1. Modified Gravity and Degravitation
ICHEP
July, 23rd 2010
Modified Gravity and Degravitation
Thank for the invitation, it’s a pleasure to be here. Today I talk about the 2 hierarchy problems that exist at the interface between particle physics and cosmology, more precisely when putting together the 3 standard forces with gravity and show how extra dimensions, and in particular 2 or more large extra dimensions provide very fruitful new directions to tackle them
Work with
Gia Dvali, Gregory Gabadadze, Justin Khoury, Stefan Hofmann, Oriol Pujolas, Michele Redi, Andrew Tolley
Claudia de Rham
Université de Genève

2. What is Dark Energy ?
Is it a Cosmological Constant ???

3. What is Dark Energy ? Is it a Cosmological Constant ???
 120 orders of magnitude discrepancy!

4. What is Dark Energy ? Is it a Cosmological Constant ??? OR
 120 orders of magnitude discrepancy! OR
Is Dark Energy is Dynamical ??
 new form of energy
eg. Quintessence,…
 or self-accelerating Universe
eg. F(R), DGP

5. What is Dark Energy ? Is it a Cosmological Constant ??? OR
 120 orders of magnitude discrepancy! OR
Is Dark Energy is Dynamical ??
 new form of energy
eg. Quintessence,…
 or self-accelerating Universe
eg. F(R), DGP
These degrees of freedom must be extremely light!

6. What is Dark Energy ? Is it a Cosmological Constant ???
 120 orders of magnitude discrepancy!
Is the cosmological constant small or does it have a small effect on the geometry

7. Degravitation
Can Gravity be modified at Large Distances such that the CC gravitates more weakly?
One naïve way to modify gravity is to promote the Newton’s constant GN to a high pass filter operator,
k: 4d momentum
L-2
Arkani-Hamed, Dimopoulos, Dvali &Gabadadze, ‘02
Dvali, Hofmann & Khoury, ‘07

8. Massive Gravity
Filtering gravity is effectively a theory of massive gravity
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like k2 m2

9. Massive Gravity
Filtering gravity is effectively a theory of massive gravity
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like k2 m2

10. Worries But our Universe is accelerating !!!
For pure vacuum energy, the metric remains flat
But our Universe is accelerating !!!
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like

11. Relaxation mechanism L H2 time
The degravitation mechanism is a causal process.
1/m H2
time
Phase transition L
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like

12. Relaxation mechanism L H2 time
The degravitation mechanism is a causal process.
1/m H2
time
Phase transition L
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like

13. Worries But our Universe is accelerating !!!
For pure vacuum energy, the metric remains flat
How does it help with the tuning issue?
But our Universe is accelerating !!!
The Universe keeps accelerating while relaxing towards the static solution
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like

14. Tuning / Fine-tuning
From naturalness considerations, we expect a vacuum energy of the order of the cutoff scale (Planck scale).
But observations tell us
For the degravitation mechanism to work, the mass of the graviton should be
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like

15. but technically natural
Tuning / Fine-tuning
The amount of tuning is the same
But the graviton mass remains stable against quantum corrections
we recover a symmetry in the limit m
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like
The theory is tuned
but technically natural
‘t Hooft naturalness argument

16. but technically natural
Worries
For a CC, the effective Newton Constant vanishes
How does it help with the tuning issue?
How many degrees of freedom is there ?
But our Universe is accelerating !!!
The Universe keeps accelerating while relaxing towards the static solution
Another worry, which is I think more serious is to do with the tuning issue which was the reason why we introduced this model in the first place.
The theory is tuned
but technically natural

17. Massive Gravity A massless spin-2 field in 4d, has 2 dof
A massive spin-2 field, has 5 dof
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like

18. Graviton mass To give the graviton a mass, include the interactions
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like

19. Graviton mass To give the graviton a mass, include the interactions
Mass for the fluctuations around flat space-time
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like
Tensor
Stückelberg field

20. Fierz-Pauli mass To give the graviton a mass, include the interactions
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like
Ghost-like contribution:
Disappear for the Fierz-Pauli choice:
This choice can be made to all orders !

21. Ghost-free decoupling limit
Keeping this procedure to all orders in the decoupling limit with the scale fixed, we get pl
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like

22. Ghost-free decoupling limit
Keeping this procedure to all orders in the decoupling limit with the scale fixed, we get with pl
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like

23. Properties Keeping this procedure to all orders,
The Bianchi identity requires
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like

24. Properties Keeping this procedure to all orders,
The Bianchi identity requires
Beyond 3rd order, all the transverse tensors at ith order in p vanish identically.
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like

25. Properties Keeping this procedure to all orders,
The Bianchi identity requires
Beyond 3rd order, all the transverse tensors at ith order in p vanish identically.
is at most 2nd order in time derivative !
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like

26. Properties Keeping this procedure to all orders,
The Bianchi identity requires
Beyond 3rd order, all the transverse tensors at ith order in p vanish identically.
is at most 2nd order in time derivative !
The linear and quadratic mixings can be removed by a local field redefinition
To see that, we can simply look at the linearized Einstein equation.
If I omit the tensor structure for a second, the Einstein equation looks like

27. The Galileon
For a stable theory of massive gravity, the decoupling limit is
The interactions have 3 special features:
They are local
They possess a Shift and a Galileon symmetry
They have a well-defined Cauchy problem (eom remain 2nd order)
Before looking at how these interactions affect observations, let me emphasize 3 very important characteristics of this model.
First of all the interactions are local, are derive from an action.
Second they have a specific symmetry which is a shift symmetry inheritated from 5d diff invariance,
And finally all the terms in the eom are at most (and actually exactly) 2nd order in derivative, which means that the system will have a well-defined cauchy problem and no ghost like instabilities.
Correspond to the Galileon family of interactions
Luty, Porrati, hep-th/
CdR, Gabadadze,
Nicolis, Rattazzi and Trincherini,

28. EFT and relevant operators
Higher derivative interactions are essential for the viability of this class of models.
Within the solar system, p reaches the scale L*, yet, we are still within the regime of validity of the theory
What is interesting to notice in this model is that we are working in a regime where the interactions are important, but quantum corrections are still well under control, and so the theory is not breaking down.
In particular the field can have a large velocity but the amplitude of the quantum corrections small and still remain under control. This is very important to be able to trust the theory at the scale \Lambda_\star.
And actually this is very similar to what happens in another model which is to due this time with the very early Universe.
Luty & Porrati, hep-th/
Nicolis & Rattazzi, hep-th/
Vainshtein, Phys. Lett. B 39 (1972) 393
Babichev, Deffayet & Ziour,  
CdR &Tolley,

29. EFT and relevant operators
Higher derivative interactions are essential for the viability of this class of models.
Within the solar system, p reaches the scale L*, yet, we are still within the regime of validity of the theory
What is interesting to notice in this model is that we are working in a regime where the interactions are important, but quantum corrections are still well under control, and so the theory is not breaking down.
In particular the field can have a large velocity but the amplitude of the quantum corrections small and still remain under control. This is very important to be able to trust the theory at the scale \Lambda_\star.
And actually this is very similar to what happens in another model which is to due this time with the very early Universe.
Luty & Porrati, hep-th/
Nicolis & Rattazzi, hep-th/
Vainshtein, Phys. Lett. B 39 (1972) 393
Babichev, Deffayet & Ziour,  
CdR &Tolley,

30. EFT and relevant operators
Higher derivative interactions are essential for the viability of this class of models.
Within the solar system, p reaches the scale L*, yet, we are still within the regime of validity of the theory
The breakdown of the EFT is not measured by but by itself gradients should be small
So we can trust a regime where as long as
What is interesting to notice in this model is that we are working in a regime where the interactions are important, but quantum corrections are still well under control, and so the theory is not breaking down.
In particular the field can have a large velocity but the amplitude of the quantum corrections small and still remain under control. This is very important to be able to trust the theory at the scale \Lambda_\star.
And actually this is very similar to what happens in another model which is to due this time with the very early Universe.
Luty & Porrati, hep-th/
Nicolis & Rattazzi, hep-th/
CdR &Tolley,

31. Dirac Born Infeld
One of the most attractive model of inflation is provided by the DBI action
Which describes the dynamics of a probe-brane in a extra dimension
In particular, one of the most attractive models of inflation comes from the DBI action, which describes the dynamics of a probe brane in an extra dimension. Where the parameter pi is associated to the brane position along the extra dimension, and we see that if the brane is moving relativistically, we should introduce the Lorentz factor here.
Kabat and Lifschytz, hep-th/

32. DBI - Galileon DBI is similar to the Galileon in that
It relies on higher derivative interactions,
While keeping quantum corrections under control
It exhibits a 5d Poincaré or AdS symmetry
It has a well-defined Cauchy problem
This DBI model which is a priori completely disconnected from the Galileon, shares a lot of commun features with the Galileon.
The first one is that higher derivative interactions are important, but quantum corrections are still kept under control.
It also has a surprising underlying symmetry which it inheritated from the 5d underlying theory.
And most importantly, the equations of motion remain 2nd order in derivative, so the system has a well-defined Cauchy problem, and does not have any ghost like instability.

33. Cosmological Puzzles Current Universe Early Universe
Massive gravity is one of the only model tackling the cosmological constant problem
The DBI brane model provides an attractive realization of inflation
Both models rely on specific higher derivative interactions that remain under control at the quantum level
They have non-linearly realized symmetries and well-defined Cauchy problem
Both these models rely on the existence of higher derivative interactions which remain under control at the quantum level,
And have a non-lilearly realized symmetry, inheritated from the extra dimension.

34. They can be seen as 2 limits of the same underlying theory
Cosmological Puzzles
Current Universe
Early Universe
Massive gravity is one of the only model tackling the cosmological constant problem
The DBI brane model provides an attractive realization of inflation
And what I have shown is that both these models are actually simply 2 different limits of the same underlying theory.
They can be seen as 2 limits of the same underlying theory

35. Observational Signatures
Such models lead to specific observational signatures Advance of the perihelion (LLR) Structure formation Lyman-a forest (excess of power) - CMB (excess power at short scales) - large bulk flows in velocity surveys kinetic Sunyaev-Zeldovich ISW cross-correlation (larger effect)
Due to extra scalar field dof
Due to modified Friedman eq.
This model is also very rich in observational signatures.
One of first difference from standard GR is that there extra scalar dof. they are strongly coupled close to a massive source so they lead to very weak signatures within the solar system, but the advance of the perihelion of the moon for instance around the earth is measured with such an accuracy that they are just on the edge of being observable at LLR experiments. these extra scalar modes will also lead to some effects in structure formation.
Another source of signature comes from the fact that the Friedman eq. is modified and therefore can lead to different observational signatures which are also just on the edge of being observable, some of which have even been observed within 2 sigma.
Lue, Scoccimarro & Starkman, ’04
Khoury & Wyman, ’09
Lue & Starkman, ’04
Chan & Scoccimarro, ‘09
Lue, ’05
Bognat, CdR, Wyman, to appear
Afshordi, Geshnizjani & Khoury, ’08
Dutta & CdR, in progress
Scoccimarro, ’09