Testing General Relativity Testing General Relativity Using Gravitational Waves:Using Gravitational Waves:
A TutorialA Tutorial
Clifford WillWashington University, St. Louis
Gravitational Wave Tests of Alternative Theories of Gravity in the Advanced Detector Era
University of Wisconsin, Milwaukee, 26 May, 2010
GW probe new sectors of the dynamics of a theory New phenomena can occur (negative energy flux,
dipole radiation, v ≠1 propagation), even in theories that agree with GR at PN order
With compact objects as sources, additional effects can be present (violations of SEP, composition- dependent effects)
Lessons of PPN formalism not obviously applicable No obvious “parametrization” of GW
PPE templates (Yunes & Pretorius arXiv:0909.3328) Mishra, Arun, Iyer & Sathyaprakash (arXiv:1005.0304)
Must one plow through theory by theory?
Testing General Relativity Using Testing General Relativity Using Gravitational Waves: A TutorialGravitational Waves: A Tutorial
Testing General Relativity Using Testing General Relativity Using Gravitational Waves: A TutorialGravitational Waves: A Tutorial
Introduction Propagation of gravitational waves: polarization Propagation of gravitational waves: speed Generation of gravitational waves: compact binaries
General relativity Scalar-tensor gravity Other alternative theories
Tests of alternative theories using matched filtering
University of Wisconsin, Milwaukee, 26 May, 2010
Propagation of Gravitational Waves:Propagation of Gravitational Waves:PolarizationPolarization
Ψ2
Im Ψ4Re Ψ4
Φ2 2
Re Ψ3 Im Ψ3
Invariant classification ofComponents of Riemann tensor
Eardley et al, PRL 30,884, 1973TEGP, 10.2
Specific transformationsunder null rotations
θ
φ
ψ
Interferometer output
z
x y
Measuring polarization with interferometersMeasuring polarization with interferometersWave propagation coordinates (z direction)
Beam pattern functions
How many interferometers are required to measure or
bound all 6 modes?
How many if 5/6 are present (Ψ2 = 0)?
How many if 3/6 are present (Ψ2 = 0, Ψ3 = 0)?
What is gained by a 4th Asian/Southern Hemisphere IO? Is the Wen-Schutz “redundancy veto” useful?
Measuring polarization with interferometersMeasuring polarization with interferometers
See bibliography in Sec 6.2 of CBGRE
Testing General Relativity Using Testing General Relativity Using Gravitational Waves: A TutorialGravitational Waves: A Tutorial
Introduction Propagation of gravitational waves: polarization Propagation of gravitational waves: speed Generation of gravitational waves: compact binaries
General relativity Scalar-tensor gravity Other alternative theories
Tests of alternative theories using matched filtering
University of Wisconsin, Milwaukee, 26 May, 2010
Why Speed could differ from “1”
massive graviton: vg2 = 1 - (mg/Eg)2
gµν coupling to background fields: vg = F(φ,Kα,Hαβ )
gravity waves propagate off the braneExamples
General relativity. For λ<<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.
Possible Limits
�
1- vg � 2 � 10- 13 200Mpc
DDta - (1+ Z)Dte[ ]
D = distance of source, Z = redshift, Dta (Dte ) = time difference in hours
Propagation of Gravitational Waves: SpeedPropagation of Gravitational Waves: Speed
Bounding the graviton mass using inspiralling binariesBounding the graviton mass using inspiralling binaries
t
x
Detector
Source
CW, PRD 57, 2061 (1998)
∆ta
∆te
Effect of graviton mass on GW phasingEffect of graviton mass on GW phasingRobertson-Walker metric
Massive graviton
Dispersion of arrival times
Effect on phase of h(f)
χ=0χ= χe
Testing General Relativity Using Testing General Relativity Using Gravitational Waves: A TutorialGravitational Waves: A Tutorial
Introduction Propagation of gravitational waves: polarization Propagation of gravitational waves: speed Generation of gravitational waves: compact binaries
General relativity Scalar-tensor gravity Other alternative theories
Tests of alternative theories using matched filtering
University of Wisconsin, Milwaukee, 26 May, 2010
Generation of gravitational waves: GRGeneration of gravitational waves: GRRewrite Einstein’s equations
Define the fieldImpose Lorentz gauge (coordinate condition)Einstein’s equations become:
DIRE: Direct integration of the relaxed Einstein equationsDIRE: Direct integration of the relaxed Einstein equations
Formal solution
Lorentz gauge implies:
Can be shown to be equivalent to theequations of motion
Why “relaxed”? solve for h as a functional of source variables solve equations of motion to get source variables as functions of time determine h(t,x)
The The post-Newtonianpost-Newtonian approximation: Near zone approximation: Near zone
�
ε ~ (v /c)2 ~ (Gm /rc 2) ~ ( p /r c 2) ~ [(� /� t) /(� /� x)]2
gmn = h mn + eh(1)mn + e2h(2)
mn +K
Near zone and far zone λ
Near zone
Far zone
Expand the retarded time (near zone)
Baryonic mass density:
Matter stress-energy tensor
The post-Newtonian approximation: Near zoneThe post-Newtonian approximation: Near zone
ε εε 2 ε 2ε 2
We need to calculate
Recall the action for a geodesic
The post-Newtonian limit of general relativityThe post-Newtonian limit of general relativity
PN equations of motion for compact binariesPN equations of motion for compact binaries
�
a = -m
r3 x +1PN +1PNSO +1PNSS + 2PN + 2.5PN
+ 3PN
+ 3.5PN
+ 3.5PNSO
+ 3.5PNSS
B F SB F S
W B W B
W W
WW
B = Blanchet, Damour, Iyer et alF = Futamase, ItohS = Schäfer, Jaranowski W = WUGRAV
Post-Minkowski expansion: Far ZonePost-Minkowski expansion: Far Zone
Multipole moments
From Lorentz gauge
So for the waveform valid toO(1/R) we only need hij
Post-Minkowski expansion: Far ZonePost-Minkowski expansion: Far Zone
Dominant m=0 term:
Useful identity from
The quadrupole formulafor gravitational waves
Global conservation laws and fluxes to infinityGlobal conservation laws and fluxes to infinity
In the far zone:
where
Integrating over all angles
Energy loss and binary pulsarsEnergy loss and binary pulsarsFor a two-body system
Average over an orbit
Fractional change in E per orbit:
a 2.5 PN effect
The Hulse-Taylorbinary pulsar
A test to 0.3 %
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
Case study: GW in scalar-tensor gravityCase study: GW in scalar-tensor gravityMirshekari, Yunes and CMW
Goals: work in generalized ST theory ω(φ) compact bodies with self gravity (Eardley method) use the “relaxed” ST equations derive the 2PN equations of motion, non spinning derive the 2PN gravitational waveform derive the 2PN energy and angular momentum flux and convert to phasing answer the conjecture: ST binary black holes are observably indistiguishable from GR BBH up to 2PN order in EOM and waves (cmw) analyse tests using ground and space interferometers
The “relaxed” Scalar-Tensor EquationsThe “relaxed” Scalar-Tensor Equations
Field equations
Gravitationally bound “point” masses:
Equation of motion (not-quite geodesic)
The “relaxed” Scalar-Tensor EquationsThe “relaxed” Scalar-Tensor EquationsField definitions and gauge condition
Relaxed ST equations
constructed using
ηµν -hµν
1. Tensor field
2. Scalar field
Effect of compact objectsEffect of compact objects
Scalar field solution: single static body
For neutron stars:
For a stationary black hole (Price, Dykla, Hawking):
To lowest order in the fields,
Another argument:
Effect of compact objects: RadiationEffect of compact objects: RadiationScalar field solution: compact binary system
Dipole field
Dipole energy flux
Contribution to the phase Ψ(f)
Other theories of interestOther theories of interest
Einstein-Æther Theory (Jacobson et al) vector-tensor theory special case of Helling-Nordtvedt-Will class of theories vector field is timelike with unit norm designed to explore violations of Lorentz invariance in gravity
TeVeS Theory (Bekenstein et al) scalar-vector-tensor theory designed to exhibit MOND behavior at galactic scales
Chern-Simons Theory (Yunes et al) scalar-vector-tensor theory designed to exhibit MOND behavior at galactic scales
DGP (Dvali-Gabadadze-Poratti) Theory extra dimensions, inspired by braneworlds can binary system & GW calculations be done?
Testing General Relativity Using Testing General Relativity Using Gravitational Waves: A TutorialGravitational Waves: A Tutorial
Introduction Propagation of gravitational waves: polarization Propagation of gravitational waves: speed Generation of gravitational waves: compact binaries
General relativity Scalar-tensor gravity Other alternative theories
Tests of alternative theories using matched filtering
University of Wisconsin, Milwaukee, 26 May, 2010
�
Ψ( f ) = 2pftc - F c - p /4
+3
128u- 5 / 3 1[
+209
743336
+114
h�
� �
�
� � h - 2 / 5u2 / 3
- 16pu
+10305673
1016064+
54291008
h +617144
h 2�
� �
�
� � h - 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 η = m1m2/M2 M = η3/5M
u = πMf ~ v3
�
Ψ( f ) = 2pftc - F c - p /4
+3
128u- 5 / 3 1[
+209
743336
+114
h�
� �
�
� � h - 2 / 5u2 / 3
- 16pu
+10305673
1016064+
54291008
h +617144
h 2�
� �
�
� � h - 4 / 5u4 / 3
+ O(u5) ]
GW Phasing: Bounding scalar-tensor gravityGW Phasing: Bounding scalar-tensor gravity
N
1PN
1.5PN
2PN
�
− 584
Ds2
wh 2 / 5u- 2 / 3
“Sensitivity” difference
Coupling constant
M = m1+m2 η = m1m2/M2 M = η3/5M
u = πMf ~ v3
Testing scalar-tensor theory with AdLIGO/AdVirgoTesting scalar-tensor theory with AdLIGO/AdVirgo
CMW, PRD 57, 2061 (1993)Updated by K. G. Arun
Testing scalar-tensor theory with ETTesting scalar-tensor theory with ET
From Einstein Telescope Design study: Vision DocumentFigure by K. G. Arun
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)
�
Ψ( f ) = 2pftc - F c - p /4
+3
128u- 5 / 3 1[
+209
743336
+114
h�
� �
�
� � h - 2 / 5u2 / 3
- 16pu
+10305673
1016064+
54291008
h +617144
h 2�
� �
�
� � h - 4 / 5u4 / 3
+ O(u5) ]
GW Phasing: Bounding the graviton massGW Phasing: Bounding the graviton mass
N
1PN
1.5PN
2PN
�
−128p 2DM
3l g2 (1+ Z)
h 3 / 5u2 / 3
“Distance”
Compton wavelength
M = m1+m2 η = m1m2/M2 M = η3/5M
u = πMf ~ v3
Bounding the graviton mass using inspiralling binariesBounding the graviton mass using inspiralling binariesEffect of spin precessions
Effect of higher harmonics
106 & 106 Msun @ z=0.55Stavridis & CMW 2009
Arun & CMW 2009
Bounding the graviton mass using inspiralling binariesBounding the graviton mass using inspiralling binaries
m1 m2 Distance(Mpc) Bound on λg (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 λg (km)Solar system 1/r2 law Assumes direct link
between static λg and wave λg
3 X 1012
Galaxies & clusters Same 6 X 1019
CWDB phasing LISA (Cutler et al) 1 X 1014
Berti, Buonanno, Yunes, Arun, Stavridis, CW
Testing General Relativity Using Testing General Relativity Using Gravitational Waves: A TutorialGravitational Waves: A Tutorial
Introduction Propagation of gravitational waves: polarization Propagation of gravitational waves: speed Generation of gravitational waves: compact binaries
General relativity Scalar-tensor gravity Other alternative theories
Tests of alternative theories using matched filtering
University of Wisconsin, Milwaukee, 26 May, 2010