Black Holes and Compact Binaries in Alternative

Theories

Nicolas YunesAssistant Professor,

Montana State University

August 31st, 2011NR-HEP Workshop

Standing on the Shoulders of...Professors:Clifford Will, Jim Gates, David Spergel, Frans Pretorius, Neil Cornish, Stephon Alexander, Abhay Ashtekar, Sam Finn, Ben Owen, Bernd Bruegman, Carlos Sopuerta, Pablo Laguna, Emanuele Berti, Alessandra Buonanno.

Post-Docs:Victor Taveras, Bence Kocsis, Daniel Grumiller.

Graduate Students:Kent Yagi, Laura Sampson, Leo Stein, Sarah Vigeland.

Do Black Holes Exist?

MPI movie

There is a dense object at the center of our galaxy, but how do we know that it can be completely described by the Kerr metric?

What else could it be? Can we test the Kerr hypothesis?

Road Map

I. Black Holes in Alternative Theories

II. Compact Binaries in Alternative Theories

III. Gravitational Wave Tests of Alternative Theories

I. Black Holes in Alternative Theories

Black Holes in General Relativity

z

No-Hair TheoremsAll stationary, vacuum BH solutions to the Einstein equations have spherical horizon topology (Hawking)

All stationary, axisymmetric vacuum solutions to the Einstein equations are uniquely characterized by 2 parameters: M and S -> Kerr BHs are unique. (Carter)

All stationary, vacuum, BH solutions to the Einstein equations are static or axisymmetric (Hawking) All static, vacuum, spherically symmetric BH solutions to the Einstein equations are Schwarzschild (Birhoff theorem, Israel)

Black Holes need not be Kerr if:

(i) They are not in vacuum (eg. plasma, accretion disk).(ii) They are not stationary (eg. flows in disks).(iii) They are not solutions to the Einstein equations.

Black Holes in Alternative Theories

z

We will focus on stationary, vacuum solutions to modified gravity field equations in four-dimensions.

We will study theories with curvature expansions of the form:

Simplifications

We will search for “small” deformations away from GR solutions because we want (i) stable solutions, (ii) solutions that pass weak-field tests and (iii) analytic solutions.

Theta is a spacetime function and the alpha’s are coupling constants.

S = SGR + SAT + SKin

SGR =

∫d4x

√−g κ R SKin =

∫d4x

√−g β (∂aϑ)(∂aϑ)

SAT =

∫d4x

√−g(α1ϑR

2 + α2ϑRabRab + α3ϑRabcdR

abcd + α4ϑRabcd∗Rabcd

)

Black Holes in DCSG

z

valid to O(a^2/M^2) and to O(alpha_4)vvaalliidd ttoo OO((aa^̂22//MM^̂22)) aanndd ttoo OO((aallpphhaa 44))

This modified gravity deformation corrects the dragging of inertial frames (the gravitomagnetic sector of the metric only).

ds2 = ds2Kerr +2Ma

rsin2 θdt dφ

[5

16

(α24

βκM4

)M3

r3

(1 +

12

7

M

r+

27

10

M2

r2

)]

ϑ =5

8

α4

β

a

M

cos θ

r2

(1 +

2M

r+

18M2

5r2

)

Spherically symmetric metrics are not modified in DCSG (Birkhoff theorem holds) but non-spherically symmetric ones are, coupling to parity breaking terms.

(Yunes and Pretorius ’09)

A “hairy” solution in that it has a pseudo-scalar charge that acts as a magnetic dipole.

Black Holes in EDGB

Spherically symmetric metrics are modified, Birkhoff Theorem does not hold (neither do the no-hair theorems).

(Yunes and Stein ’10)

A “hairy” solution in that it has a scalar charge that acts as an electric monopole.

ϑ =α3

β

2

Mr

(1 +

M

r+

4M2

3r2

)

gtt = gSchwtt − 1

3

α23

βκ

M3

r3

(1 +

26M

r+ . . .

)

grr = gSchwrr − α2

3

βκ

(1− 2M

r

)−2M2

r2

(1 +

M

r+ . . .

)

Modified Gravity Bumpy Black Holes

Can we construct a metric that parametrically deviates from Kerr such that the previous metrics can be reproduced in some limit,

while also approximately preserving the Kerr Killing symmetries?

(Vigeland, Stein and Yunes ’10)

where hab is given by some complicated functions of coordinates and bump coefficients. gab satisfies the Killing equations.

gab = gKerrab + hab

The resulting metric is stationary and axially symmetric, but it does not solve the Einstein equations.

This metric is valid for small deformations away from Kerr (hab<<1)

With this, we can now study how EMRIs evolve in a parametrically deformed, bumpy spacetime.

II. Compact Binaries in Alt. Theories

Compact Binaries in GR

Gμν = 8πTμνgμν = ημν + hμν �ηhμν = τμν [h2]→

metric tensor

flatGW metric pert.

Einstein tensor Stress-Energy

tensorFlat-space,

diff. wave op.Annoying

non-linearities

∇μ (Gμν − 8πTμν) = 0

Metric Perturbation

Acceleration

Assumptions Perfect Fluid Tμν = (ρ+ p)uμuν + pgμν

Point Particles ρ = m δ3(xi − yi)

Four-Velocity PressureDensity

MassParticle Location

ε :=v

c=

√m

r� 1

orbital velocity

speed of light

total massorbital

separation

Solve Perturbatively

An Example: Equations of Motion

d

Leading term: Newtonian gravity.

1 PN (Relativity corrections)

2 PN (more corrections ... )

and craziness ensues...

d

[Blanchet 2006, Liv Rev Rel 9, 4, Eq. (168)]

3 PN (yet more corrections ... )

Impressed yet ... ?

Compact Binaries in Alt. Theories

d

and now you can use the same PN tools as always to solve the above wave equations (see eg. the DIRE approach or dim regularization).

Gμν + Cμν = 8πTμν

�ηhμν = τμν [h2] + σμν [h

2, ∂nh]

�ηδhμν = τμν [h2GR] + σμν [h

2GR, ∂

nhGR]

Start with the modified field equations

Linearize about Minkowski

Linearize about GR

But be careful!!

The point-particle description of BHs works in GR (in part due to the Birkhoff theorem), but this need not be so in Alternative Theories. In fact, usually one must compensate for violations of this description.

(Yagi, Stein, Yunes and Tanaka ’10)

Some Preliminary Results

d

Based on this, one expects the scalar fields to radiate like an electric-type monopole (-1PN) and like a magnetic-type dipole (2 PN).

This was confirmed through the methods described previously.

Going back to EDGB and DCSG, recall that the

scalar fields act like an electric monopole and a

magnetic dipole.

(Yagi, Stein, Yunes and Tanaka ’10)

ϑ =α3

β

2

Mr

(1 +

M

r+

4M2

3r2

)

ϑ =5

8

α4

β

a

M

cos θ

r2

(1 +

2M

r+

18M2

5r2

)

In both cases, this radiation modifies the rate of change of the energy-momentum lost by the binary, strongly impacting waves.

In the DCSG case, not including the spin-dependent background scalar field leads to a huge suppression of the modification.

III. Gravitational Wave Tests

Gravitational Waveform in GR

h×(t) ∝ μ

Rcos ι [MF (t)]2/3 cos 2φ + . . .×

22//33

gravitational wave

reduced mass distance to

the sourceinclination

angletotal mass

orbital freq.

orbital phase

Gravitational Waves contain information

about the system that generates them.

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

Parameterized post-Einsteinian Framework (ppE)

Promote the response function to a non-GR response, with parameters that control “well-motivated” deformations

ppE parameters

GR:

BD: (α, a, β, b) = (0, a, βBD,−7/3)

PV: (α, a, β, b) = (αCS , 1, 0, b)

(α, a, β, b) = (0, a, 0, b)

(Yunes & Pretorius ’09)

Extremely Simple Eg: Inspiral ppE template

Match filter with this new response function and let the data decide what these ppE parameters are.

You cannot test GR by assuming GR templates a priori

PPPromottte ttthhhe response fffunctttiiion ttto a non-GGGRRR response wiiittthhh

YYYYoooouuuu ccccaaaannnnnnnnooootttt tttteeeesssstttt GGGGRRRR bbbbyyyy aaaassssssssuuuummmmiiiinnnngggg GGGGRRRR tttteeeemmmmppppllllaaaatttteeeessss aaaa pppprrrriiiioooorrrriiii

h̃ = h̃GR (1 + α ηc ua) ei β ηdub

MMMaaatttccchhh fififilllttteeerrr wwwiiittthhh ttthhhiiisss nnneeewww rrreeessspppooonnnssseee fffuuunnnccctttiiiooonnn aaannnddd llleeettt ttthhheee dddaaatttaaa decide what these ppE parameters are.

Questions for ppEGiven a GW detection, how sure are we it was a GR event?

Statistically significant anomalies in the signal?

Can we test for deviations from/consistency with GR, without explicitly building templates banks for all conceivable theories?

How would we mischaracterize the universe if GR was close but not quite the correct theory of nature? (“fundamental bias”)

Templates/Theories GR ppE

GR

Not GR

Business as usual

Quantify the likelihood of GR being the underlying theory

describing the detected event, within the class of alt. theories

captured by ppE

Understand the bias that could be introduced filtering non-GR

events with GR templates

Measure deviations from GR characterized by non-GR ppE

parameters.

(Yunes and Pretorius ‘09)

Constraining Phase Deviations

(Cornish, Sampson, Yunes & Pretorius (2011)h̃ = h̃GR (1 + αfa) eiβfb

Strong FieldWeak Field

GR Signal/ppE Templates, 3-sigma constraints, SNR = 20

(Yunes and Hughes, PRD 82 (2010)

1e-08

1e-06

0.0001

0.01

1

100

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5

β’

b

PulsaraLIGO binary, mass ratio 2:1aLIGO binary, mass ratio 3:1

massive graviton, solar systemmassive graviton 1e-08

1e-06

0.0001

0.01

1

100

10000

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

β’

b

PulsarLISA binary, z = 1LISA binary, z = 3

massive graviton, solar systemPN bounds

massive graviton

Parameter Bias

Non GR injection, extracted with GR templates (dashed) and ppE templates (solid). GR template extraction is “wrong” by much more

than the systematic (statistical) error -> “Fundamental Bias”

Non-GR Signal/GR and ppE Templates, SNR = 20

(Cornish, Sampson, Yunes & Pretorius, 2011)

30 40 50 60 70 80 90 100 110DL (Gpc)

BF = 1 ppEGR

injected value

30 40 50 60 70 80 90 100 110DL(Gpc)

BF = 53ppEGR

injected value

2.75 2.8 2.85 2.9 2.95 3ln(M)

BF = 0.3β’ = 1

injected value

2.7 2.75 2.8 2.85 2.9 2.95 3ln(M)

BF = 5.6β’ = 5

injected value

2.65 2.7 2.75 2.8 2.85 2.9 2.95 3ln(M)

BF = 322β’ = 10

injected value

2.4 2.5 2.6 2.7 2.8 2.9 3 3.1ln(M)

BF = 3300β’ = 20

injected value

aLIGO LISA

a=0.5, z=7, alpha=(2.5, 3)

Conclusions

The full exploit of GW astrophysics will require the strong collaboration between relativists, astrophysicists, data analysts &

high-energy theorists. The future is around the corner!

Compact binary evolution is also modified in alternative theories of gravity, not only due to strong-field modifications to the individual

BHs, but also due to new dynamical effects.

Several analytic examples exist (EDGB, DCSG) for non-Kerr, “hairy” BH solutions in alternative theories.

Black holes need not be described by the Kerr metric in alternative theories of gravity, even if these are stationary and axisymmetric.

TTTThhhheeee ffffuuuullllllll eeeexxxxppppllllooooiiiitttt ooooffff GGGGWWWW aaaassssttttrrrroooopppphhhhyyyyssssiiiiccccssss wwwwiiiillllllll rrrreeeeqqqquuuuiiiirrrreeee tttthhhheeee ssssttttrrrroooonnnngggg collaboration between relativists, astrophysicists, data analysts &

high-energy theorists. The future is around the corner!

A new (ppE) waveform parameterization has been proposed to capture generic, model-independent GR deformations.

But What Theory Do We Pick?

It’s not easy to fool Mother Nature! (Wald)

A Minimal (?) Set of Criteria:

1. Weak-Field Consistency (existence and stability of physical solutions, satisfaction of precision tests).2. Strong-Field Inconsistency (deviations only where experiments cannot currently rule out modifications)

Other Nice Criteria:3. Well motivated from fundamental physics.4. Well-posed theory ?? This is hard to do...

IIIItttt’’’’ssss nnnnooootttt eeeeaaaassssyyyy ttttoooo ffffoooooooollll MMMMooootttthhhheeeerrrr NNNNaaaattttuuuurrrreeee!!!! ((((WWWWaaaalllldddd))))

A Proposed Recipe

(2) Go back to your data and study whether you have missed something or whether the data is consistent with GR:

(1) For low SNR sources, GWs are buried in noise. Construct Templates and extract via matched filtering, assuming GR is right. After all, Solar System/Binary Pulsar Tests have confirmed GR in the weak-field limit, so the early inspiral must be right.

If it is consistent If it is not consistent

Cross-Correlate with other

detectors to eliminate inst. and astroph. artifacts

Test GRPlace a constraint on how large Phase and Amplitude deviations

could be given uncertainties.

Characterize any Phase or Amplitude

deviation. Trace back to a specific

modification to GR.

Unavoidable Correlations

a = 0

b = 0

b = -5/3

Identifying GR Deviations

Filter an injected ppE signal (a,alpha,b,beta)=(a,0,-1.25,0.1) with a ppE template family. For a given beta’, integrate the posterior to

find the evidence or Bayes factor or odds Ratio.

Non-GR Signal/ppE Templates, aLIGO, SNR = 20

Eg. if beta’ = 0.175, there would be a

100:1 odds that GR is wrong.

(Cornish, Sampson, Yunes & Pretorius, 2011)

b beta

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

- F

F

Bay

es F

acto

r

β’

Bayes FactorFitting Factor

Identifying GR Deviations

Filter an injected ppE signal (b,beta,eta,beta’)=(-1.25,10.0,0.204,13.57) with ppE templates. The marginalized posterior for beta shows a preference away from GR.

Non-GR Signal/ppE Templates, aLIGO, SNR = 20

With a single detection, you

cannot break the eta and beta

degeneracy, so you can constrain beta’

(Cornish, Sampson, Yunes & Pretorius, 2011)

b beta-1.2503 -1.25 -1.2497

b 5 10 15 20 25 30

β

0.19 0.2 0.21 0.22

η 13.3 13.4 13.5 13.6 13.7 13.8

β’