Gravitational Wave Tests of General Relativityberti/TestGRTalks/Yunes.pdf · Gravitational Wave...

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...


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?


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�+ . . .


I. Data Analysis !

II. Theoretical Analysis !

III. Parametrized post-Einsteinian Framework

Road Map


Data Analysis


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


How do we estimate “things” ?Mapping the highest peak in Gallatin range


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


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.


Theoretical Analysis

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]


ppE Framework


(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


(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


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


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





= (1− FF2)SNR2

2+ ∆ logO . (25)


E. Parameter Biases


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




V. CONCLUSION


12

0.01

0.1

10 20 30 40 50

beta

unc

erta

inty

SNR

actual values1/SNR


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





= (1− FF2)SNR2

2+ ∆ logO . (25)


E. Parameter Biases


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




V. CONCLUSION


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


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


Stage I

0

0.5

1

1.5

2

2.5

3


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

YunesGW Tests of GR Vallisneri & Yunes, 2013

Stealth Bias

Fundamental Bias that we can’t detect!SNR needed to detect a

GR deviationSNR needed for fundamental bias error

to be larger than statistical error

Overt BiasNegligible Bias

Stealth Bias

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