Clifford WillWashington University

John Archibald WheelerInternational School on Astrophysical RelativityTheoretical Foundations of Gravitational-Wave Astronomy: A Post-Newtonian Approach

Interferometers Around The WorldLIGO Hanford 4&2 kmLIGO Livingston 4 kmGEO Hannover 600 mTAMA Tokyo300 mVirgo Cascina 3 km

LISA: a space interferometerfor 2015

Inspiralling Compact Binaries - Strong-gravity GR tests?Fate of the binary pulsar in 100 MyGW energy loss drives pair toward merger

LIGO-VIRGO Last few minutes (10K cycles) for NS-NS40 - 700 per year by 2010BH inspirals could be more numerousLISAMBH pairs(105 - 107 Ms) in galaxiesWaves from the early universeLast 7 orbitsA chirp waveform

The advent of gravitational-wave astronomyThe problem of motion & radiation - a historyPost-Newtonian gravitational radiationTesting gravity and measuring astrophysical parameters using gravitational wavesGW recoil - Do black holes get kicked out of galaxies?Interface between PN gravity and numerical relativityClifford Will, J. A. Wheeler School on Astrophysical RelativityTheoretical Foundations of Gravitational-Wave Astronomy: A Post-Newtonian Approach

The problem of motion & radiation Geodesic motion 1916 - Einstein - gravitational radiation (wrong by factor 2) 1916 - De Sitter - n-body equations of motion 1918 - Lense & Thirring - motion in field of spinning body 1937 - Levi-Civita - center-of-mass acceleration 1938 - Eddington & Clark - no acceleration 1937 - EIH paper & Robertson application 1960s - Fock & Chandrasekhar - PN approximation 1967 - the Nordtvedt effect 1974 - numerical relativity - BH head-on collision 1974 - discovery of PSR 1913+16 1976 - Ehlers et al - critique of foundations of EOM 1976 - PN corrections to gravitational waves (EWW) 1979 - measurement of damping of binary pulsar orbit 1990s - EOM and gravitational waves to HIGH PN orderDriven by requirements for GW detectors(v/c)12 beyond Newtonian gravity

The advent of gravitational-wave astronomyThe problem of motion & radiation - a historyPost-Newtonian gravitational radiationTesting gravity and measuring astrophysical parameters using gravitational wavesGW recoil - Do black holes get kicked out of galaxies?Interface between PN gravity and numerical relativityClifford Will, J. A. Wheeler School on Astrophysical RelativityTheoretical Foundations of Gravitational-Wave Astronomy: A Post-Newtonian Approach

The post-Newtonian approximation

DIRE: Direct integration of the relaxed Einstein equationsEinsteins EquationsRelaxed Einsteins Equations

DIRE: Direct integration of the relaxed Einstein equationsEinsteins EquationsRelaxed Einsteins Equations

PN equations of motion for compact binariesB F SW B W W (in progress) B = Blanchet, Damour, Iyer et alF = Futamase, ItohS = Schfer, Jaranowski W = WUGRAV

Gravitational energy flux for compact binariesWB W B WB B = Blanchet, Damour, Iyer et alF = Futamase, ItohS = Schfer, Jaranowski W = WUGRAVB B Wagoner & CW 76

The advent of gravitational-wave astronomyThe problem of motion & radiation - a historyPost-Newtonian gravitational radiationTesting gravity and measuring astrophysical parameters using gravitational wavesGW recoil - Do black holes get kicked out of galaxies?Interface between PN gravity and numerical relativityClifford Will, J. A. Wheeler School on Astrophysical RelativityTheoretical Foundations of Gravitational-Wave Astronomy: A Post-Newtonian Approach

Gravitational Waveform and Matched FilteringQuasi-Newtonian approximationFourier transformMatched filtering - schematic

GW Phasing as a precision probe of gravityN1PN1.5PN2PNMeasure chirp mass MMeasure m1 & m2Tail term - test GRTest GR

M = m1+m2h = m1m2/M2M = h3/5Mu = pMf ~ v3

GW Phasing: Bounding scalar-tensor gravityN1PN1.5PN2PNSelf-gravity differenceCoupling constant

M = m1+m2h = m1m2/M2M = h3/5Mu = pMf ~ v3

Bounding masses and scalar-tensor theory with LISANS + 103 Msun BHSpins aligned with LSNR = 10104 binary Monte Carlo____ = one detector------ = two detectorsSolar system boundBerti, Buonanno & CW (2005)

Speed of Waves and Mass of the GravitonWhy Speed could differ from 1massive graviton: vg2 = 1 - (mg/Eg)2gmn coupling to background fields: vg = F(f,Ka,Hab)gravity waves propagate off the braneExamplesGeneral relativity. For l

Bounding the graviton mass using inspiralling binariestxDetectorSource(CW, 1998)

Bounding the graviton mass using inspiralling binaries

m1m2Distance(Mpc)Bound on lg (km)Ground-Based (LIGO/VIRGO)1.41.43004.6 X 1012101015006.0 X 1012Space-Based (LISA)10710730006.9 X 101610510530002.3 X 1016

Other methodsCommentsBound on lg (km)Solar system 1/r2 lawAssumes direct link between static lg and wave lg3 X 1012Galaxies & clustersDitto6 X 1019CWDB phasingLISA (Cutler et al)1 X 1014

The advent of gravitational-wave astronomyThe problem of motion & radiation - a historyPost-Newtonian gravitational radiationTesting gravity and measuring astrophysical parameters using gravitational wavesGW recoil - Do black holes get kicked out of galaxies?Interface between PN gravity and numerical relativityClifford Will, J. A. Wheeler School on Astrophysical RelativityTheoretical Foundations of Gravitational-Wave Astronomy: A Post-Newtonian Approach

Radiation of momentum and the recoil of massive black holesGeneral Relativity Interference between quadrupole and higher moments Peres (62), Bonnor & Rotenberg (61), Papapetrou (61), Thorne (80) Newtonian effect for binaries Fitchett (83), Fitchett & Detweiler (84) 1 PN correction term Wiseman (92) AstrophysicsMBH formation by mergers could terminate if BH ejected from early galaxiesEjection from dwarf galaxies or globular clustersDisplacement from center could affect galactic core Merritt, Milosavljevic, Favata, Hughes & Holz (04) Favata, Hughes & Holz (04)

Radiation of momentum to 2PN orderBlanchet, Qusailah & CW (2005) Calculate relevant multipole moments to 2PN order quadrupole, octupole, current quadrupole, etcCalculate momentum flux for quasi-circular orbit [x=(mw)2/3(v/c)2] recoil = -fluxIntegrate up to ISCO (6m) for adiabatic inspiralMatch quasicircular orbit at ISCO to plunge orbit in SchwarzschildIntegrate with respect to proper w to horizon (x -> 0)

Recoil velocity as a function of mass ratioX = 0.38Vmax = 250 50km/sV/c 0.043 X2Blanchet, Qusailah & CW (2005)X=1/10V = 70 15 km/s

Radiation of momentum to 2PN orderBlanchet, Qusailah & CW (2005)Checks and testsVary matching radius between inspiral and plunge (5.3m-6m) --- 7%Vary matching method --- 10 %Vary energy damping rate from N to 2PN --- no effect Vary cutoff: (a) r=2(m+m) (b) r=2m --- 1%Add 2.5PN, 3PN and 3.5PN terms: a2.5PNx5/2 + a3PNx3 + a3.5PNx7/2 and vary coefficients between +10 and -10 --- 30 % or an rms error of 20 %

Maximum recoil velocity: Range of Estimates0100 200 300 400Favata, Hughes & Holtz (2004)

Campanelli (Lazarus) (2005)

Blanchet, Qusailah & CW (2005)

Damour & Gopakumar (2006)

Baker et al (2006)

Getting a kick from numerical relativityBaker et al (GSFC), gr-qc/0603204

The advent of gravitational-wave astronomyThe problem of motion & radiation - a historyPost-Newtonian gravitational radiationTesting gravity and measuring astrophysical parameters using gravitational wavesGW recoil - Do black holes get kicked out of galaxies?Interface between PN gravity and numerical relativityClifford Will, J. A. Wheeler School on Astrophysical RelativityTheoretical Foundations of Gravitational-Wave Astronomy: A Post-Newtonian Approach

The end-game of gravitational radiation reactionEvolution leaves quasicircular orbitdescribable by PN approximationNumerical models start with quasi-equilibrium (QE) stateshelical Killing vector stationary in rotating framearbitrary rotation states (corotation, irrotational)How well do PN and QE agree?surprisingly well, but some systematic differences existDevelop a PN diagnostic for numerical relativityelucidate physical content of numerical modelssteer numerical models toward more realistic physicsT. Mora & CMW, PRD 66, 101501 (2002) (gr-qc/0208089) PRD 69, 104021 (2004) (gr-qc/0312082)

Ingredients of a PN Diagnostic3PN point-mass equationsDerived by 3 different groups, no undetermined parameters

Finite-size effectsRotational kinetic energy (2PN) Rotational flattening (5PN)Tidal deformations (5PN)Spin-orbit (3PN)Spin-spin (5PN)

Eccentric orbits in relativistic systems. IIRelativistic GravityDefine measurable eccentricity and semilatus rectum:Plusses: Exact in Newtonian limit Constants of the motion in absence of radiation reaction Connection to measurable quantities (W at infinity) Easy to extract from numerical data e 0 naturally under radiation reaction Minuses Non-local Gauge invariant only through 1PN order

Eccentric orbits in relativistic systems. III3PN ADMenergy andangular momentumat apocenter

Tidal and rotational effectsUse Newtonian theory; add to 3PN & Spin resultsstandard textbook machinery (eg Kopal 1959, 1978)multipole expansion -- keep l=2 & 3direct contributions to E and Jindirect contributions via orbit perturbationsDependence on 4 parameters

Corotating Black Holes - Meudon Data

Energy of Corotating Neutron Stars - Numerical vs. PNSimulations by Miller, Suen & WUGRAV

Energy of irrotational neutron stars - PN vs Meudon/TokyoData from Taniguchi& GourgoulhonPRD 68, 124025 (2003)G=2q=8.3

Energy of irrotational neutron stars - PN vs Meudon/TokyoG=1.8q=7.1G=2.0q=7.1G=2.25q=7.1G=2.5q=7.1

Energy of irrotational neutron stars - PN vs Meudon/TokyoG=2q=8.3G=2q=7.1G=2q=6.25G=2q=5.6

Concluding remarks PN theory now gives results for motion and radiation through 3.5 PN order Many results verified by independent groups Spin and finite-size effects More convergent series? Measurement of GW chirp signals may give tests of fundamental theory and astrophysical parametersPN theory may provide robust estimates of strong-gravity phenomena Kick of MBH formed from merger Initial states of compact binaries near ISCO

It is difficult to think of any occasion in the history of astrophysics when three stars at once shone more brightly in the sky than our three stars do at this conference today. First is X-ray astronomy. It brings rich information about neutron stars. It begins to speak to us of the first identifiable black hole on the books of science. Second is gravitational-wave astronomy. It has already established upper limits on the flux of gravitational waves at selected frequencies. At the fantastic new levels of sensitivity now being engineered, it promises to pick up signals every few weeks from collapse events in nearby galaxies. Third is black-hole physics. It furnishes the most entrancing applications we have ever seen of Einsteins geometric account of gravitation. It offers for our study, both theoretical and observational, a wealth of fascinating new effects. John A. Wheeler, IAU Symposium, Warsaw, 1973

The advent of gravitational-wave astronomyThe problem of motion & radiation - a historyPost-Newtonian gravitational radiationTesting gravity and measuring astrophysical parameters using gravitational wavesGW recoil - Do black holes get kicked out of galaxies?Interface between PN gravity and numerical relativityClifford Will, J. A. Wheeler School on Astrophysical RelativityTheoretical Foundations of Gravitational-Wave Astronomy: A Post-Newtonian Approach