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Gravitational Wave Tests of General Relativity
Nico Yunes Montana State University
OleMiss Workshop, Mississippi 2014 (see Yunes & Siemens, Living Reviews in Relativity, 2013)
GW Tests of GR Yunes
An incomplete summary of what and how GWs will tell us about the gravitational interaction
Clifford Will, Stephon Alexander, Sam Finn, Ben Owen, Emanuele Berti, Vitor Cardoso, Leonardo Gualtieri, David Spergel, Frans Pretorius, Neil
Cornish, Scott Hughes, Carlos Sopuerta, Takahiro Tanaka, Jon Gair, Paolo Pani, Antoine Klein, Kent Yagi, Laura Sampson, Leo Stein, Sarah Vigeland,
Katerina Chatziioannou, Philippe Jetzer, Leor Barack, Curt Cutler,
Kostas Glampedakis, Stanislav Babak, Ilya Mandel, Chao Li, Eliu Huerta, Chris Berry, Alberto Sesana, Carl Rodriguez, Georgios Lukes-Gerakopoulos, George Contopoulus, Chris van den Broeck, Walter del Pozzo, John Veitch,
Nathan Collins, Deirdre Shoemaker, Bangalore Sathyaprakash, Michalis Agathos, Tjonnie Li, Salvatore Vitale, Alberto Vecchio, Justin Alsing, Enrico
Barausse, Cliff Burgess, Michael Horbatsch, Saeed Mirshekari, Richard O’Shaughnessy, Hajime Sotani, Norbert Wex, etc.
Standing on the Shoulders of...
GW Tests of GR Yunes
10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100
ε=M/r10-1310-1210-1110-1010-910-810-710-610-510-410-310-210-1
ξ1/2 =(
M/r3 )1/
2 [km
-1]
Double Binary Pulsar
Lunar Laser Ranging
LIGO BH-BH Merger
Sun's SurfaceEarth's Surface
LISA IMBH-IMBH Merger
Perihelion Precession of Mercury
LIGO NS-NS Merger
IMRIs IMBH-SCO
LAGEOSLISA SMBH-SMBH Merger
EMRIs SMBH-SCO
Pulsar Timing Arrays Field Strength
Curvature Strength
GWs can probe the non-linear, dynamical, strong-field regime
Strong Field Tests
Weak Field Tests
Will, Liv. Rev., 2005, Psaltis, Liv. Rev., 2008, Siemens & Yunes, Liv. Rev. 2013.
Why should we test GR?
GW Tests of GR Yunes
gravitational wave template
symmetric mass ratio distance to
the sourceinclination
angletotal mass
orbital freq.
orbital phase
Gravitational Waves contain information about the system that generates them.
Learn About Astrophysics, Black holes, Neutron stars. Test General Relativity and search for GR Deviations.
Why should we use GWs?
h⇥(t) ⇠⌘M
DLcos ◆ (M!)2/3 cos 2�+ . . .
GW Tests of GR Yunes
I. Data Analysis !
II. Theoretical Analysis !
III. Parametrized post-Einsteinian Framework
Road Map
GW Tests of GR Yunes
C. Hanna, LSC/PI
signal-to-noise ratio
(SNR)
detector noise (spectral noise
density)
data
template (projection of GW metric perturbation)
template param that characterize system
Matched Filtering:
Maximize the likelihood
(SNR) over all template
parameters
How do we detect “things” ?
⇢2 ⇠Z
s(f)h(f,�µ)
Sn(f)df
GW Tests of GR Yunes
How do we test GR ?Compare 2 hypotheses (H0 and H1) by constructing a measure that determines
which one is better supported by the data
Bayes’ Theorem
posterior belief on H0 given data
prior belief on H0
marginalized likelihood of H0 over sys params.
Bayes’ Factor
If BF>>1, then H1 is much preferred over H0
H0 is nested in H1, with H0=H1 when w=0
prior
posterior
0
2
4
6
8
10
12
-0.4 -0.2 0 0.2 0.4t
BF = 1
0
2
4
6
8
10
12
-0.4 -0.2 0 0.2 0.4t
BF = 10
0
2
4
6
8
10
12
-0.4 -0.2 0 0.2 0.4t
BF = 0.25
0
2
4
6
8
10
12
-0.4 -0.2 0 0.2 0.4t
BF = 1
0
2
4
6
8
10
12
-0.4 -0.2 0 0.2 0.4t
BF = 10
0
2
4
6
8
10
12
-0.4 -0.2 0 0.2 0.4t
BF = 0.25
0
2
4
6
8
10
12
-0.4 -0.2 0 0.2 0.4t
BF = 1
0
2
4
6
8
10
12
-0.4 -0.2 0 0.2 0.4t
BF = 10
0
2
4
6
8
10
12
-0.4 -0.2 0 0.2 0.4t
BF = 0.25
GW Tests of GR Yunes
How do we control UnCeRtAiNtIeS?We DoN’t !Statistical Systematic
• Origin: noise fluctuations. • Origin: modeling or instrumental.
PN Astrophysics Non GR
• Measure: Fisher or Posterior Width
• Control: Increase SNR
Louder signal Lower Noise
• Measure: Fisher or Bayes’ Factor
• Control: ✴PN: Resum or higher PN order. ✴Astrophysics: Include Astro. ✴Non-GR: Allow for GR Deviations
• Effect: Same as statistical, but potential confusion problem!
• Effect: ✴Error in parameter estimation. ✴Loss of detection efficiency.
YunesGW Tests of GR
Template Family = GW (far away from source) as a function of time (or freq) and system parameters
How do you build a template family?
• Solve the field equations in the far-zone.
• Solve the field equations in the near-zone.
• Solve the (dissipative) GW back-reaction.
• Solve for wave propagation speed
• Construct time-domain response.
• Fourier transform time-domain response.
YunesGW Tests of GR
No!
Does this work for all sources?
80 100 120 140 160 180 200 220 240 260 280 300t/M
-2
-1
0
1
2
h+
InspiralMerger Ring down
Post-Newtonian
Num. Rel.BH Pert. Theory
20 mins later…
post-Newtonian Inspiral
Numerical Relativity Merger
BH Pert. Theory
Ringdown
(20 mins, 10,000 cycles) (secs, 30 cycles) (secs, 5 cycles)
NS/NS:
In Modified Gravity:
hard, too many theories, about 10 theories analyzed.
very hard, 2 theories analyzed.
same as inspiral
YunesGW Tests of GR
Which part is more important?Depends on the Source.
[Disclaimer: Talk is LIGO-centric, left out EMRIs]
GW Tests of GR Yunes
(i) Scalar-Tensor theories:
(iii) Gravitational Parity Violation:
because of dipolar energy emission
GW freq. inversely related to the BD coupling parameter
related to CS coupling
[Alexander, Finn & Yunes, PRD 78, 2008. Yunes, et al, PRD 82, 2010. Alexander and Yunes, Phys. Rept. 480, 2009]
[Will, PRD 50, 1994. Scharre & Will, PRD 65, 2002. Will & Yunes, CQG 21, 2004. Berti, et al. PRD 71, 2005. Alsing et al, 2011.]
(ii) Massive Graviton Theories:
related to graviton Compton wavelength
(iv) G(t) theories:
related to G variability
[Yunes, Pretorius, & Spergel, PRD 81, 2010.]
[Will, PRD 57, 1998, Will & Yunes, CQG 21, 2004 Stavridis & Will, PRD 80, 2009. Arun & Will, CQG 26, 2009.]
Alternative Theory Zoo
h = hGR ei �BD⌘2/5f�7/3
h = hGR ei �MG⌘0f�1
h = hGR
�1 + ↵PV ⌘
0f1�
h = hGR
⇣1 + ↵G⌘
3/5f�8/3⌘
ei �G⌘3/5f�13/3
GW Tests of GR Yunes
(v) Quadratic Gravitybecause it’s a higher curvature correction
related to theory couplings
[Yunes & Stein, PRD 83, 2011
We have still not found any theories whose predicted gravitational wave cannot be mapped to these.
(vi) Lorentz-Violating GW Propagation:[Mirshekari, Yunes & Will, PRD 85, 2012] related to degree of
Lorentz violation
Yagi, Stein, Yunes & Tanaka, accepted in PRD.]
Gravitational Waves in Alternative Theories
h = hGR ei �QG⌘�4/5f�1/3
h = hGR ei �LV ⌘0f↵�1
(vii) Shielded Theories[Damour & Esposito-Farese, Barausse, et al (spontaneous scalarization), Alsing, et al + Berti, et al (massive scalar)] scalar mass
YunesGW Tests of GR
Yunes & Pretorius, PRD 2009 Mirshekari, Yunes & Will, PRD 2012 Chatziioannou, Yunes & Cornish, PRD 2012
I. Parametrically deform the Hamiltonian.
II. Parametrically deform the RR force.
III. Deform waveform generation.
IV. Parametrically deform g propagation.
h = hGR (1 + �fa) ei�fb
Result: To leading PN order and leading GR deformation
Parameterized post-Einsteinian Framework
A = AGR + �A�AH,RR = ↵H,RRv
aH,RR
h = F+h+ + F⇥h⇥ + Fshs + . . .
E2g = p2gc
4 + ↵p↵g
h(f) = hGR(f) (1 + ↵fa) ei�fb
GW Tests of GR Yunes
Templates/Theories GR ppE
GR Business as usualQuantify the statistical significance that
the detected event is within GR. Anomalies?
Not GRQuantify fundamental bias
introduced by filtering non-GR events with GR templates
Can we measure deviations from GR characterized by non-GR signals?
Model Evidence.
Parameterized post-Einsteinian Framework
h = hGR (1 + �fa) ei�fb
All current efforts to test GR with GWs use some flavor of the ppE framework.
(see talks by van den Broeck, Li and Cornish later in this workshop)
YunesGW Tests of GR
Non GR injection, extracted with GR templates (blue) and ppE templates (red). GR template extraction is “wrong” by much more than the systematic
(statistical) error. “Fundamental Bias”
Non-GR Signal/GR Templates, SNR = 20
12
0.01
0.1
10 20 30 40 50
beta
unc
erta
inty
SNR
actual values1/SNR
FIG. 14: The scaling of the parameter estimation error inthe ppE parameter β for an aLIGO simulation with ppE pa-rameters (a,α, b,β) = (0, 0,−1.25, 0.1). The parameter errorsfollow the usual 1/SNR scaling.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.005 0.01 0.015 0.02 0.025 0.03
1
100
10000
1e+06
1.0
- FF
Baye
s Fa
ctor
β
Bayes FactorFitting Factor
FIG. 15: The log Bayes factors and (1 − FF) plotted as afunction of β for a ppE injection with parameters (a,α, b,β) =(0, 0,−1.25,β). The predicted link between the fitting factorand Bayes factor is clearly apparent.
the log Bayes factor is equal to
logB = χ2min/2 + ∆ logO
= (1− FF2)SNR2
2+ ∆ logO . (25)
Thus, up to the difference in the log Occam factors,∆ logO, the log Bayes factor should scale as 1−FF whenFF ∼ 1. This link is confirmed in Figure 15.
E. Parameter Biases
If we assume that nature is described by GR, but intruth another theory is correct, this will result in therecovery of the wrong parameters for the systems we arestudying. For instance, when looking at a signal thathas non-zero ppE phase parameters, a search using GRtemplates will return the incorrect mass parameters, asillustrated below.
2.8 2.82 2.84 2.86 2.88 2.9 2.92 2.94ln(M)
BF = 0.3β = 1
2.75 2.8 2.85 2.9 2.95ln(M)
BF = 5.6β = 5
2.65 2.7 2.75 2.8 2.85 2.9 2.95ln(M)
BF = 322β = 10
2.4 2.5 2.6 2.7 2.8 2.9ln(M)
BF = 3300β = 20
FIG. 16: Histograms showing the recovered log total massfor GR and ppE searches on ppE signals. As the source getsfurther from GR, the value for total mass recovered by theGR search moves away from the actual value.
Perhaps the most interesting point to be made withthis study is that the GR templates return values of thetotal mass that are completely outside the error rangeof the (correct) parameters returned by the ppE search,even before the signal is clearly discernible from GR. Werefer to this parameter biasing as ‘stealth bias’, as it isnot an effect that would be easy to detect, even if onewere looking for it.
This ‘stealth bias’ is also apparent when the ppE αparameter is non-zero. As one would expect, when a GRtemplate is used to search on a ppE signal that has non-zero amplitude corrections, the parameter that is mostaffected is the luminosity distance. We again see the biasof the recovered parameter becoming more apparent asthe signal differs more from GR. In this study, becausewe held the injected luminosity distance constant insteadof the injected SNR, the uncertainty in the recovered lu-minosity distance changes considerably between the dif-ferent systems. In both cases shown, however, the re-covered posterior distribution from the search using GRtemplates has zero weight at the correct value of lumi-nosity distance, even though the Bayes factor does notfavor the ppE model over GR.
V. CONCLUSION
The two main results of this study are that the ppEwaveforms can constrain higher order deviations from GR(terms involving higher powers of the orbital velocity)much more tightly than pulsar observations, and thatthe parameters recovered from using GR templates torecover the signals from an alternative theory of gravitycan be significantly biased, even in cases where it is notobvious that GR is not the correct theory of gravity. Wealso see that the detection efficiency of GR templates canbe seriously compromised if they are used to characterizedata that is not described by GR.
Cornish, Sampson, Yunes & Pretorius, 2011, Vallinseri & Yunes, 2013, Vitale & Del Pozzo, 2013.
What is Fundamental and Stealth Bias?12
0.01
0.1
10 20 30 40 50
beta
unc
erta
inty
SNR
actual values1/SNR
FIG. 14: The scaling of the parameter estimation error inthe ppE parameter β for an aLIGO simulation with ppE pa-rameters (a,α, b,β) = (0, 0,−1.25, 0.1). The parameter errorsfollow the usual 1/SNR scaling.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.005 0.01 0.015 0.02 0.025 0.03
1
100
10000
1e+06
1.0
- FF
Baye
s Fa
ctor
β
Bayes FactorFitting Factor
FIG. 15: The log Bayes factors and (1 − FF) plotted as afunction of β for a ppE injection with parameters (a,α, b,β) =(0, 0,−1.25,β). The predicted link between the fitting factorand Bayes factor is clearly apparent.
the log Bayes factor is equal to
logB = χ2min/2 + ∆ logO
= (1− FF2)SNR2
2+ ∆ logO . (25)
Thus, up to the difference in the log Occam factors,∆ logO, the log Bayes factor should scale as 1−FF whenFF ∼ 1. This link is confirmed in Figure 15.
E. Parameter Biases
If we assume that nature is described by GR, but intruth another theory is correct, this will result in therecovery of the wrong parameters for the systems we arestudying. For instance, when looking at a signal thathas non-zero ppE phase parameters, a search using GRtemplates will return the incorrect mass parameters, asillustrated below.
2.8 2.82 2.84 2.86 2.88 2.9 2.92 2.94ln(M)
BF = 0.3β = 1
2.75 2.8 2.85 2.9 2.95ln(M)
BF = 5.6β = 5
2.65 2.7 2.75 2.8 2.85 2.9 2.95ln(M)
BF = 322β = 10
2.4 2.5 2.6 2.7 2.8 2.9ln(M)
BF = 3300β = 20
FIG. 16: Histograms showing the recovered log total massfor GR and ppE searches on ppE signals. As the source getsfurther from GR, the value for total mass recovered by theGR search moves away from the actual value.
Perhaps the most interesting point to be made withthis study is that the GR templates return values of thetotal mass that are completely outside the error rangeof the (correct) parameters returned by the ppE search,even before the signal is clearly discernible from GR. Werefer to this parameter biasing as ‘stealth bias’, as it isnot an effect that would be easy to detect, even if onewere looking for it.
This ‘stealth bias’ is also apparent when the ppE αparameter is non-zero. As one would expect, when a GRtemplate is used to search on a ppE signal that has non-zero amplitude corrections, the parameter that is mostaffected is the luminosity distance. We again see the biasof the recovered parameter becoming more apparent asthe signal differs more from GR. In this study, becausewe held the injected luminosity distance constant insteadof the injected SNR, the uncertainty in the recovered lu-minosity distance changes considerably between the dif-ferent systems. In both cases shown, however, the re-covered posterior distribution from the search using GRtemplates has zero weight at the correct value of lumi-nosity distance, even though the Bayes factor does notfavor the ppE model over GR.
V. CONCLUSION
The two main results of this study are that the ppEwaveforms can constrain higher order deviations from GR(terms involving higher powers of the orbital velocity)much more tightly than pulsar observations, and thatthe parameters recovered from using GR templates torecover the signals from an alternative theory of gravitycan be significantly biased, even in cases where it is notobvious that GR is not the correct theory of gravity. Wealso see that the detection efficiency of GR templates canbe seriously compromised if they are used to characterizedata that is not described by GR.
12
0.01
0.1
10 20 30 40 50
beta
unc
erta
inty
SNR
actual values1/SNR
FIG. 14: The scaling of the parameter estimation error inthe ppE parameter β for an aLIGO simulation with ppE pa-rameters (a,α, b,β) = (0, 0,−1.25, 0.1). The parameter errorsfollow the usual 1/SNR scaling.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.005 0.01 0.015 0.02 0.025 0.03
1
100
10000
1e+06
1.0
- FF
Baye
s Fa
ctor
β
Bayes FactorFitting Factor
FIG. 15: The log Bayes factors and (1 − FF) plotted as afunction of β for a ppE injection with parameters (a,α, b,β) =(0, 0,−1.25,β). The predicted link between the fitting factorand Bayes factor is clearly apparent.
the log Bayes factor is equal to
logB = χ2min/2 + ∆ logO
= (1− FF2)SNR2
2+ ∆ logO . (25)
Thus, up to the difference in the log Occam factors,∆ logO, the log Bayes factor should scale as 1−FF whenFF ∼ 1. This link is confirmed in Figure 15.
E. Parameter Biases
If we assume that nature is described by GR, but intruth another theory is correct, this will result in therecovery of the wrong parameters for the systems we arestudying. For instance, when looking at a signal thathas non-zero ppE phase parameters, a search using GRtemplates will return the incorrect mass parameters, asillustrated below.
2.8 2.82 2.84 2.86 2.88 2.9 2.92 2.94ln(M)
BF = 0.3β = 1
2.75 2.8 2.85 2.9 2.95ln(M)
BF = 5.6β = 5
2.65 2.7 2.75 2.8 2.85 2.9 2.95ln(M)
BF = 322β = 10
2.4 2.5 2.6 2.7 2.8 2.9ln(M)
BF = 3300β = 20
FIG. 16: Histograms showing the recovered log total massfor GR and ppE searches on ppE signals. As the source getsfurther from GR, the value for total mass recovered by theGR search moves away from the actual value.
Perhaps the most interesting point to be made withthis study is that the GR templates return values of thetotal mass that are completely outside the error rangeof the (correct) parameters returned by the ppE search,even before the signal is clearly discernible from GR. Werefer to this parameter biasing as ‘stealth bias’, as it isnot an effect that would be easy to detect, even if onewere looking for it.
This ‘stealth bias’ is also apparent when the ppE αparameter is non-zero. As one would expect, when a GRtemplate is used to search on a ppE signal that has non-zero amplitude corrections, the parameter that is mostaffected is the luminosity distance. We again see the biasof the recovered parameter becoming more apparent asthe signal differs more from GR. In this study, becausewe held the injected luminosity distance constant insteadof the injected SNR, the uncertainty in the recovered lu-minosity distance changes considerably between the dif-ferent systems. In both cases shown, however, the re-covered posterior distribution from the search using GRtemplates has zero weight at the correct value of lumi-nosity distance, even though the Bayes factor does notfavor the ppE model over GR.
V. CONCLUSION
The two main results of this study are that the ppEwaveforms can constrain higher order deviations from GR(terms involving higher powers of the orbital velocity)much more tightly than pulsar observations, and thatthe parameters recovered from using GR templates torecover the signals from an alternative theory of gravitycan be significantly biased, even in cases where it is notobvious that GR is not the correct theory of gravity. Wealso see that the detection efficiency of GR templates canbe seriously compromised if they are used to characterizedata that is not described by GR.
YunesGW Tests of GR Sampson, 2013
What Happens if you Ignore Fundamental Bias?injection =
(not-ruled out) ppE template=GR
Fitting Factor Deteriorates
Physical Parameters Completely Biased
YunesGW Tests of GR
Strong FieldWeak Field
GR Signal/ppE Templates, 3-sigma constraints, SNR = 20
Yunes & Hughes, 2010, Cornish, Sampson, Yunes & Pretorius, 2011 Sampson, Cornish, Yunes 2013.
Newt 1PN 1.5 2 2.5 3 3.5
aLIGO projected bounds
Double Binary Pulsar bounds
Can we Constrain GR deviations or not?
h(f) = hGR(f) (1 + ↵fa) ei�fb
YunesGW Tests of GR
Sampson, Cornish & Yunes, 2013
Bayes Factor between a 1-parameter ppE template and a GR template (red) and between a 2-parameter ppE template and a GR template (blue), given a non-GR
injection with 3 phase deformations, as a function of the magnitude of the leading-order phase deformation.
Do we need more complicated ppE Models?
YunesGW Tests of GR Sampson, 2013
What about BH Coalescence Events?
0
0.5
1
1.5
2
2.5
3
27.5 28 28.5 29 29.5 30 30.5 31Mtotal (solar masses)
Stage I
0
0.5
1
1.5
2
2.5
3
27.5 28 28.5 29 29.5 30 30.5 31Mtotal (solar masses)
Stage II
0
1
2
3
4
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8`1.5PN
Stage I
0
1
2
3
4
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8`1.5PN
Stage II
Stage 1: use standard inspiral only search, using all the data!
Stage II: use inspiral analysis, stopping at 10M (with M from stage I)
0
0.5
1
1.5
2
2.5
3
27.5 28 28.5 29 29.5 30 30.5 31Mtotal (solar masses)
Stage I
0
0.5
1
1.5
2
2.5
3
27.5 28 28.5 29 29.5 30 30.5 31Mtotal (solar masses)
Stage II
0
1
2
3
4
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8`1.5PN
Stage I
0
1
2
3
4
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8`1.5PN
Stage II
YunesGW Tests of GR
Does the point-particle approximation hold for BH and NS binaries?
Is the non-linear and dynamical sector of the Einstein equations correct at astrophysical black hole horizon scales?
GW Observations of compact binary inspirals will provide unparalleled information about the gravitational interaction
in the dynamical, non-linear regime.
Do GWs have only two massless polarizations?
Doveryai, no proveryai
Gravitational waves will allow us to constrain deviations from General
Relativity in the “strong-field” to unparalleled levels.
What does it all mean?
hBD(f ;~�GR,�BD)
”◆0”
hD>4(f ;~�GR,�D>4)
hLV (f ;~�GR,�LV )
hppE(f ;~�GR,~�ppE)
1PN
2PN
3PN
4PN
-1PN
-2PN
-3PN
-4PN
0.5PN
1.5PN
2.5PN
3.5PN
-0.5PN
-1.5PN
-2.5PN
-3.5PN
0PNCurrent
ConstraintsGW
Constraints
GRBD MG
EDGB
CS
Gdot
LV
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