Varying c cosmological fluctuations and the quest for

(Varying c), cosmological fluctuations and the quest for quantum gravity João Magueijo 2016 Imperial College, London

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

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

What we see…

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

The higher order “holy grail” n Very specific departures from scale invariance: n 3 -point function (non-Gaussianities) n Gravity waves (tensor modes)

The focus:

A critique of “evidence for inflation” Bayesian evidence is a tool for playing the lottery or investing in the stock market: hedging the bets works! n It favors non-predictive theories, because you get a power-law fine for spreading your bets, but an exponential one for making a prediction and missing the data. n

Science is about falsifiability A theory should not only have large evidence, but its evidence should be exceptional, given what it would have been had the data been different. n See ar. Xiv: 1506. 09143. n

The connection with quantum gravity Deformed dispersion relations (DDRs) are already a central tool in quantum gravity phenomenology. n They may be used to explain the CMB fluctuations directly. n There is a connection with the mounting evidence for a 2 D UV fixed point. n

Exact scale-invariance is associated with the “sweet” DDR:

The Big Bang horizon problem: If n (with Dominates at late times )

How inflation solves the problem: n With Dominates earlier Dominates later

How theories with a varying c solve the problem: n With with but we still get: Dominates earlier Dominates later

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

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

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

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

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

A cosmological constant in the matter C frame leads to the (anti)DBI action 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.

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… n including a varying speed of sound. n This is the so-called K-inflation (an inflaton with non-quadratic kinetic terms). n

Using C the K-essence toolbox Constant w solutions for mass potentials

A remarkable result: n For ALL equations of state This scaling law seems to be the equivalent of de Sitter space in bimetric models

Even more remarkable! This law can be realized by an anti-DBI model in the Einstein frame (DBI with wrong sign of the coupling), which… n turns out to be the minimal dynamics associated with a bimetric VSL (Klein-Gordon n equations in matter frame, constant B in relation between metrics)

An alternative derivation n This can be understood from the following recasting of the final result:

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

Do we go above Planck? Yes, we do. n We have a direct trans-Planckian problem and we embrace it. n Inflation, instead, was designed to insulate the observable Universe from quantum gravity. n

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

Is this then another “theory of C anything”? No! n No gravity waves… n but a possibility for a “consistency relation” is to look into the bispectrum (3 -point function):

A consistency relation between shape C of the 3 -point function and n_s

What about thermal fluctuations? (see Afshordi and Magueijo, 1603. 03312) n Implicit in all previous “power spectra” is the multiplicative factor: n But what if the state is a thermal state?

This changes the spectral index n In all regimes of interest n Therefore vacuum quantum fluctuations and thermal fluctuations have spectra related by: (i. e. we get white noise where usually scale-invariance in found)

What speed of sound profile would lead to thermal scale-invariance? n For ALL equations of state we find that we need a very fast phase transition in n Amplitude of the fluctuation is now fixed by the temperature at which the phase transition occurs:

Exact scale-invariance is unreachable, because the limit is discontinuous

The critical model has a running spectrum: And contains a unique prediction of n_s from the amplitude

We should get brownie points on this plot:

Also there is a string theory interpretation: n This solution has a simple geometrical interpretation based on the action of a probe 3 -brane embedded in an EAd. S 2 X E_3 geometry.

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

Conclusion: this dispersion relation is very interesting indeed! This is the DDR equivalent of de. Sitter spacetime for inflation n [Or of for bimetric models. ] n

BUT there are some further interesting robust connections: The dimensional reduction inference is made via UV spectral dimension of “spacetime” as computed with deformed dispersion relations n This is the same as the Hausdorff dimension of momentum space in units in which the DDRs are trivialized. n

Specifically: n We can always find “linearizing units” which trivialize DDRs. In our case: n This “dual picture” shifts the non-trivial effects elsewhere, e. g. the measure:

In the dual picture the gravity is no longer Einstein! The metric becomes “rainbow”, i. e. kdependent. n Gravity switches off (or rather, everything is conformally coupled) just as for radiation under Einstein gravity: n n What does this mean for QG? . . . .

Beyond this calculation, the following concepts seem closely related: Dimensional reduction to 2 dimensions in the UV n Scale-invariance of vacuum quantum fluctuations. n Gravity switching off; or everything coupling conformally to gravity (so that there is no inside/outside horizon). n

What about “the devil in the detail”? n Six sigma evidence that the spectrum is red. n Tensor modes… n Non-Gaussianities…

Gravitational waves/ tensor modes n 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). n The UV ratio of the speed of light and the speed of gravity

n The amplitude of the spectrum depends on this: n If you work out all the factors the result is:

Not very predictive at this stage, but… It plugs CMB observables directly into quantum gravity phenomenology (instead of insulating one from the other). n More than “consistency relations” in principle n_S and r could be fully fixed by QG! n

Conclusions: You can and you should talk about a varying speed of light n It could be an essential tool in the quest for quantum gravity n It could lead to the most predictive theory of the initial conditions in our Universe. n