Varying constants, cosmic structure and the quest for quantum gravity

1. Varying constants, cosmic structure and the quest for quantum gravity João Magueijo 2013 Imperial College, London La Sapienza, Roma

3. “Look what happens to people when they get married” (Niels Bohr)

4. Bekenstein (1982) By now some varying-constants have become run of the mill Brans-Dicke (1961) ETC… Potential damage to Lorentz invariance guaranteed

5. c(x,t) c(E) Varying c theories [JM, Rept. Prog. Phys. 66] • Covariant and Lorentz invariant [Moffat,Magueijo, etc, etc] • Bimetric theories [Moffat, Clayton, Drummond, etc, etc] • Preferred frame [Albrecht, Magueijo,Barrow,etc,etc] • Deformed dispersion relations [Amelino-Camelia, Mavromatos, Magueijo & Smolin, etc, etc]

6. The focus here will be on cosmology/early Universe

7. A non-inflationary solution to the horizon problem

8. Quantum Gravity and Cosmology: mismatched socks?

9. Quantum gravity and cosmology: will they ever meet? • The quantum gravity community is either: • obsessed with its mathematical navel and treats data and the real world as a venereal disease. • or, shows a distinct lack of sociological guts and tries to force contact with mainstream cosmology: i.e. inflation.

10. Unrequited love… • Inflation typically doesn’t need quantum gravity for anything. It already works wonders without it. • Quantum gravity would connect better to cosmology without appealing to inflation. • Via models better connected to it, e.g. VSL. • Directly!

11. The problem is not the horizon problem. Here’s the real problem:

12. The zero-th order “holy grail” of cosmology: • Near scale-invariance • Amplitude

13. c(x,t) c(E) Varying c theories [JM, Rept. Prog. Phys. 66] • Covariant and Lorentz invariant [Moffat,Magueijo, etc, etc] • Bimetric theories [Moffat, Clayton, Drummond, etc, etc] • Preferred frame [Albrecht, Magueijo,Barrow,etc,etc] • Deformed dispersion relations [Amelino-Camelia, Mavromatos, Magueijo & Smolin, etc, etc]

14. Bimetric theories A metric for gravity (Einstein frame): A metric for matter (matter frame):

15. This is a rather conservative thing to do… If the two metrics are conformal, we have a varying-G (Brans-Dicke) theory If they are disformal we have a VSL theory The speed of light differs from the speed of gravity (larger if B>0, with )

16. What sort of fluctuations come out of these theories? • If we project onto the Einstein frame, we end up with the same formalism usually used for inflation, but… • including a varying speed of sound. • This is the so-called K-inflation (an inflaton with non-quadratic kinetic terms).

17. How to compute fluctuations: I II

18. Why the horizon problem leads to a real problem: Dominates at late times If If (with )

19. How inflation solves the problem: Dominates earlier Dominates later With But why do we get scale invariance?

20. Follow up vacuum quantum fluctuations Consider first the regime With this normalization when we second quantize the amplitudes become creation/annihilation operators

21. A miracle happens near deSitter (w=-1) Compute the vacuum expectation value In the limit we get:

22. How a varying speed of light solves the problem: Dominates earlier Dominates later With but with we still get:

23. But could this lead to scale-invariance? Consider first the regime With this normalization when we second quantize the amplitudes become creation/annihilation operators

24. Can solve for a generic w and c_s Compute the vacuum expectation value take the limit and see when we get:

25. A remarkable result (!!!!!!!!!!!!!!) For ALL equations of state This scaling law for c seems to be uniquely associated with scale invariance.

26. Where does the amplitude come from? Obviously the variations in c must be cut off at low energies: The cut-off scale fixes the amplitude:

27. The minimal bimetric VSL theory C A subtlety with the variational calculus problem: The KG Lagrangian in the matter frame does NOT give the KG equation.

28. Something truly cool… C Gives a Klein-Gordon equation in matter frame

29. A cosmological constant in the matter frame leads to the (anti)DBI action C Specifically need a positive Lambda in the Einstein frame balanced by a negative lambda in the matter frame, to get the right low-energy limit: with f = –B < 0.

30. Apply the k-essence tool box to (anti)DBI to find that… C

31. So our remarkable result is even more remarkable • Not only is it possible to identify a universal varying speed of sound law associated with scale invariance… • but this law can be realized by an anti-DBI model (in the Einstein frame), which… • turns out to be the minimal dynamics associated with a bimetric VSL

32. Beyond the “zeroth order” holy grail • If the relation between the two metrics is then we obtain a tilted spectrum

33. Summary in terms of (if you really must!) C Standard inflation VSL DBI inflation

34. Is this then another “theory of anything”? No. A consistency relation. C

35. c(x,t) c(E) Varying c theories • Covariant and Lorentz invariant [Moffat,Magueijo, etc, etc] • Bimetric theories [Moffat, Clayton, Drummond, etc, etc] • Preferred frame [Albrecht, Magueijo,Barrow,etc,etc] • Deformed dispersion relations (DDRs) [Amelino-Camelia, Mavromatos, Magueijo & Smolin, etc, etc]

36. Mounting evidence for a 2D UV fixed point • Causal dynamical triangulations • Asymptotically safe Quantum Einstein Gravity • Horava-Lifschitz theory (z=3) • Space-time non-commutativity • Spin foams • Etc…

37. A bandwagon effect?...Or the answer to the quantum gravity puzzle.

38. Measures of the dimensionality of space-time: • Topological dimension (standard stuff we learn in analysis but don’t define properly) • Hausdorff dimension (balls of size R have volume R^D; allows for fractional D) • Spectral dimension… (a popular dimension in QG, because it’s “easy” to compute)

39. Running of spectral dimensions can be modelled with DDR • If we plug into the “spectral dimension” formula the DDR: • we get:

40. This dispersion relation is very interesting indeed:

41. What we did with bimetric VSL can also be done with DDR • Deformed dispersion relations can give a frequency dependent speed of light, e.g: • The speed of light/sound would then also vary in time, by proxy, via expansion:

42. We still obtain the same equation: … assuming the theory is represented as a HOD theory in which only spatial gradients are higher order than 2 … assuming Einstein gravity, for background and perturbations

43. The calculation follows through: Initial condition: Leads to the full solution

44. Again we find a universal law! Scale-invariance derives universally from Furthermore the right amplitude can be obtained with

45. Deviations from strict scale-invariance • It could be that gamma isn’t exactly 2, so that we run into a dimension slightly higher than 2

47. It could also be that we have a very slow transient into the UV • By slow, I mean really slow: • Then we can compute that

48. Or, • Or, restoring the w dependence:

49. How about gravitational waves? • In general they could have different DDRs to scalar modes. For example: • (and even if gamma is the same, b could be different from 1). • The UV ratio of the speed of light and the speed of gravity

50. The amplitude of the spectrum depends on this! • So: