1. Dark Energy and Modified Gravity
Philippe Brax, IphT Saclay, France
SPhN January 2011

2. Outline
1 – Modified gravity and Dark Energy
2- Large Scale Structures

4. The Big Puzzle

5. Acceleration of the Universe
Observations interpreted assuming that the theory of gravity is General Relativity, described in a Lagrangian way:
This action depends on the Ricci curvature R and Newton’s constant. Matter couples in a minimal way:

6. The Universe is homogeneous and isotropic on large scales?
CMB ?

7. Evidence: The Hubble Diagram
The explosion of high red-shift SN Ia (standard candles):
Within General Relativity, link to matter and dark energy
Dark Energy must exist!

8. The Cosmic Microwave Background
Fluctuations of the CMB temperature across the sky lead to acoustic peaks and troughs, snapshot of the plasma oscillations at the last scattering surface when the universe became transparent
The position of the first peak:
The universe is spatially flat
WMAP data

9. Underlying hypothesis: the validity of General Relativity.
What does this mean?

10. Dark Energy Really?
In fact we are not absolutely certain that the acceleration of the universe is due to dark energy. On the contrary, the acceleration of the expansion of the universe may be interpreted in four different ways:
1) The acceleration is entirely due to the presence of a constant vacuum energy (cosmological constant).

11. Nernst and Pauli first worried about gravity and vacuum energy, later revived by Zeldovich. From an effective field theory point of view, the vacuum energy is a relevant quantity receiving two types of different contributions:
Vacuum energy at very high energy, maybe in a UV completion (string theory ?)
Decoupling of all the species which have been « integrated out »
We observe the acceleration of the universe in a regime where E is less than the electron mass. We know that the decoupling of all the electroweak and QCD effect leads to a huge contribution. Somehow the high energy theory in the UV must know about all these low energy effects in the IR:
HOW?

12. In fact, the notion of vacuum is ambiguous in field theory in the presence of accelerated frames (Rindler…) and/or curvature effects:
We are just lucky enough to be in a situation where gravitational fields in galaxy clusters are small and the expansion rate of the universe is rather slow: we can get away with field theory in an almost Minkowskian background.
We know that the QCD vacuum gravitates and that quantum vacuum fluctuations have physical effects (Casimir). It could be argued that vacuum properties have always been associated and probed with matter. What about the quantum vacuum per se? Does it really gravitate?

13. Weinberg’s theorem
It would be natural if the small vacuum energy were due to a slightly broken symmetry. A good candidate is scale invariance:
Assuming that the vacuum is determined by the minimum of the potential of a scalar field theory. Scale invariance selects:
Vacua are determined by N equations for the (N-1) z fields: only compatible if there is a relation between the couplings.
The field is not determined, i.e. a flat direction of vacua connected to the origin in field space. Along this direction fermions get a mass. Under renormalisation, the couplings become scale dependent and in general the relation between them does not hold anymore: the non trivial vacua are lifted. Only remains the trivial vacuum which does not give masses to fermions: excluded.
Non trivial relation between couplings requires a new symmetry

14. Supersymmetry
The supersymmetry algebra implies that the vacuum has vanishing energy
as long the vacuum is supersymmetric:
Supersymmetry has to be broken. Moreover, supergravity does not guarantee that the unbroken vacuum has vanishing energy!
The amount of supersymmetry breaking has to be tuned to lift this negative value to zero.

15. Dark Energy Really?
In fact we are not absolutely certain that the acceleration of the universe is due to dark energy. On the contrary, the acceleration of the expansion of the universe may be interpreted in four different ways:
1) The acceleration is entirely due to the presence of a constant vacuum energy (cosmological constant).
2) The acceleration results from the existence of a new type of matter: dark energy.

16. Dark Energy Planck scale now
Dark energy « forgets » the vacuum energy problem and concentrates on the small energy density required to generate the acceleration of the universe…
Planck scale now
Field rolling down a runaway potential, reaching large values now (typically Planck scale)
Almost Flat potential for an almost decoupled field.

17. How Flat? very fast roll slow roll cosmological constant
Energy density and pressure:
very fast roll
slow roll cosmological constant
gentle roll dark energy

18. The problem is that even if you discard the vacuum energy problem, scalar field models face serious difficulties when coupled to matter. Indeed if a scalar couples to fermions with a large mass M:
This correction is too large for standard model particles unless the coupling constant satisfies:
Really unnatural value of the coupling unless justified by a symmetry.

19. Pseudo-Goldstone Bosons
Imagine that a U(1) global symmetry is broken, leaving a massless field in the spectrum (a Goldstone boson).
There is a residual shift symmetry at energies below the symmetry breaking scale corresponding to the original U(1) symmetry which forbids a direct coupling to fermions:
A non-trivial potential can be generated if the global symmetry is slightly broken
This is a thawing model of dark energy: the field stays at the maximum due to Hubble friction and eventually rolls down. Imposing acceleration and a mass of the order of the Hubble rate now:

20. Derivative couplings to matter are present. To lowest order:
The mass correction is very small:
These models may be embedded in proper particle physics. Need some grand unification scale at least where a symmetry is broken and a link between the the explicit breaking parameter and the neutrino mass scale.

21. Dark Energy Really?
In fact we are not absolutely certain that the acceleration of the universe is due to dark energy. On the contrary, the acceleration of the expansion of the universe may be interpreted in four different ways:
1) The acceleration is entirely due to the presence of a constant vacuum energy (cosmological constant).
2) The acceleration results from the existence of a new type of matter: dark energy.
3) What is seen as acceleration is in fact a misinterpretation of data and really we must face a modification of gravity at large enough scales.

22. Dark Energy Really?
In fact we are not absolutely certain that the acceleration of the universe is due to dark energy. On the contrary, the acceleration of the expansion of the universe may be interpreted in four different ways:
1) The acceleration is entirely due to the presence of a constant vacuum energy (cosmological constant).
2) The acceleration results from the existence of a new type of matter: dark energy.
3) What is seen as acceleration is in fact a misinterpretation of data and really we must face a modification of gravity at large enough scales.
4) There is no real acceleration. We just live in a void surrounded by more matter. No copernican principle stands.

23. New Scales in Physics
Mass of the scalar field on cosmological scales. Fifth force.
Dark energy scale
The dark energy scale is tantalizingly close to the neutrino mass scale and the scale at which gravity has been tested…

24. Deviations from Newton’s law are parametrised by:
For fields of zero mass or the order of
the Hubble rate now, the tightest constraint
on β comes from the Cassini probe measuring
the Shapiro effect (time delay):

25. Three known mechanisms can accomodate usual gravity locally with deviations on larger scales:
i) The Vainshtein mechanism in the case of DGP gravity (and similar models like the Galileon). In this case, gravity is like GR locally, like a scalar-tensor theory at larger scales (and 5d at very large scales).
ii) The chameleon mechanism for scalar-tensor theories and f(R) models. Gravity is locally like GR, deviates from GR at intermediate distances and is like GR far away.
iii)The Damour-Polyakov mechanism (dilaton at strong coupling) or symmetron. Gravity is like GR locally and modified at all scales further away.

26. Modifying Gravity Modifying gravity is particularly difficult!
The simplest modification is massive gravity (Pauli-Fierz):
Pauli-Fierz gravity is ghost free (negative kinetic energy terms) . Unfortunately, a massive graviton carries 5 polarisations when a massless one has only two polarisations.
The massless limit does not gives GR! (van Dam-Velman-Zakharov discontinuity). The extra polarization is lethal.
It has been argued that it becomes non-linear at short distance and cured by the Vainshtein mechanism.

27. Despite the Vainshtein mechanism which prevents strong deviations from Newton’s law in the solar system, massive gravity has a ghost in curved backgrounds (cosmology).
An infinite class of modified gravity models can be considered:
These Lagrangian field theories fall within the category of higher derivative theories.
Ostrosgradski’s theorem states that these theories are generically plagued with ghosts. Quantum mechanically, this implies an explosive behaviour with particles popping out of the vacuum continuously. In particular an excess in the gamma ray background.

28. A large class is ghost-free though, the f(R) models:
Unfortunately …

29. f(R) theories
f(R) totally equivalent to an effective field theory with gravity and scalars
The potential V is directly related to f(R).
Same problems as dark energy: cosmological constant value + corrections to the mass.
Let us assume that the potential is the effective potential obtained after integration overall
the high energy phenomena and valid at low energy. There is still a problem:
a large coupling to matter!
Not so terrible…

30. f(R) models:

31. When coupled to matter, scalar fields have a matter dependent effective potential
Environment dependent minimum
Chameleon effect

32. The force mediated by the scalar is:
The field outside a compact body of radius R interpolates between the minimum inside and outside the body
Inside the solution is nearly constant up to the boundary of the object and jumps over a thin shell
Outside the field is given by:

33. The force on a test particle outside a spherical body is shielded:
When the shell is thin, the deviation from Newtonian gravity is small.
The size of the thin-shell is:
Small for large bodies (sun etc..) when Newton’s potential at the surface of the body is large enough.

34. F(R) Cosmology Electron kick during BBN Late time acceleration
Lurking cosmological constant

35. At the background level, these models behave like a pure cosmological constant.
Fortunately, this is not the case at the perturbation level where the growth factor evolves like:
The new factor in the brackets is due to a modification of gravity depending on the comoving scale k.
This is equivalent to a scale dependent Newton constant.

36. General Relativity Modified gravity
In all these models, the effective range of the fifth force is less than the Hubble size as m>> H.
Inside the Compton wavelength of the scalar field, structures grow at a modified rate.
General Relativity Modified gravity
z=z*

37. Everything depends on the comoving Compton length:
Determining this scale in definite models is crucial
Gravity acts in an usual way for scales larger than the Compton length
Gravity is modified inside the Compton length with a growth:

38. The Dilaton
Another way of evading gravity tests has been advocated in a string context by Damour and Polyakov: a field dependent coupling.
As an example the string dilaton couples to matter in an universal way and would lead to the existence of a fifth force.
This could be avoided in the strong string coupling limit if the coupling function had a minimum:
Naturally, cosmology would drive the dilaton to this minimum where the coupling to matter vanishes:
Minimum of the effective potential

39. Gasperini, Piazza and Veneziano advocated that in the strong coupling limit of string theory, the dilaton has a runaway potential:
This can play the role of dark energy for large values of the field. Still the value of the coupling must be small to accomodate gravity tests!
In the following, we will combine the Damour-Polyakov coupling and the exponential coupling.

40. The dilaton is not canonically normalised so the coupling to gravity:
Cosmologically, the dilaton is attracted towards the minimum of the effective potential for which:
The coupling to matter is suppressed in dense environments

41. Locally, in the presence of an overdensity, the coupling is suppressed compared to the coupling in the cosmological background.
Imposing that local tests are satisfied amounts to a bound on the magnitude of the coupling. This bound is only verified provided the variation of the coupling is bounded by Newton’s potential:
More stringent constraints are obtained by numerically integrating the Klein-Gordon equation in a Navarro-Frenkel-White profile.

42. Typically we find that:

43. The local tests imply the existence of an upper bound on the range of the force mediated by the dilaton. On cosmological scales, we find that the range (the inverse mass) is bounded:
This is typically the size of galaxy clusters, hence the possibility of seeing effects of the dilaton on structure formation.

44. An indicator of modified gravity can be obtained by comparing the weak lensing measurements to the peculiar velocities of galaxies :
In General Relativity, this ratio is:

45. One can go beyond linear perturbation theory and study non-Gaussianities such as the bi-spectrum.
At the level of second order perturbation theory:
The bi-spectrum can be expressed as:
Modified gravity alters the shape of the bi-spectrum.

46. The late time structure of the solution:
In GR and the Einstein-de Sitter limit:
The effect of modified gravity when ε is constant:

47. Bispectrum
General Relativity
Modified gravity.
ε=1
With F. Bernardeau, to appear
Bispectrum for equilateral configurations (i.e. normalised three point function) at second order in perturbation theory (non-Gaussianity).

48. Squeezed limit at the transition kc
Result without the extra couplings.
Full result. Deviation of a few per cent
As a rule, always deviations from GR of order of a few per cent.

49. Numerical Simulation
Modification of the N-body code MLAPM. Simulations of sizes 32 and 64 hˉ¹Mpc.
Simulation of 4 different models with (A2,λ)=( ,2) (400000,10) (200000,100) ( ,30) on the verge of the models allowed by local tests.
The particle number is 256x256x256. The domain grid is 128x128x128 and the finest refined grids have cells on each side corresponding to 6 hˉ¹kpc resolution.
The dilaton models are strongly constrained by local gravitational tests. Large scale deviations from GR would be at the (few) percent level.

50. Numerical simulation of the coupling β

51. The power spectrum

52. Number of halos

53. Conclusions
Acceleration of the Universe may be due to Dark Energy or Modified Gravity
Pseudo-Goldstone bosons are good DE candidates
Gravity may be altered in models of cosmic acceleration.
Locally, Newtonian gravity must be restored.
Effects on large scale structures: exciting prospects.

54. The dilaton string frame action:
In the Einstein frame:

55. The coupling constants:
The strong string coupling limit: