1. DARK ENERGY PARTICLE PHYSICS PROBLEMS STEEN HANNESTAD UNIVERSITY OF AARHUS NBI, 27 AUGUST 2007

2. WHAT ARE THE PARTICLE PHYSICS PROBLEMS RELATED TO DARK ENERGY? A WELL-KNOWN PROBLEM IN COSMOLOGY IS THAT VACUUM ENERGY IS INFINITE, OR AT LEAST GIVEN BY SOME ULTRAVIOLET CUT-OFF IMPOSED ON THE THEORY FURTHERMORE IT IS PROPORTIONAL TO VOLUME, I.E. IT DOES NOT DILUTE WITH THE COSMOLOGICAL EXPANSION. IN PRINCIPLE THE VACUUM ENERGY PROBLEM COULD BE A QUESTION OF RENORMALIZATION, BUT THAT REQUIRES EXTREME FINE TUNING

3. THE FIELD THEORY VIEW OF DARK ENERGY Dark energy is associated with a very low energy scale Such a low energy scale is extremely hard to realise in realistic particle physics models of dark energy

4. Possible ”solutions” are: A slowly rolling scalar field – Quintessence (does not solve the underlying problem) Solutions from string theory – KKLT, holography (again not solutions strictly in field theory) Dark energy is not a field theory prolem Modified gravity – DGP, f(R), etc.

5. Problems in quintessence: In order to have w ~ -1 the scalar field mass must be extremely small (i.e. the potential is extremely flat) Why a new scalar field with characteristic energy scale 10 -3 eV? How can such a small mass be radiatively stable? Here are a few examples of models which at least seemingly avoid this problem

6. Coupling neutrinos to dark energy - An example of how to get around this fine tuning The neutrino mass scale is suspiciously close to the required quintessence energy scale There are models on the market in which neutrino masses are generated by the interaction with a new scalar field (Majoron type models) Normally the neutrino mass comes from the breaking of a U(1) symmetry at some high energy scale However, what if the breaking scale is put by hand to be very low?

7. We assume the existence of a new scalar field interacting with neutrinos The scalar field is not at its global minimum, but rolling in a potential In this case the neutrino mass becomes a dynamical quantity, calculable from the VEV of the scalar field -> Mass varying neutrinos (Fardon, Nelson, Weiner 2003)

8. The Lagrangian of the combined system is then given by where the neutrino mass is now given from the scalar field VEV Why is this good? The effective scalar field potential now has a contribution from the neutrino energy density

9. If chosen properly the additional neutrino contribution can provide a minimum of the effective scalar field potential This in turn means that it is possible to have while still maintaining the slow-roll condition

10. Only the non-relativistic regime is important for understanding the scenario We will assume that the field evolves adiabatically so that at any given point

13. IT IS POSSIBLE TO CONSTRUCT STABLE MODELS OF MASS VARYING NEUTRINOS. (O.E. BJAELDE ET AL. 2007) INTERESTINGLY, THEY WILL HAVE OBSERVABLE CONSEQUENCES IN THE FORM OF VARYING EOS ETC. IT IS POSSIBLE TO MAKE PHANTOM-LIKE DARK ENERGY BY THIS METHOD

15. Introduction Quintessence: Requires an extremely small scalar mass  Technically unnatural unless protected by some symmetry  Requires a new light scale  0.001eV << 1TeV where the symmetry is broken

16. Loop corrections will in general give large contributions to scalar masses  Λ 2  Symmetry needed! SUSY protects scalar masses, but is broken at a TeV  SUSY masses naturally of order TeV Shift symmetry φ -> φ + const. forbids a mass term for φ A symmetry breaking term Λ 4 f(φ/M) introduces a technically natural small mass Successful quintessence requires a new light mass scale Λ ~ 0.001eV of explicit sym. breaking Example: pNBG quintessence Frieman et al. 95

17. Example: Neutrino mass seesaw The neutrino mass matrix: Has a small neutrino mass eigenvalue: Different approach: Seesaw Is it possible to get a small quintessence mass from a scalar mass seesaw?

18. Problems with a scalar mass seesaw: The neutrino seesaw mass m 2 /M is never small enough if m > TeV ! No chiral symmetries to protect zero’s in diagonal However - for 8x8 matrices we do have matrices with: Mass eigenvalues m 5 /M 4 Only M’s in the diagonal  Consider a potential of the type: where m ij is 0, m or M

19. A CONCRETE EXAMPLE

20. The Matrix can be diagonalized, yielding where the a i ’s are all of order one. Negative eigenvalues  tachyonic instabilities (for Fermions these can be cured by chiral rotation) Adding Yukawa’s in tachyonic directions will in general also lift the mass of the light direction  we need to be more careful!

21. The light mass eigenstate  8 can be written in terms of the original interaction eigenstates: It turns out that  The light direction  8 has a very suppressed contribution of  2  We can lift the tachyonic directions by adding Yukawas in the  2 direction - without harming our light mass eigenstate

22. Since the m ‘s break all discrete symmetries, we may worry if we can protect the zero’s in the matrix from obtaining values of order m Brane configuration: Suppose each of the scalar fields are quasi-localized on each their brane with wave functions in the extra dimensions proportional to If the branes are on top of each other, one has bilinear mixing terms of the type However, if the branes are geographically separated in the extra dimensions, the overlap of the wavefunctions are exponentially suppressed  when the extra dimension is integrated out, the bilinear interactions of the effective theory are exponentially suppressed The Origin of the Matrix

23. Now, assume that all the elements (m 2 ) i j are given by M 2, The suppression of the bilinear interaction terms is given by M 2 exp(-Mr) with M  M GUT The branes can be taken to lie on top of each other in the directions where bilinear elements in the mass matrix of scale M are induced While the branes are separated by r  60/M in the directions where bilinear elements at the soft scale m  v are induced. In the directions where there are zeros in the off-diagonal, the branes are separated by a distance r >> 262/M.

24. Assume that there are three brane fixed points, A, B and C, in each dimension A and B are separated from each other by r  60/M C separated from A and B at a distance r >> 262/M, This leads to the following brane configuration for eight branes in six extra dimensions:

25. FINALLY, AN EXAMPLE OF A STRING INSPIRED MODEL: WHAT IS HOLOGRAPHY IN COSMOLOGY? A WELL-KNOWN PROBLEM IN COSMOLOGY IS THAT VACUUM ENERGY IS INFINITE, OR AT LEAST GIVEN BY SOME ULTRAVIOLET CUT-OFF IMPOSED ON THE THEORY FURTHERMORE IT IS PROPORTIONAL TO VOLUME, I.E. IT DOES NOT DILUTE WITH THE COSMOLOGICAL EXPANSION THE SECOND FACT COULD BE POINTING TO A SOLUTION TO THE PROBLEM

26. IDEA:SINCE WE KNOW THAT THE ENTROPY OF A BLACK HOLE IS PROPORTIONAL TO ITS AREA (BEKENSTEIN, HAWKING) IT COULD BE CONJECTURED THAT THERE IS AN UPPER BOUND TO THE ENTROPY IN A GIVEN VOLUME, GIVEN BY THE ENTROPY OF A BLACK HOLE OF THE SAME SIZE (BEKENSTEIN, ’THOOFT, SUSSKIND). THIS CONJECTURE IS KNOWN AS THE SPHERICAL ENTROPY BOUND

27. THIS CONJECTURE IMMEDIATELY COMES INTO CONFLICT WITH LOCAL FIELD THEORY. WHY? IN A GIVEN VOLUME THE ENTROPY IN Z FIELDS IS GIVEN ROUGHLY BY BY TAKING Z LARGE ENOUGH ONE CAN ALWAYS VIOLATE THE SPHERICAL ENTROPY BOUND. IN PRACTISE THIS IS HARD, BUT IN PRINCIPLE IT CAN BE DONE. WHERE IS THE PROBLEM? IN FIELD THEORY OR IN GRAVITATION?

28. THE HOLOGRAPHIC PRINCIPLE STATES THAT THE PROBLEM IS IN THE CONCEPT OF LOCAL FIELD THEORY. THE IDEA IS THAT FOR A REGION R IN D DIMENSIONS, ANY PHYSICAL PROCESS IN R CAN BE DESCRIBED BY PROCESSES ON THE D-1 DIMENSIONAL BOUNDARY. THIS IS COMPATIBLE WITH THE ENTROPY BOUND. SINCE ENTROPY CAN ONLY SCALE AS AREA, THE ANSWER FROM LOCAL FIELD THEORY MUST BE WRONG. THIS WAS FIRST POINTED OUT BY ’THOOFT IN 1993 AND CALLED DIMENSIONAL REDUCTION IN QUANTUM GRAVITY IN 1995 IT WAS PUT INTO THE SPHERICAL ENTROPY BOUND FORMULATION BY SUSSKIND

29. IN 2000 BOUSSO FORMULATED IT AS THE ”COVARIANT ENTROPY BOUND” WHICH CAN BE USED IN COSMOLOGY FOR FRW METRICS. ITS CONSEQUENCES ARE NOT IMMEDIATELY CLEAR, BUT THE PRINCIPLE CLEARLY STATES THAT THE EVOLUTION OF ANY FRW METRIC IS GIVEN BY ITS BOUNDARY CONDITIONS. IN THE COVARIANT ENTROPY BOUND THIS WOULD (PRESUMABLY) MEAN ITS FUTURE EVENT HORIZON. ONE POSSIBLE AND VERY INTERESTING CONSEQUENCE WOULD BE IF VACUUM ENERGY IS ALSO BOUNDED FROM ABOVE. THAT THIS SHOULD BE TRUE DOES NOT FOLLOW FROM THE ENTROPY BOUND BECAUSE VACUUM ENERGY IN ITSELF DOES NOT CONTAIN ENTROPY. HOWEVER (COHEN ET AL., LI, …) A HOLOGRAPHIC BOUND ON THE VACUUM ENERGY CAN BE CONJECTURED FROM THE ASSUMPTION THAT THE VACUUM ENERGY IN A REGION CANNOT BE LARGER THAN THE MASS IN THE SAME REGION

30. WHICH REGION IS THAT? THE FUTURE EVENT HORIZON IS THE MOST STRAIGHTFORWARD ANSWER (BUT OF COURSE NOT NECESSARILY TRUE). THAT LEADS TO SIMPLE PREDICTIONS FOR THE EQUATION OF STATE ETC (LI 2003). ANOTHER INTERESTING QUESTION IS COSMOLOGICAL PERTURBATIONS. IF PHYSICS IN A VOLUME IS GIVEN BY EFFECTS ON ITS BOUNDARY ONLY, THEN PRESUMABLY SUPER-HORIZON PERTURBATIONS CANNOT CONTRIBUTE TO THE EVOLUTION OF LOCAL FLUCTUATIONS. THIS LEADS TO A NATURAL CUT-OFF IN POWER AROUND THE PRESENT HORIZON (WHICH IS CLOSE TO THE FUTURE EVENT HORIZON), WHICH COULD BE COMPATIBLE WITH WMAP (ENQVIST, HANNESTAD, SLOTH 2005)

31. CONCLUSIONS: PLUS SIDE:MOST (BUT NOT ALL) OF THE MORE EXOTIC DARK ENERGY MODELS HAVE OBSERVABLE SIGNATURES DIFFERENT FROM LAMBDA MINUS:TOO MANY MODELS WITHOUT ADDITIONAL INPUT FROM THEORY OR OTHER EXPERIMENTS A TIME-VARYING EOS DOES NOT NECESSARILY POINT TO A SPECIFIC MODEL