Recent Progress in Testing Alternate Theories of Gravity

Jeremy Sakstein Institute of Cosmology and Gravitation, Portsmouth

Institute for Theoretical Physics University of Heidelberg

20th May 2015

If it ain’t broke, don’t fix itWhy modify gravity?

The universe is accelerating, and we don’t know why!

Can’t GR accelerate?GR can accelerate with a cosmological constant:

Get acceleration if eV4

So, what’s the problem?Naturalness!

E.g. electron decoupling

Okay, so what happens next?

Have to fine-tune a classical number against a quantum correction to 56 decimal places!

This is not natural

We have to do this for every particle:

Technical naturalness to the rescue

E.g. fermion masses are natural because of chiral symmetry

A small fermion mass is natural.

Symmetry restored when

How about gravity?Can we find a theory of gravity with a technically natural

vacuum?

• No naturalness problem

• No fine-tuning

• Self acceleration?

But wait? Ins’t gravity already constrained?

Yes and no!

Gravity tested in:

• Solar system - Newtonian and post-Newtonian

• Binary Pulsars - post-Newtonian

The fully relativistic structure has not been probed!

Local testsE.g. Cassini measures light bending by the Sun

“How much space is curved by a unit rest mass”

What do local tests mean?E.g. scalar-tensor theory - new scalar graviton

Cassini: Theory is GR on all scales

Screening mechanisms to the rescue

Non-linear effects decouple cosmological scales from the solar system

solar system astrophysics cosmology

screened partially screened unscreened

This talk

• Astrophysics only partially screened

• Identify novel probes in stars and galaxies

• Place new constraints

1. Vainshtein mechanism

This talk

• Can constrain cosmology using solar system tests

• Interesting cosmological phenomenology

• Still many unsolved mysteries

2. Disformal Gravity

The Vainshtein mechansimRecall the problem:

L = � 1

16⇡G@µ�@

µ�+ ↵�T

r2� = 8↵⇡G⇢

The Vainshtein mechanismTry to fix this by adding new kinetic terms

E.g. cubic galileon:

L = � 1

16⇡G@µ�@

µ�� 1

16⇡G⇤3@µ�@

µ⇤�+ ↵�T

r2�+1

⇤3r2d

dr

⇣r�02

⌘= 8↵⇡G⇢

Vainshtein MechanismWe can integrate this once:

- Vainshtein radius

Vainshtein Screening

Astrophysical Screening

Exhibited in:

• DGP - tension with data • Covariant galileons - too much ISW • Massive gravity - no FRW solutions • Massive bigravity - unstable (or is it?) • Beyond Horndeski - new and unexplored

Mechanism is partially broken in beyond Horndeski

Vainshtein Breaking

d�

dr=

GM(r)

r

d

dr=

GM(r)

rGR:

ds2 = � (1 + 2�) dt2 + (1� 2 ) �ij dxi dxj

Motion of NR matter Bending of Light

Vainshtein Breaking

d

dr=

GM(r)

r�5⌥G

4r

dM(r)

dr

d�

dr=

GM(r)

r+⌥G

4

d2M(r)

dr2 ⌥ =

�̇0

⇤

!4

Stars and satellites behave differently

BH:

Light bent differently

Cosmological field

Potential probes

• Stellar structure

• Galactic rotation curves

• Gravitational lensing

Stellar Structure TestsMain idea:

• Stars burn fuel to stave off gravitational collapse

• Changing gravity changes the burning rate

• This alters the temperature, luminosity and life time

Vainshtein Stars

Gravity weaker

Slower burning rate

Dimmer and cooler stars that live longer

Polytropic stars

Balls of gas that collapse under gravity - no physics

• No nuclear burning, convection etc.

• Can isolate new effects of MG

• Not realistic enough to compare with data

Mass-G-Luminosity relation

Gas pressure -

Radiation pressure -

L / G4M3

L / GM

High-mass stars are more radiation pressure-supported

Vainshtein Polytropes

20 40 60 80 100MM☉

0.2

0.4

0.6

0.8

LMGLGR ⌥ = 0.1

⌥ = 0.3

⌥ = 0.5

Koyama & JS 2015

Realistic starsWe have modified MESA to include MG:

• Fully consistent treatment of stellar structure

• No approximations

• Includes burning, convection, mass loss etc.

• Can compare with data

3.603.653.703.75Log Teff

0.0

0.5

1.0

1.5

2.0

Log L

⌥ = 0.5⌥ = 0.3

Dimmer + cooler on main-sequence

No change on red giant

Koyama & JS 2015

Vainshtein vs. GR

• Main-sequence cooler and dimmer

• No change to red giant phase

• MS degenerate with GR + more metals

May be detectable by comparing MS and RG fits to globular clusters.

Galactic rotation curves

Circular velocity:

d�

dr=

GM(r)

r+⌥G

4

d2M(r)

dr2

New features in rotation curves?

Galactic rotation curves

NFW density profile:

⇢(r) =⇢s

rrs

⇣1 + r

rs

⌘2

v2circ =4⇡Gr3s ⇢s

r

"ln

✓1 +

r

rs

◆�

⇣1 +

rsr

⌘�1+⌥

4

�rsr � 1

��1 + rs

r

�3

#

Vainshtein rotation curves

GR

⌥ = 1

⌥ = 0.5

Koyama & JS 2015

Vainshtein rotation curves

Deviations in 21 cm region compared with stellar prediction

Measure using 21 cm

Measure using stellar motions

Gravitational lensingLensing of light (relative to NR force ):

GR: �+ = 2�

�+ =8⇡Gr3s ⇢s

r

2

6664ln

✓1 +

r

rs

◆

| {z }2�GR

+⌥

8

�6 + rs

r

��1 + rs

r

�2

3

7775BH:

Vainshtein lensing

Deviations from GR in the strong lensing regime

GR

⌥ = 0.5

⌥ = 1

⌥ = 0.2

Koyama & JS 2015

Disformal Gravity

36

S =

Zd4x

p�g

M

2pl

2

R

2� 1

2rµ�rµ

�� V (�)

�+ Sm[g̃µ⌫ ]

Matter moves on geodesics of notg̃µ⌫ gµ⌫

) fifth-force

g̃µ⌫|{z}Jordan frame

= A2(�)| {z }conformal

2

664 gµ⌫|{z}Einstein frame

+B2(�)

⇤2

| {z }disformal

@µ�@⌫�

3

775

Beckenstein 1992

Potential Problem

37

Jordan frame metric can become singular:

“Natural pathology resistance”

Solutions slow down to avoid this, but no one knows why.

p�g̃p�g

= A4

s

1 +B2 (@�)2

⇤2= A4

s

1� �̇0

⇤2

Koivisto, Mota & Zumalacarregui 2012

Metric Singularity?

38

4.2 4.4 4.6 4.8 5.0N=ln a

20

40

60

80

100

1010- g̃

-g

-4 -2 2 4N=ln a

2×1094×1096×1098×1091×1010

- g̃

-g

Natural Pathology Resistance

39

Is this okay? No*, e.g. disformal only:

N2 = 1� �̇2

⇤2dT = Ndt

Lapse -> 0 but so what? Two speeds:c2tensors

c2light

= N2

Don’t know what the true non-relativistic limit is!

ds̃2 = �N2 dt2 + · · ·

Proper time -> 0

Non-relativistic limit

40

EOM is horrible:

�⇤�� 8⇡GB2

⇤2Tµ⌫m rµr⌫� =

� 8⇡↵GTm � 8⇡GB2

⇤2(↵� �)Tµ⌫

m @µ�@⌫�+ �V (�),�

� = 1 +B2(@�)2

⇤2↵ =

d lnA

d�� =

d lnB

d�

JS 2014

Non-relativistic limitBut we have calculated the PPN parameters:

g̃ij = (1 + 2�PPNU) �ij

Light bending

g̃0i = �1

2(· · ·+ ↵1 � ↵2 + · · · )Vi �

1

2(· · ·+ ↵2 + · · · )Wi

Preferred frameg̃00 = 1 + 2U � 2�U2 + · · ·

Orbital precession

Non-relativistic limit

g̃0i = �1

2(· · ·+ ↵1 � ↵2 + · · · )Vi �

1

2(· · ·+ ↵2 + · · · )Wi

�⌥ (GR = 0) �4⌥ (GR = 0)

g̃00 = 1 + 2U � 2�U2 + · · ·

1�⌥ (GR = 1)

g̃ij = (1 + 2�PPNU) �ij

1�⌥ (GR = 1)⌥ =

B�̇0

⇤

!2

ConstraintsStrongest constraint comes from ↵2

↵2 < 10�7

Simple model: V (�) / e���

�2

✓H0

⇤

◆2

< 4⇥ 10�7

B = 1,

Constraints

Ip, Schmidt & JS 2015

What does this mean?

�2

✓H0

⇤

◆2

< 4⇥ 10�7

Need ✓H0

⇤

◆2

⇠ O(1) for novel cosmology

Cosmology is identical to

Van de Bruck & Morrice 2015

⇤CDM

Cosmology

46

Friedmann equations same as GR

Field equations are coupled:

⇢̇+ 3H⇢ = Q0�̇1

�̈1 + 3H�̇1 + V 0(�1) = �Q0

Q0 = 8⇡G⇢↵+ B2

⇤2

⇣[� � ↵] �̇2

1 � 3H�̇1 � V,�

⌘

1 + B2

⇤2

⇣8⇡G⇢� �̇2

1

⌘

Dynamical Systems

47

-1.0 -0.5 0.0 0.5 1.0x

0.2

0.4

0.6

0.8

1.0

y

x =�

0p6

y =

pVp3H

Different initial conditions, common late-time behaviour

Why is this useful?

48

Classify the entire solution space

Know cosmological properties at the fixed points

Identify models that have the properties we want

Tells us which models to focus future efforts on

What we want

49

• Dark energy domination - match observations

• Finite metric determinant - well-defined NR limit

Fixed points with

Recap: Conformal Case

50

V (�) = m20e

��� A(�) = e↵� N = ln a

Phase space is 2-dimensional

x =�

0p6

y =

pVp3H

� = �V 0

V= constant

�̈1 + 3H�̇1 + V 0(�) = 8⇡↵G⇢

Conformal phase space

51

-1.0 -0.5 0.0 0.5 1.0x

0.2

0.4

0.6

0.8

1.0

y

Disformal System

52

What happens when we include a disformal coupling?

Old attractor is a saddle point - what’s going on?

-1.0 -0.5 0.0 0.5 1.0x

0.2

0.4

0.6

0.8

1.0y

Disformal System

53

x =�

0p6

y =

pVp3H

� = �V 0

V= constant

�̈1 + 3H�̇1 + V 0(�1) = �Q0

Q0 = 8⇡G⇢↵+ B2

⇤2

⇣[� � ↵] �̇2

1 � 3H�̇1 � V,�

⌘

1 + B2

⇤2

⇣8⇡G⇢� �̇2

1

⌘

Can’t eliminate in terms of andx

yB(�)

Phase space is 3D

z =BH

⇤=

Z

Z + 1B(�) = e��

0 Z 1 � constant

Stability is altered

54

Perturbations in the z - direction are unstable

(3 eigenvalues instead of 2, one can be positive)

New dark energy dominated fixed point atx = 0, y = 1, Z = 1

But: p

�g̃ = 0

Metric singularity at late times

Special Case

55

� = �/2 ) zy = const

New fixed point with interesting properties

Can always match any measurement without fine-tuning

BUT:

!DE = !DE

✓⇤

m0

◆⌦DE = ⌦DE

✓�,

⇤

m0

◆

p�g̃ = 0

What does this mean?

56

Only viable models are identical to conformal theories!*

* Possible loopholes are models whose dynamics are not described by this analysis

Summary

57

Vainshtein

• Vainshtein broken in beyond Horndeski

• Main-sequence stars are dimmer and cooler

• Circular velocity is lower

• Deviations from GR in strong lensing regime

Summary

58

Disformal

• Metric singularity not well-understood

• Solar system constraints give ΛCDM

• Cosmological solutions evolve towards singularity

• Only viable models have no disformal properties

• Still a lot to understand!

Thank you! (and to my collaborators)

Vainshtein

Kazuya Koyama (ICG)

Papers1502.06872

DisformalFabian Schmidt (MPA)

Hiu Yan Ip (MPA)Papers

15xx.xxxxx 1409.7296 1409.1734