Clifford Will Washington University John Archibald Wheeler

Theoretical Foundations of Gravitational-Wave Astronomy: A Post-Newtonian Approach Clifford Will Washington University John Archibald Wheeler International School on Astrophysical Relativity

Interferometers Around The World LIGO Hanford 4&2 km GEO Hannover 600 m TAMA Tokyo 300 m Virgo Cascina 3 km LIGO Livingston 4 km

LISA: a space interferometer for 2015

Inspiralling Compact Binaries - Strong-gravity GR tests? • Fate of the binary pulsar in 100 My • GW energy loss drives pair toward merger LIGO-VIRGO • Last few minutes (10K cycles) for NS-NS • 40 - 700 per year by 2010 • BH inspirals could be more numerous LISA • MBH pairs(105 - 107 Ms) in galaxies • Waves from the early universe A chirp waveform Last 7 orbits

Theoretical Foundations of Gravitational-Wave Astronomy: A Post-Newtonian Approach • The advent of gravitational-wave astronomy • The problem of motion & radiation - a history • Post-Newtonian gravitational radiation • Testing gravity and measuring astrophysical parameters using gravitational waves • GW recoil - Do black holes get kicked out of galaxies? • Interface between PN gravity and numerical relativity Clifford Will, J. A. Wheeler School on Astrophysical Relativity

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 order • Driven by requirements for GW detectors • (v/c)12 beyond Newtonian gravity

The post-Newtonian approximation

DIRE: Direct integration of the relaxed Einstein equations Einstein’s Equations “Relaxed” Einstein’s Equations

PN equations of motion for compact binaries B F S B = Blanchet, Damour, Iyer et al F = Futamase, Itoh S = Schäfer, Jaranowski W = WUGRAV W B W W (in progress)

Gravitational energy flux for compact binaries Wagoner & CW 76 W B W B = Blanchet, Damour, Iyer et al F = Futamase, Itoh S = Schäfer, Jaranowski W = WUGRAV B W B B B

Gravitational Waveform and Matched Filtering Quasi-Newtonian approximation Matched filtering - schematic Fourier transform

GW Phasing as a precision probe of gravity N Measure chirp mass M 1PN Measure m1 & m2 1.5PN “Tail” term - test GR 2PN Test GR

GW Phasing: Bounding scalar-tensor gravity Self-gravity difference N Coupling constant 1PN 1.5PN 2PN

Bounding masses and scalar-tensor theory with LISA Solar system bound • NS + 103 Msun BH • Spins aligned with L • SNR = 10 • 104 binary Monte Carlo • ____ = one detector • ------ = two detectors Berti, Buonanno & CW (2005)

Speed of Waves and Mass of the Graviton • Why Speed could differ from “1” • massive graviton: vg2 = 1 - (mg/Eg)2 • gmn coupling to background fields: vg = F(f,Ka,Hab) • gravity waves propagate off the brane Examples • General relativity. For l<<R, GW follow geodesics of background spacetime, as do photons (vg = 1) • Scalar-tensor gravity. Tensor waves can have vg ≠ 1, if scalar is massive • Massive graviton theories with background metric. Circumvent vDVZ theorem. Visser (1998), Babak & Grishchuk (1999,2003) Possible Limits

Bounding the graviton mass using inspiralling binaries (CW, 1998) Source Detector t x

Bounding the graviton mass using inspiralling binaries

Radiation of momentum and the recoil of massive black holes • General Relativity • Interference between quadrupole and higher momentsPeres (62), Bonnor & Rotenberg (61), Papapetrou (61), Thorne (80) • “Newtonian effect” for binariesFitchett (83), Fitchett & Detweiler (84) • 1 PN correction termWiseman (92) • Astrophysics • MBH formation by mergers could terminate if BH ejected from early galaxies • Ejection from dwarf galaxies or globular clusters • Displacement from center could affect galactic coreMerritt, Milosavljevic, Favata, Hughes & Holz (04) Favata, Hughes & Holz (04)

Radiation of momentum to 2PN order Blanchet, Qusailah & CW (2005) • Calculate relevant multipole moments to 2PN order quadrupole, octupole, current quadrupole, etc • Calculate momentum flux for quasi-circular orbit [x=(mw)2/3≈(v/c)2] recoil = -flux • Integrate up to ISCO (6m) for adiabatic inspiral • Match quasicircular orbit at ISCO to plunge orbit in Schwarzschild • Integrate with respect to “proper w” to horizon (x -> 0)

Recoil velocity as a function of mass ratio X = 0.38 Vmax = 250 ± 50km/s V/c ≈ 0.043 X2 X=1/10 V = 70 ± 15 km/s Blanchet, Qusailah & CW (2005)

Radiation of momentum to 2PN order Blanchet, Qusailah & CW (2005) Checks and tests • Vary 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 Estimates Favata, Hughes & Holtz (2004) Campanelli (Lazarus) (2005) Blanchet, Qusailah & CW (2005) Damour & Gopakumar (2006) Baker et al (2006) 0 100 200 300 400

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

The end-game of gravitational radiation reaction • Evolution leaves quasicircular orbit • describable by PN approximation • Numerical models start with quasi-equilibrium (QE) states • helical Killing vector • stationary in rotating frame • arbitrary rotation states (corotation, irrotational) • How well do PN and QE agree? • surprisingly well, but some systematic differences exist • Develop a PN diagnostic for numerical relativity • elucidate physical content of numerical models • “steer” numerical models toward more realistic physics T. Mora & CMW, PRD 66, 101501 (2002) (gr-qc/0208089) PRD 69, 104021 (2004) (gr-qc/0312082)

Ingredients of a PN Diagnostic • 3PN point-mass equations • Derived by 3 different groups, no undetermined parameters • Finite-size effects • Rotational kinetic energy (2PN) • Rotational flattening (5PN) • Tidal deformations (5PN) • Spin-orbit (3PN) • Spin-spin (5PN)

“Eccentric” orbits in relativistic systems. II Relativistic Gravity Define “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 reactionMinuses • Non-local • Gauge invariant only through 1PN order

“Eccentric” orbits in relativistic systems. III 3PN ADM energy and angular momentum at apocenter

Tidal and rotational effects • Use Newtonian theory; add to 3PN & Spin results • standard textbook machinery (eg Kopal 1959, 1978) • multipole expansion -- keep l=2 & 3 • direct contributions to E and J • indirect contributions via orbit perturbations • Dependence on 4 parameters

Corotating Black Holes - Meudon Data

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

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

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

Energy of irrotational neutron stars - PN vs Meudon/Tokyo G=2 q=8.3 G=2 q=7.1 G=2 q=6.25 G=2 q=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 parameters • PN 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 Einstein’s 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

Clifford Will Washington University John Archibald Wheeler