Testing General Relativity Using Gravitational Waves: A · PDF file ·...

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


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


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






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






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?






�

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






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