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Clifford WillWashington University
John Archibald WheelerInternational School on Astrophysical Relativity
Theoretical Foundations of Theoretical Foundations of Gravitational-Wave Astronomy: Gravitational-Wave Astronomy:
A Post-Newtonian ApproachA Post-Newtonian Approach
Interferometers Around The WorldInterferometers Around The WorldLIGO Hanford 4&2 km
LIGO Livingston 4 km
GEO Hannover 600 m
TAMA Tokyo300 m
Virgo Cascina 3 km
LISA: a space interferometer
for 2015
Inspiralling Compact Binaries - Strong-gravity GR tests?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 numerous
LISAMBH pairs(105 - 107 Ms) in galaxiesWaves from the early universe
Last 7 orbits
A chirp waveform
•The advent of gravitational-wave astronomy
•The problem of motion & radiation - a history
•Post-Newtonian gravitational radiation
•Testing gravity and measuring astrophysical parametersusing gravitational waves
•GW recoil - Do black holesget kicked out of galaxies?
•Interface between PN gravity and numerical relativity
Clifford Will, J. A. Wheeler School on Astrophysical Relativity
Theoretical Foundations of Theoretical Foundations of Gravitational-Wave Astronomy: Gravitational-Wave Astronomy:
A Post-Newtonian ApproachA Post-Newtonian Approach
The problem of motion & radiationThe 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 advent of gravitational-wave astronomy
•The problem of motion & radiation - a history
•Post-Newtonian gravitational radiation
•Testing gravity and measuring astrophysical parametersusing gravitational waves
•GW recoil - Do black holesget kicked out of galaxies?
•Interface between PN gravity and numerical relativity
Clifford Will, J. A. Wheeler School on Astrophysical Relativity
Theoretical Foundations of Theoretical Foundations of Gravitational-Wave Astronomy: Gravitational-Wave Astronomy:
A Post-Newtonian ApproachA Post-Newtonian Approach
The post-Newtonian approximationThe post-Newtonian approximation
€
ε ~ (v /c)2 ~ (Gm /rc 2) ~ ( p /ρc 2)
gμν = η μν + εh(1)μν + ε 2h(2)
μν +K
Gμν = 8πTμν (G = c =1)
Tμν = ρ uμ uν + p(uμ uν + gμν )
DIRE: Direct integration of the relaxed Einstein equationsDIRE: Direct integration of the relaxed Einstein equations
€
Gμν = 8πTμν
€
hμν ≡ η μν − −ggμν
∂ν hμν = 0
q hμν = −16π (−g)(T μν + t μν )
hμν = 4τ μν (t− | x − ′ x |, ′ x )
| x − ′ x |C∫ d3 ′ x
∇ν T μν = 0, or ∂ν τ μν = 0
Einstein’s Equations
“Relaxed” Einstein’s Equations
DIRE: Direct integration of the relaxed Einstein equationsDIRE: Direct integration of the relaxed Einstein equations
€
Gμν = 8πTμν
€
hμν ≡ η μν − −ggμν
∂ν hμν = 0
q hμν = −16π (−g)(T μν + t μν )
hμν = 4τ μν (t− | x − ′ x |, ′ x )
| x − ′ x |C∫ d3 ′ x
∇ν T μν = 0, or ∂ν τ μν = 0
Einstein’s Equations
“Relaxed” Einstein’s Equations
PN equations of motion for compact binariesPN equations of motion for compact binaries
€
a = −m
r3x +1PN +1PNSO +1PNSS + 2PN + 2.5PN
+ 3PN
+ 3.5PN
+ 3.5PNSO
+ 3.5PNSS
B F SB F S
W B W B
W W
W W (in progress)(in progress)
B = Blanchet, Damour, Iyer et alF = Futamase, ItohS = Schäfer, Jaranowski W = WUGRAV
Gravitational energy flux for compact binariesGravitational energy flux for compact binaries
€
˙ E = ˙ E quad +1PN
+1PNSO +1PNSS
+1.5PN
+ 2PN
+ 2.5PN
+ 3PN
+ 3.5PN
WW
B W B W
B WB W
B B
B = Blanchet, Damour, Iyer et alF = Futamase, ItohS = Schäfer, Jaranowski W = WUGRAV
B B
B B
Wagoner & CW 76Wagoner & CW 76
•The advent of gravitational-wave astronomy
•The problem of motion & radiation - a history
•Post-Newtonian gravitational radiation
•Testing gravity and measuring astrophysical parametersusing gravitational waves
•GW recoil - Do black holesget kicked out of galaxies?
•Interface between PN gravity and numerical relativity
Clifford Will, J. A. Wheeler School on Astrophysical Relativity
Theoretical Foundations of Theoretical Foundations of Gravitational-Wave Astronomy: Gravitational-Wave Astronomy:
A Post-Newtonian ApproachA Post-Newtonian Approach
€
h(t) ≈ A(t) e iΦ(t )
€
˜ h ( f ) = h(t)e2πiftdt−∞
∞
∫ ≈ ˜ A ( f )e iΨ( f )
Gravitational Waveform and Matched FilteringGravitational Waveform and Matched Filtering
Quasi-Newtonian approximation
Fourier transform
€
h(t) s(t) + n(t)[ ]
Matched filtering - schematic
€
Ψ( f ) = 2πftc − Φc − π /4
+3
128u−5 / 3 1[
+20
9
743
336+
11
4η
⎛
⎝ ⎜
⎞
⎠ ⎟η −2 / 5u2 / 3
−16πu
+10305673
1016064+
5429
1008η +
617
144η 2 ⎛
⎝ ⎜
⎞
⎠ ⎟η −4 / 5u4 / 3
+ O(u5) ]
GW Phasing as a precision probe of gravityGW Phasing as a precision probe of gravity
N
1PN
1.5PN
2PN
Measure chirp mass M
Measure m1 & m2
“Tail” term - test GR
Test GR
M = m1+m2 = m1m2/M2 M = 3/5M
u = Mf ~ v3
€
Ψ( f ) = 2πftc − Φc − π /4
+3
128u−5 / 3 1[
+20
9
743
336+
11
4η
⎛
⎝ ⎜
⎞
⎠ ⎟η −2 / 5u2 / 3
−16πu
+10305673
1016064+
5429
1008η +
617
144η 2 ⎛
⎝ ⎜
⎞
⎠ ⎟η −4 / 5u4 / 3
+ O(u5) ]
GW Phasing: Bounding scalar-tensor gravityGW Phasing: Bounding scalar-tensor gravity
N
1PN
1.5PN
2PN
€
−5
84
ΔS2
ωη 2 / 5u−2 / 3
Self-gravity difference
Coupling constant
M = m1+m2 = m1m2/M2 M = 3/5M
u = Mf ~ v3
Bounding masses and scalar-tensor theory with LISABounding masses and scalar-tensor theory with LISA
NS + 103 Msun BHSpins aligned with LSNR = 10104 binary Monte Carlo____ = one detector------ = two detectors
Solar system bound
Berti, Buonanno & CW (2005)
Speed of Waves and Mass of the GravitonSpeed of Waves and Mass of the Graviton
Why Speed could differ from “1”
massive graviton: vg2 = 1 - (mg/Eg)2
g coupling to background fields: vg = F(,K,H)
gravity waves propagate off the brane
ExamplesGeneral relativity. For <<R, GW follow geodesics of background spacetime, as do photons (vg = 1)
Scalar-tensor gravity. Tensor waves can have vg ≠ 1, if scalar is massiveMassive graviton theories with background metric. Circumvent vDVZ theorem. Visser (1998), Babak & Grishchuk (1999,2003)
Possible Limits
€
1− vg ≈ 5 ×10−17 200Mpc
DΔta − (1+ Z)Δte[ ]
D = distance of source, Z = redshift, Δta (Δte ) = time difference in seconds
Bounding the graviton mass using inspiralling binariesBounding the graviton mass using inspiralling binaries
t
x
Detector
Source
(CW, 1998)
Bounding the graviton mass using inspiralling binariesBounding the graviton mass using inspiralling binaries
m1 m2Distance(Mpc)
Bound on g (km)
Ground-Based (LIGO/VIRGO)
1.4 1.4 300 4.6 X 1012
10 10 1500 6.0 X 1012
Space-Based (LISA)
107 107 3000 6.9 X 1016
105 105 3000 2.3 X 1016
Other methods Comments Bound on g (km)
Solar system 1/r2 law
Assumes direct link between static g
and wave g
3 X 1012
Galaxies & clusters Ditto 6 X 1019
CWDB phasing LISA (Cutler et al) 1 X 1014
•The advent of gravitational-wave astronomy
•The problem of motion & radiation - a history
•Post-Newtonian gravitational radiation
•Testing gravity and measuring astrophysical parametersusing gravitational waves
•GW recoil - Do black holesget kicked out of galaxies?
•Interface between PN gravity and numerical relativity
Clifford Will, J. A. Wheeler School on Astrophysical Relativity
Theoretical Foundations of Theoretical Foundations of Gravitational-Wave Astronomy: Gravitational-Wave Astronomy:
A Post-Newtonian ApproachA Post-Newtonian Approach
Radiation of momentum and the recoil of Radiation of momentum and the recoil of massive black holesmassive black holes
General 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)
Astrophysics
MBH 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 orderRadiation of momentum to 2PN orderBlanchet, Qusailah & CW (2005)
Calculate relevant multipole moments to 2PN order quadrupole, octupole, current quadrupole, etc
Calculate momentum flux for quasi-circular orbit [x=(m)2/3≈(v/c)2]recoil = -flux
€
dP
dt= −
464
105
δm
mη 2x11/ 2 1+ −
452
87−
1139
522η
⎛
⎝ ⎜
⎞
⎠ ⎟x +
309
58πx 3 / 2
⎡
⎣ ⎢
+ −71345
22968+
36761
2088η +
147101
68904η 2 ⎛
⎝ ⎜
⎞
⎠ ⎟x 2
⎤
⎦ ⎥λ
Integrate up to ISCO (6m) for adiabatic inspiralMatch quasicircular orbit at ISCO to plunge orbit in SchwarzschildIntegrate with respect to “proper ” to horizon (x -> 0)
Recoil velocity as a function of mass ratioRecoil velocity as a function of mass ratio
€
=m1m2
(m1 + m2)2
=X
(1+ X)2
X = 0.38Vmax = 250 ± 50km/s
V/c ≈ 0.043 X2
Blanchet, Qusailah & CW (2005)
X=1/10V = 70 ± 15 km/s
Radiation of momentum to 2PN orderRadiation 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+) (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 EstimatesMaximum recoil velocity: Range of Estimates
0 100 200 300 400
Favata, Hughes & Holtz (2004)
Campanelli (Lazarus) (2005)
Blanchet, Qusailah & CW (2005)
Damour & Gopakumar (2006)
Baker et al (2006)
Getting a kick from numerical relativityGetting a kick from numerical relativity
Baker et al (GSFC), gr-qc/0603204
•The advent of gravitational-wave astronomy
•The problem of motion & radiation - a history
•Post-Newtonian gravitational radiation
•Testing gravity and measuring astrophysical parametersusing gravitational waves
•GW recoil - Do black holesget kicked out of galaxies?
•Interface between PN gravity and numerical relativity
Clifford Will, J. A. Wheeler School on Astrophysical Relativity
Theoretical Foundations of Theoretical Foundations of Gravitational-Wave Astronomy: Gravitational-Wave Astronomy:
A Post-Newtonian ApproachA Post-Newtonian Approach
The end-game of gravitational radiation reactionThe end-game of gravitational radiation reaction
Evolution leaves quasicircular orbitdescribable by PN approximation
Numerical 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 exist
Develop a PN diagnostic for numerical relativityelucidate 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)
€
∂ /∂t + Ω∂ /∂φ
Ingredients of a PN DiagnosticIngredients 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)€
ERot ≈ mR2ω2 ≈ ENq2(m /r)2
€
EFlat ≈ δIω2 ≈ ω4R5 ≈ ENq5(m /r)5
€
ETide≈(′ δ m)2/R≈ENq
5(m/r)
5
€
ESO ≈ LS /r3 ≈ ENq2(m /r)3
€
ESS ≈ S1S2 /r3 ≈ ENq4 (m /r)5
€
E = EN 1+m
r+
m
r
⎛
⎝ ⎜
⎞
⎠ ⎟2
+m
r
⎛
⎝ ⎜
⎞
⎠ ⎟3 ⎧
⎨ ⎩
⎫ ⎬ ⎭
J = JN 1+m
r+
m
r
⎛
⎝ ⎜
⎞
⎠ ⎟2
+m
r
⎛
⎝ ⎜
⎞
⎠ ⎟3 ⎧
⎨ ⎩
⎫ ⎬ ⎭
““Eccentric” orbits in relativistic systems. IIEccentric” orbits in relativistic systems. II
Relativistic GravityDefine “measurable” eccentricity and semilatus rectum:
€
e ≡Ω p − Ωa
Ω p + Ωa
ζ ≡m
p≡
mΩ p + mΩa
2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
4 / 3
=mΩa
(1− e)2
⎛
⎝ ⎜
⎞
⎠ ⎟2 / 3
Plusses: Exact in Newtonian limit Constants of the motion in absence of radiation reaction
Connection to “measurable” quantities ( 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. IIIEccentric” orbits in relativistic systems. III
3PN ADMenergy andangular momentumat apocenter
Tidal and rotational effectsTidal and rotational effects
Use 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 perturbations
Dependence on 4 parameters
€
=I / MR2
q = R / M
k2,k3 : apsidal constants
kl =0 , point mass3
4(l −1), homogeneous
⎧ ⎨ ⎪
⎩ ⎪
Corotating Black Holes - Meudon DataCorotating Black Holes - Meudon Data
Energy of Corotating Neutron Stars - Numerical vs. PNEnergy of Corotating Neutron Stars - Numerical vs. PN
Simulations by Miller, Suen & WUGRAV
Energy of irrotational neutron stars - PN vs Meudon/TokyoEnergy of irrotational neutron stars - PN vs Meudon/Tokyo
Data from Taniguchi& GourgoulhonPRD 68, 124025 (2003)
=2q=8.3
Energy of irrotational neutron stars - PN vs Meudon/TokyoEnergy of irrotational neutron stars - PN vs Meudon/Tokyo
=1.8q=7.1
=2.0q=7.1
=2.25q=7.1
=2.5q=7.1
Energy of irrotational neutron stars - PN vs Meudon/TokyoEnergy of irrotational neutron stars - PN vs Meudon/Tokyo
=2q=8.3
=2q=7.1
=2q=6.25 =2
q=5.6
Concluding remarksConcluding 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 It is difficult to think of any occasion in the history of astrophysics when three stars at once shone more brightly in astrophysics when three stars at once shone more brightly in the sky than our three stars do at this conference today. First is the sky than our three stars do at this conference today. First is X-ray astronomy. It brings rich information about neutron stars. X-ray astronomy. It brings rich information about neutron stars. It begins to speak to us of the first identifiable black hole on the It begins to speak to us of the first identifiable black hole on the books of science. Second is gravitational-wave astronomy. It books of science. Second is gravitational-wave astronomy. It has already established upper limits on the flux of gravitational has already established upper limits on the flux of gravitational waves at selected frequencies…. At the fantastic new levels of waves at selected frequencies…. At the fantastic new levels of sensitivity now being engineered, it promises to pick up signals sensitivity now being engineered, it promises to pick up signals every few weeks from collapse events in nearby galaxies. Third every few weeks from collapse events in nearby galaxies. Third is black-hole physics. It furnishes the most entrancing is black-hole physics. It furnishes the most entrancing applications we have ever seen of Einstein’s geometric account applications we have ever seen of Einstein’s geometric account of gravitation. It offers for our study, both theoretical and of gravitation. It offers for our study, both theoretical and observational, a wealth of fascinating new effects. observational, a wealth of fascinating new effects.
John A. Wheeler, IAU Symposium, Warsaw, 1973
•The advent of gravitational-wave astronomy
•The problem of motion & radiation - a history
•Post-Newtonian gravitational radiation
•Testing gravity and measuring astrophysical parametersusing gravitational waves
•GW recoil - Do black holesget kicked out of galaxies?
•Interface between PN gravity and numerical relativity
Clifford Will, J. A. Wheeler School on Astrophysical Relativity
Theoretical Foundations of Theoretical Foundations of Gravitational-Wave Astronomy: Gravitational-Wave Astronomy:
A Post-Newtonian ApproachA Post-Newtonian Approach