1. Pedro G. Ferreira University of Oxford
On Modifying Gravity
Pedro G. Ferreira
University of Oxford

2. Collaborators: Celine Boehm (Annecy) D. Mota (Heidelberg)
C. Skordis (Perimeter Institute)
G. Starkman (Case Western Reserve)
T. Zlosnik (Oxford)

3. Roadmap The problem with gravity (dark matter or modified gravity)
Bekenstein’s proposal for a covariant theory: TeVeS
Perturbations in TeVes
The time-like vector: the “Aether”
Conclusions

4. NGC 3198
Keplerian:
Begeman 1987

6. Density Perturbations
Three Regimes
Tight coupling
Recombination
Free Streaming

7. Dark Matter If

8. Adiabatic baryonic universe
Adiabatic CDM universe
Isocurvature baryonic universe

9. Alternative solution to rotation curve problem
Milgrom, 83
constant

10. Modified theory of gravity
Rewrite as:

11. Many successes but problems:
Clusters mass to light is too large
Dwarf Galaxies have large tidal radii
No Birkhoff theory (how do you do N-body?)
Silk damping?
Non-relativistic

12. How to build relativistic theory: basic idea
Dynamics
Gravity

13. Relativistic version Dynamics Gravity (geodesic equation)
(Einstein equations)
Common metric

14. “Bimetric” theories Use two different metrics “Physical” in
Geodesic equations
“Geometric” metric
in Einstein equations
Bekenstein, 04

15. A theory (TeVeS)
constants
Lagrange multipliers

16. How do we get modified gravity?
Physical potential Geometric potential
Scalar Field Equation
Lagrange multiplier
Low energy/non relativistic limit is MOND

17. Cosmology Two metrics: two scale factors … where: Physical Geometric
Skordis et al, 2006
Two metrics: two scale factors …
Physical Geometric
where:

18. Cosmological TrackingIf
then
Rad. era
Matter or
c.c. era
BBN:
Not Dark Matter

19. Perturbations: definitions
Scalar Field
Vector Field
(scalar modes)
Metric

20. Perturbations: evolution

21. Solution to vector Approximation: with where So growth of is enhanced.
Result: growth of and
Dodelson and Liguori, 2006

25. Need
to fit peaks

26. Main points:
Can generate large scale structure (in a flat universe, need neutrinos)
Vector field plays an essential role
may be smoking gun for these theories

27. “Simplify” this theory
Solve constraint:
Rewrite action solely in terms of physical metric and vector field
Zlosnik, Ferreira & Starkman, 2006

28. The Aether
Timelike vector

29. Equations of motion Simplest case:
Jacobson, Mattingley, Forster, Carroll, Lim ( )

30. How do we modify gravity?
Take non-relativistic, weak field limit:
Examples:
Rescale G
MONDian

31. Tests MONDian behaviour Solar system constraints
Causal, non-tachyon, subluminal propagation
Cosmological (FRW behaviour)
In Progress:
PPN calculation
Weak Lensing
Cosmological perturbations

32. Conclusions and comments
Vector-Tensor theory can have growing mode (TeVeS)
Perturbation in vector, , MUST be considered
Working out GENERAL features
Is the smoking gun?
Does not satisfy Birkhoff’s theorem
Severe tests it must pass: clusters and satellites

33. Stars
Globular Cluster
Dwarf Galaxy