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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!
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 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⇢
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
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
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
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,
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⇢
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!