , a ( t ) = φ c − 2 [( t c − t ) ] 5 / 8 , φ ( t ) = 5 M 5 M 256( t c − t )] 3 / 8 . π M F ( t ) = [ v 3 = πmF Invariant Velocity v : M = ν 3 / 5 m Chirp Mass : Total mass : m Symmetric mass ratio : ν

GW from ICB • Computations of GW from ICB requires control of 3 independent modules 1. Motion: Given a binary system, iterate EE to discuss Conservative motion of this system 2. Generation: Given the motion of a bi- nary on a fixed orbit (circular), iterate EE to compute Multipoles of the Grav Field and thus GW flux 3. Radiation Reaction: Given the flux of radiated energy (AM), use balance arguments to compute the effect on the orbit

MOTION • Status of PN EOM satisfactory Agreement between different approaches and techniques • 2.5PN Damour, Deruelle: Harmonic coords, Riesz regularisation Schafer : ADM, Hadamard partie finie Kopeijkin and Grischuk: Physical computation using self gravitating extended bodies Blanchet, Faye and Ponsot: Direct PN iteration, Matching Itoh, Futamase and Asada: Variant of surface integral approach of EIH

• 3PN Jaranowski, Schafer and Damour: ADM coords, Hadamard regularisation, EOM has an arbitrary parameter ω static Blanchet, Faye and Andrade: Harmonic coords, (extended) Hadamard regularisation, EOM has an arbitrary parameter λ λ = − 3 11 ω static − 1987 3080 • The undetermined constant reflects the incompleteness of the Hadamard regularisation • Hadamard regularisation does not satisfy distributivity of products ( FG ) 1 � = ( F ) 1 ( G ) 1 . Violates Leibniz rule for differentiation of a product

• Dimensional Regularisation preserves the gauge symmetry of perturbative GR underlying the link between Bianchi identities and EOM and hence respects ALL basic properties of algebraic and differential calculus of ordinary functions • Damour, Jaranowski and Schafer: Dimensional regularisation gives ω static = 0 so that λ = − 1987 3080 = − . 645 .. • 3PN EOM and ALL conserved quantities available for General Orbits

WORK IN PROGRESS • Computation without regularisation: Calculate 3PN EOM for extended bodies taking into account internal structure (pressure, density..) and then take its limit as ‘radius’ goes to zero. Compare with the point mass regularised result 2PN: Kopeijkin and Grischuk implemented this and showed effacement of internal structure 3PN: Can one determine λ ??? Is this consistent with ω static = 0 (Blanchet, Esposito-Farese, Poujade) • Can one compute EOM in harmonic coords using dimensional regularisation and determine λ ?

• Upto now one has been discussing the conservative motion of the binary • The radiative part of the EOM is available only upto leading order (2.5PN) • Deriving the full relative 3PN/3.5PN radiation reaction ie absolute 5.5PN/6PN contributions is impossible with present technology • Thus we move to the second module

GENERATION • Apply wave generation formula to compute the work done by radiation reaction force i.e. total energy flux at null infinity Computation of Source multipole moments I L and J L Determination and control of Tails and non-linear effects relating source moments to Radiative moments • 2PN Blanchet, Damour, BRI, Will and Wiseman BDI - Multipolar Post Minkowskian method, Hadamard/Riesz self-field regularisation WW - Direct Integration of Relaxed Ein- stein (DIRE) ; Epstein-Wagoner-Thorne + retarded integral

• Mathematical Equivalence of Both approaches ( Blanchet) • 3PN Instantaneous part Blanchet, BRI, Joguet; Circular Orbits; Harmonic coords + Hadamard regularn of infinite self-field General orbits ( In Progress: Blanchet, BRI) • Hereditary part : Blanchet Tails : 1.5PN, 2.5PN, 3.5PN Tails of Tails, (Tail) 2 : 3PN • 3 undetermined constants in the the Mass Quadrupole combine to a single undetermined constant θ in GW Luminosity in addition to the λ coming from EOM

DETERMINATION of θ ?? • Can θ be computed by an Extended Body Computation?? ( Blanchet, BRI..) • Can we formulate the wave generation in ADM coords?? 2PN , Tails, ???? • Can self-fields in harmonic coordinates be controlled by the Dimensional Regularn in the generation problem?? Need to first discuss EOM in Harmonic coordinates with Dimensional Regularn Setting up the entire MPM generation formalism in d dimensions seems non-trivial Rotation group in higher dimensions, Propagator in higher dimensions, Backscattering/Tails.. Can one be smart enough to apply Dimensional regularisation only where required without setting up the whole edifice???

� 2 / 3 � � h + , × = 2 Gmη Gmω H (0) + , × + x 1 / 2 H (1 / 2) + xH (1) + , × + x 3 / 2 H (3 / 2) + x 2 H (2) � + , × + , × + , × c 2 c 3 0 r 0 H (0) − (1 + c 2 ) cos 2 ψ , = + − s δm H (1 / 2) � (5 + c 2 ) cos ψ − 9(1 + c 2 ) cos 3 ψ � = , + 8 m 1 (19 + 9 c 2 − 2 c 4 ) − η (19 − 11 c 2 − 6 c 4 ) � H (1) � = cos 2 ψ + 6 4 3 s 2 (1 + c 2 )(1 − 3 η ) cos 4 ψ , − s δm (57 + 60 c 2 − c 4 ) − 2 η (49 − 12 c 2 − c 4 ) � H (3 / 2) �� = cos ψ + 192 m 27 (73 + 40 c 2 − 9 c 4 ) − 2 η (25 − 8 c 2 − 9 c 4 ) � � − cos 3 ψ 2 625 2 (1 − 2 η ) s 2 (1 + c 2 ) cos 5 ψ � − 2 π (1 + c 2 ) cos 2 ψ , + 1 (22 + 396 c 2 + 145 c 4 − 5 c 6 ) + H (2) � = + 120 5 3 η (706 − 216 c 2 − 251 c 4 + 15 c 6 ) + − 5 η 2 (98 − 108 c 2 + 7 c 4 + 5 c 6 ) � cos 2 ψ 2 (59 + 35 c 2 − 8 c 4 ) − 5 3 η (131 + 59 c 2 − 24 c 4 ) 15 s 2 � + +5 η 2 (21 − 3 c 2 − 8 c 4 ) � cos 4 ψ − 81 40(1 − 5 η + 5 η 2 ) s 4 (1 + c 2 ) cos 6 ψ + s δm 11 + 7 c 2 + 10(5 + c 2 ) ln 2 � �� sin ψ − 5 π (5 + c 2 ) cos ψ 40 m − 27 � 7 − 10 ln(3 / 2) � (1 + c 2 ) sin 3 ψ + 135 π (1 + c 2 ) cos 3 ψ � , � ω ψ = φ − 2 Gmω � ln , c 3 ω 0 0

c 3 0 η 5 Gm ( t c − t ) , Θ = φ c − 1 � 3715 8064 + 55 Θ 3 / 8 − 3 π � Θ 5 / 8 + � 4 Θ 1 / 4 φ ( t ) = 96 η η � 9275495 14450688 + 284875 258048 η + 1855 2048 η 2 � Θ 1 / 8 � + , � 743 c 3 2688 + 11 Θ − 5 / 8 − 3 π � Θ − 3 / 8 + 0 � 10Θ − 3 / 4 ω ( t ) = 32 η 8 Gm � 1855099 14450688 + 56975 258048 η + 371 2048 η 2 � Θ − 7 / 8 � + Blanchet, Damour, BRI, Will, Wiseman

− µc 2 γ − 7 4 + 1 � � � E = 1 + 4 ν γ + 2 − 7 8 + 49 8 ν + 1 γ 2 + 8 ν 2 � � + − 235 � + 64 + [106301 − 123 64 π 2 + 22 3 ln( r ) − 22 + 3 λ ] ν + r ′ 6720 0 32 ν 2 + 5 27 64 ν 3 � γ 3 � + . − µc 2 x − 3 4 − 1 � � � = 1 + x + E 12 ν 2 − 27 8 + 19 8 ν − 1 � � x 2 + 24 ν 2 + − 675 � 209323 − 205 96 π 2 − 110 � � + 64 + ν − 9 λ 4032 155 35 96 ν 2 − 5184 ν 3 � x 3 � − . (1 − 2 x )(1 − 3 x ) − 1 / 2 − 1 E test = µc 2 � � Blanchet, Faye, (Andrade)

32 c 5 − 2927 336 − 5 � 293383 + 380 5 G γ 5 ν 2 � � � � γ + 4 πγ 3 / 2 + γ 2 L = 1 + 4 ν 9 ν 9072 − 25663 − 109 � � πγ 5 / 2 + 8 ν 672 � + 16 π 2 129386791 − 1712 105 C − 856 + 105 ln(16 γ ) 7761600 3 � r � � � + 123 π 2 − 332051 + 110 + 44 λ − 88 ν − 383 � 9 ν 2 γ 3 + ln 3 θ r ′ 0 720 3 64 � 90205 + 3772673 12096 ν + 32147 3024 ν 2 � � πγ 7 / 2 + O ( γ 4 ) + 576 32 c 5 − 1247 336 − 35 5 G x 5 ν 2 � � � x + 4 πx 3 / 2 L = 1 + 12 ν − 44711 9072 + 9271 504 ν + 65 − 8191 672 − 535 � 18 ν 2 � � � x 2 + πx 5 / 2 + 24 ν � + 16 π 2 6643739519 − 1712 105 C − 856 + 105 ln(16 x ) 69854400 3 � � � + 41 π 2 − 11497453 + 176 9 λ − 88 ν − 94403 3024 ν 2 − 775 324 ν 3 x 3 + 3 θ 272160 48 − 16285 + 176419 1512 ν + 19897 � 378 ν 2 � � πx 7 / 2 + O ( x 4 ) + 504 Blanchet, BRI, Joguet

NONLINEAR EFFECTS • The 1.5PN term is the Tail term. In addition to the (instantaneous) Linear wave emitted at retarded time, we have the (hereditary) Tail wave emitted in the past and scattered off the static (Schwarzschild) gravitational field. This is important for compact binaries. • The 2.5PN term is the Nonlinear Memory term. It is the gravitational radiation from the linear gravitions. It has poor observational consequences. • Upto this order the Instantaneous and Hereditary terms remain disjoint. No longer true at 3PN order • At 3PN we have the Tail of Tails and the (Tail) 2 terms. This is important for CB.

ECCENTRIC BINARIES • Will and Wiseman, Gopakumar and BRI 2PN Energy Flux, Waveform • Gopakumar and BRI AM Flux, Evolution of orbital elements, GW polarisations without inspiral but 2PN accurate periastron precession • GW polarisations with RR ( In progress: Damour, Gopakumar, BRI)

RADIATION REACTION • We assume a energy balance equation dE dt = L • Though physically obvious , no general proof from first principles of GR of the correct- ness of the above balance eqn beyond 1PN/1.5PN • Blanchet, Faye, BRI, Joguet 3.5PN GW Phasing