Varying constants, cosmic structure and the quest for quantum gravity

JoJoãão o MagueijoMagueijo

20132013

Imperial College, LondonImperial College, London

La Sapienza, RomaLa Sapienza, Roma

“Look what happens to people

when they get married”

(Niels Bohr)

By now some varying-constants have become run of the mill

GBrans-Dicke (1961)

Bekenstein (1982)

ETC…

Potential damageto Lorentz invarianceguaranteed

Varying c theories [JM, Rept. Prog. Phys. 66]

Covariant and Lorentz invariant Covariant and Lorentz invariant [Moffat,Magueijo, etc, etc][Moffat,Magueijo, etc, etc]

Bimetric theories Bimetric theories [Moffat, Clayton, Drummond, etc, etc][Moffat, Clayton, Drummond, etc, etc] Preferred frame Preferred frame [Albrecht, Magueijo,Barrow,etc,etc] [Albrecht, Magueijo,Barrow,etc,etc]

Deformed dispersion relations Deformed dispersion relations [Amelino-Camelia, Mavromatos, Magueijo & Smolin, etc, etc][Amelino-Camelia, Mavromatos, Magueijo & Smolin, etc, etc]

c(x,t)

c(E)

The focus here will be on cosmology/early Universe

A non-inflationary solution to the horizon problem

Quantum Gravity and Cosmology: mismatched socks?

Quantum gravity and cosmology: will they ever meet?

The quantum gravity community is either:The quantum gravity community is either: obsessed with its mathematical navel and obsessed with its mathematical navel and

treats data and the real world as a treats data and the real world as a venereal disease.venereal disease.

or, shows a distinct lack of sociological or, shows a distinct lack of sociological guts and tries to force contact with guts and tries to force contact with mainstream cosmology: i.e. inflation. mainstream cosmology: i.e. inflation.

Unrequited love…

Inflation Inflation typicallytypically doesn’t need quantum doesn’t need quantum gravity for anything. It already works gravity for anything. It already works wonders without it. wonders without it.

Quantum gravity would connect better to Quantum gravity would connect better to cosmology without appealing to inflation. cosmology without appealing to inflation. Via models better connected to it, e.g. Via models better connected to it, e.g.

VSL.VSL. Directly!Directly!

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

Near scale-invarianceNear scale-invariance

Amplitude Amplitude

The zero-th order “holy grail” of cosmology:

Varying c theories [JM, Rept. Prog. Phys. 66]

Covariant and Lorentz invariant Covariant and Lorentz invariant [Moffat,Magueijo, etc, etc][Moffat,Magueijo, etc, etc]

Bimetric theories Bimetric theories [Moffat, Clayton, Drummond, etc, etc][Moffat, Clayton, Drummond, etc, etc] Preferred frame Preferred frame [Albrecht, Magueijo,Barrow,etc,etc] [Albrecht, Magueijo,Barrow,etc,etc]

Deformed dispersion relations Deformed dispersion relations [Amelino-Camelia, Mavromatos, Magueijo & Smolin, etc, etc][Amelino-Camelia, Mavromatos, Magueijo & Smolin, etc, etc]

c(x,t)

c(E)

Bimetric theories

RgdxSg gravity 4

),,(4 etcgLgdxSg mmatter

A metric for gravity (Einstein frame):

A metric for matter (matter frame):

If the two metrics are conformal, we have a varying-If the two metrics are conformal, we have a varying-G (Brans-Dicke) theoryG (Brans-Dicke) theory

If they are disformal we have a VSL theoryIf they are disformal we have a VSL theory

The speed of light differs from the speed of gravity The speed of light differs from the speed of gravity (larger if B>0, with )(larger if B>0, with )

This is a rather conservative thing to do…

What sort of fluctuations come out of these theories?

How to compute fluctuations:

I II

IfIf If (with )If (with )

Why the horizon problem leads to a real problem:

Dominates at late times

With With

But why do we get scale invariance? But why do we get scale invariance?

How inflation solves the problem:

Dominates earlier

Dominates later

Consider first the regime Consider first the regime

With this normalization when we second quantize the amplitudes become With this normalization when we second quantize the amplitudes become creation/annihilation operators creation/annihilation operators

Follow up vacuum quantum fluctuations

A miracle happens near deSitter (w=-1)A miracle happens near deSitter (w=-1)

Compute the vacuum expectation valueCompute the vacuum expectation value

In the limit we get:In the limit we get:

With butWith but

with we still get:with we still get:

How a varying speed of light solves the problem:

Dominates earlier

Dominates later

Consider first the regime Consider first the regime

With this normalization when we second quantize the amplitudes become With this normalization when we second quantize the amplitudes become creation/annihilation operators creation/annihilation operators

But could this lead to scale-invariance?

Can solve for a generic w and c_sCan solve for a generic w and c_s

Compute the vacuum expectation valueCompute the vacuum expectation value

take the limit and see when we get:take the limit and see when we get:

For ALL equations of stateFor ALL equations of state

This scaling law for c seems to be uniquely This scaling law for c seems to be uniquely associated with scale invariance. associated with scale invariance.

A remarkable result (!!!!!!!!!!!!!!)

Obviously the variations in c must be cut Obviously the variations in c must be cut off at low energies:off at low energies:

The cut-off scale fixes the amplitude:The cut-off scale fixes the amplitude:

Where does the amplitude come from?

C

The minimal bimetric VSL theory

A subtlety with the variationalcalculus problem:

The KG Lagrangian in the matter frame does NOT give the KG equation.

C

Something truly cool…

Gives a Klein-Gordon equation in matter frame

C

Specifically need a positive Lambda in the Einstein frame Specifically need a positive Lambda in the Einstein frame balanced by a negative lambda in the matter frame, to get the balanced by a negative lambda in the matter frame, to get the right low-energy limit: right low-energy limit:

with f = –B < 0.with f = –B < 0.

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

C

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

So our remarkable result is even more remarkable

Not only is it possible to identify a universal Not only is it possible to identify a universal varying speed of sound law associated with varying speed of sound law associated with scale invariancescale invariance……

but tbut this law can be realized by an anti-DBI his law can be realized by an anti-DBI model (in the Einstein frame), whichmodel (in the Einstein frame), which……

turns out to be the minimal dynamics turns out to be the minimal dynamics associated with a bimetric VSLassociated with a bimetric VSL

Beyond the “zeroth order” holy grail

If the relation between the two metrics isIf the relation between the two metrics is

then we obtain a tilted spectrumthen we obtain a tilted spectrum

C

Summary in terms of (if you really must!)

Standard inflation

DBI inflation

VSL

C

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

Varying c theories

Covariant and Lorentz invariant Covariant and Lorentz invariant [Moffat,Magueijo, etc, etc][Moffat,Magueijo, etc, etc]

Bimetric theories Bimetric theories [Moffat, Clayton, Drummond, etc, etc][Moffat, Clayton, Drummond, etc, etc] Preferred frame Preferred frame [Albrecht, Magueijo,Barrow,etc,etc] [Albrecht, Magueijo,Barrow,etc,etc]

Deformed dispersion relations (DDRs) Deformed dispersion relations (DDRs) [Amelino-Camelia, Mavromatos, Magueijo & Smolin, etc, etc][Amelino-Camelia, Mavromatos, Magueijo & Smolin, etc, etc]

c(x,t)

c(E)

Mounting evidence for a 2D UV fixed point

Causal dynamical triangulations Causal dynamical triangulations Asymptotically safe Quantum Einstein Asymptotically safe Quantum Einstein

GravityGravity Horava-Lifschitz theory (z=3)Horava-Lifschitz theory (z=3) Space-time non-commutativitySpace-time non-commutativity Spin foamsSpin foams EEtctc……

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

Measures of the dimensionality of space-time:

Topological dimension (standard stuff we Topological dimension (standard stuff we learn in analysis but don’t define properly)learn in analysis but don’t define properly)

Hausdorff dimension (balls of size R have Hausdorff dimension (balls of size R have volume R^D; allows for fractional D)volume R^D; allows for fractional D)

Spectral dimensionSpectral dimension… (a popular dimension … (a popular dimension in QG, because it’s “easy” to compute) in QG, because it’s “easy” to compute)

Running of spectral dimensions can be modelled with DDR

If we plug into the “spectral dimension” If we plug into the “spectral dimension” formula the DDR:formula the DDR:

wewe get: get:

This dispersion relation is very interesting indeed:

What we did with bimetric VSL can also be done with DDR

Deformed dispersion relations can give a Deformed dispersion relations can give a frequency dependent speed of light, e.g: frequency dependent speed of light, e.g:

The speed of light/sound would then also The speed of light/sound would then also vary in time, by proxy, via expansion:vary in time, by proxy, via expansion:

We still obtain the same equation:

… … assuming the theory is represented as a assuming the theory is represented as a HOD theory in which only spatial gradients HOD theory in which only spatial gradients are higher order than 2are higher order than 2

… … assuming Einstein gravity, for background assuming Einstein gravity, for background and perturbationsand perturbations

The calculation follows through:

Initial condition:Initial condition:

Leads to the full solutionLeads to the full solution

Again we find a universal law!

Scale-invariance derives universally fromScale-invariance derives universally from

Furthermore the right amplitude can be Furthermore the right amplitude can be obtained with obtained with

Deviations from strict scale-invariance

It could be that gamma isn’t exactly 2, so It could be that gamma isn’t exactly 2, so that we run into a dimension slightly higher that we run into a dimension slightly higher than 2 than 2

It could also be that we have a very slow transient into the UV

By slow, I mean really slow: By slow, I mean really slow:

Then we can compute thatThen we can compute that

Or, Or,

Or, restoring the w dependence:Or, restoring the w dependence:

How about gravitational waves?

In general they could have different DDRs In general they could have different DDRs to scalar modes. For example: to scalar modes. For example:

(and even if gamma is the same, b could be (and even if gamma is the same, b could be different from 1). different from 1).

The UV ratio of the speed of light and the The UV ratio of the speed of light and the speed of gravity speed of gravity

The amplitude of the spectrum depends on The amplitude of the spectrum depends on this! this!

So:So:

Watch this space:

VSL is a good meeting point for (cosmological) VSL is a good meeting point for (cosmological) observations and quantum gravityobservations and quantum gravity

With bimetric theories we have computed With bimetric theories we have computed consistency conditions relating deviations from consistency conditions relating deviations from scale-invariance and non-Gaussianitiesscale-invariance and non-Gaussianities

Work with DDRs is still preliminary, but shows Work with DDRs is still preliminary, but shows a close relation between observables in a close relation between observables in cosmology and the dimensional reduction cosmology and the dimensional reduction phenomenon in quantum gravity. phenomenon in quantum gravity.

Conclusions:

The marriage of quantum gravity and The marriage of quantum gravity and cosmology continues to be wishful cosmology continues to be wishful thinking: varying constants may be the thinking: varying constants may be the essential link essential link

Important not to hide the missing bits: Important not to hide the missing bits:

C

(where ), just like in DBI inflation,(where ), just like in DBI inflation,

but now with a different amplitude: but now with a different amplitude:

For scale-invariant varying we obtain an equilateral bispectrum