1. Dark Energy Theories Philippe Brax IPhT Dark Day
SFP Champs et particules
25 November 2015

2. Comment le sait-on? Mesure de distances!
Dark Energy: anything which leads to the late time acceleration of the expansion of the Universe !
As for inflation, many candidates, very few plausible…

3. The acceleration of the Universe can be due to many causes:
A cosmological constant and/or vacuum energy
The cosmological dynamics of fields (new matter)
A modification of General Relativity (GR)
Crazoology

4. The acceleration of the Universe can be due to many causes:
A cosmological constant and/or vacuum energy
The cosmological dynamics of fields (new matter)
A modification of General Relativity (GR)
Crazoology

5. The acceleration of the Universe can be due to many causes:
A cosmological constant and/or vacuum energy
The cosmological dynamics of fields (new matter)
A modification of General Relativity (GR)
Crazoology

6. The acceleration of the Universe can be due to many causes:
A cosmological constant and/or vacuum energy
The cosmological dynamics of fields (new matter)
A modification of General Relativity (GR)
Crazoology
Causality…

7. The Cosmological Constant
The cosmological constant comprises two terms: a bare cosmological constant which can be seen as a boundary condition set initially at the Big Bang and the sum of all the vacuum fluctuations. Vacuum fluctuations couple to matter (the Casimir effect) but do they couple to gravity? If yes they entail a serious problem:
But what is the problem exactly?
In a Lorentz invariant and causal effective field theory, the description of the late acceleration must be captured by a (very) low energy effective theory at energies well below all the mass scales of the standard model

8. The cosmological constant and masses correspond to renormalisable (relevant) operators at low energy. Their numerical values is sensitive to UV physics and must be determined at a given scale by comparison with experiments. This is the usual renormalisation programme where “infinities” are absorbed by “infinite” counterterms, leaving a finite value which is determined experimentally. The bare value of the cosmological plays the role of such a counterterm.
In a top-down approach, every time a particle decouples, it contributes to the vacuum energy:
Phase transitions (EW breaking, QCD) also contribute.
The measurement of the dark energy now corresponds to the renormalisation condition at very low energy.

9. So what is the problem?
From a bottom-up approach, there is no problem… the dark energy is measured at low energy and one can “integrate in” all the particles of the standard model to take into account the effective vacuum energy in different eras of the Universe (before BBN, before the EW transition etc…) . In different eras, the effective cosmological constant is subdominant.
From a top-down approach, there is a hierarchy problem. Why should my “grand unified theory” at very large energy know (or care) that when all the particles of the models have decoupled the sum of the bare cosmological constant, defined by the model at high energy, and the quantum fluctuations down to the electron (neutrino) scale should be ALMOST zero?
This is a similar problem to the Higgs mass problem.
In the early Universe, if inflation takes place and then ends, this sets the initial value of the cosmological constant to vanish and imposes that the contributions from the decoupling of the particles of the theory vanish. Why?

10. Weinberg’s no-go theorem:
A generic 4d Quantum Field Theory cannot admit a Poincaré invariant vacuum with vanishing vacuum energy with non-vanishing fermion masses.
Global SUSY is a counter example.
One can violate some of the hypotheses to try to calculate the dark energy:
Use extra dimensions (the curvature of the extra dimensions absorbs the vacuum energy)
Use the dynamics of cosmological fields (quintessence)

11. A Simple Example: Pseudo-Goldstone Models
Global U(1) symmetry broken at scale f:
Symmetry broken by small term:
Low energy potential:
Cosmologically, if initially the field is small compared to f, it is frozen there until its mass becomes larger than the Hubble rate. Tuning this event to be in the recent past of the Universe and requiring that the dark energy is due to the low energy potential implies:

12. A major drawback of these models is that the phase transition must be at the Planck scale, this requires to embed this model in theories dealing with quantum gravity… maybe embedding it in string theory although no global symmetries there…
On the other hand, this model is useful to see that low energy dark energy fields couple to matter:
The derivative interaction implies no effect on static tests of gravity vs fifth forces. This an example of a disformal coupling to matter.

13. The Low Energy Dynamics
Only interested in the dynamics of the Universe below the electron mass, i.e. after BBN. One scale has to be fixed to generate the dark energy. At low energy, write down the most general action of one scalar field with second order ODE’s (Horndeski).
with 4 arbitrary functions of the field and the kinetic term:
The scalar field can couple to matter with the most general form respecting Lorentz invariance and causality:
Bekenstein
Conformal coupling
Disformal coupling

14. Disformal coupling: increases the burning rate of stars
On cosmological scales, the scalar field must be nearly massless in order to have an impact on the dynamics of the Universe. The two types of couplings have different consequences:
Disformal coupling: increases the burning rate of stars
Brax-Burrage
Conformal coupling: induces a fifth force modification of Newton’s law

15. Chameleon: screens in region of large enough Newtonian potential
Having both an effect on cosmological scales and preserving local physics requires a screening mechanism:
Chameleon: screens in region of large enough Newtonian potential
The mass of the scalar becomes density dependent:
The field generated from deep inside is Yukawa suppressed. Only a thin shell radiates outside the body. Hence suppressed scalar contribution to the fifth force.

16. Having both an effect on cosmological scales and preserving local physics requires a screening mechanism:
K-mouflage: screens in region where the gravitational acceleration is large enough
This happens inside a large radius surrounding a compact object, the K-mouflage radius

17. Having both an effect on cosmological scales and preserving local physics requires a screening mechanism:
Vainshtein: screens in region where the spatial curvature is large enough
This happens inside a large radius surrounding a compact object, the Vainshtein radius

18. How is the motion of astrophysical objects affected?
Trajectories due to gravity + scalar
Chameleons:
Unlike chameleons et al., K-mouflage and Vainshtein do not affect the charge Q:
Outside the Vainshtein-Kmouflage radius, the object feels the scalar force
outside the Vainshtein-Kmouflage radius

19. For Vainshtein, even clusters are screened.
These three types of theories affect the growth of large scale structure in different ways:
Chameleons:
Mpc
10 Mpc
Non-linear screening for large Newton potential .
linear screening outside the Compton wavelength of chameleon.
K-mouflage:
Mpc
10 Mpc
Non-linear screening inside K-mouflage radius .
No screening on cluster scales
For Vainshtein, even clusters are screened.

20. More precisely on linear scales, the growth is affected:
Chameleon:
Scale-time dependence
K-mouflage:
Time dependence only

21. Power spectrum
Brax-Valageas
Linear growth:
Number of clusters

22. Galileons Theories with the Vainshtein screening.
Non-renormalisation theorem and negligible finite corrections on large scales.
Self-accelerated solutions
The cubic Galileon:
Unfortunately, ghosts in Minkowski space: cannot be embedded in “healthy” UV completion.

23. K-mouflage K-mouflage screening.
Non-renormalisation theorem and negligible finite corrections on large scales.
The quadratic K-mouflage:
2-body scattering violates perturbative unitarity :
Such theories can be embedded in a UV completion with QFT axioms if:
No standard UV-completion

24. Massive Gravity
Mass term for a graviton
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. In the presence of matter, the graviton wave function takes the form:
The massless limit does not gives GR! (van Dam-Velman-Zakharov discontinuity). The extra polarization is lethal. Solved by Vainshtein mechanism (non-linearity).
Ghost in curved space-time!

25. Bimetric Gravity
One way to cure these problems is to consider a non-linear version of massive gravity with two dynamical metric:
where the graviton mass is of order
Only a linear combination of the vielbeins couple to matter:
Only the beginning… a lot more work to understand theoretically and then compare to data!

26. Crazoology (part one)
Models like K-mouflage seem to be low energy theories with no UV completion? Are they ruled out?
At high energy, unitarity is restored because the theory becomes non-local with the creation of finite size objects. It is said to classicalise (Dvali et al.)
This is the analogous to the creation of black holes at high energy when the “size” of the interaction region is smaller than the Schwarszchild radius. Here the same would happen when this “size” is smaller than the K-mouflage or Vainshtein radius.

27. Crazoology (part two)
The cosmological constant problem indicates a failure of the effective field theory description at low energy. Indeed, in a top-down approach, the value of the cosmological constant now in the IR depends both on the whole history of the Universe (the masses of all the particles which decouple one by one and all the phase transitions) and the UV value of the cosmological constant (the bare one).
This suggests that a theory which would explain its value would have a strong correlation between UV and IR physics. Recalling that UV=early Universe and IR=late Universe, this suggests a global breaking of causality.
Kaloper-Padilla
The function σ of the two fields Λ(x) and λ(x) is a global function. The dynamics of this theory are very close to GR:

28. The crucial difference is that the effective cosmological constant is determined globally:
This models requires that the Universe is spatially finite and that the future is not infinite. In fact the Universe must end in a big crunch singularity.
The effective value of the cosmological constant violates causality globally as it depends on the whole space-time, even our finite future.
The vacuum contribution, from loops etc…, does not gravitate!
where the effective cosmological constant depends on the average of the matter energy densities and is set by a future boundary condition:

29. Conclusions
The discovery of dark energy is a fantastic opportunity to reappraise all the tenets of our standard description of the Universe.
It raises deep questions about Quantum Field Theory: UV sensitivity, UV completion, causality …
It raises deep questions about gravity: modified gravity, massive gravity …
It is linked to exciting new experiments from laboratory tests to cosmology