Quantum gravity phenomenology via Lorentz violations: concepts, constraints and lessons from

analogue models

Quantum gravity phenomenology via Lorentz violations: concepts, constraints and lessons from

analogue models

Stefano LiberatiSISSA/INFN

Trieste

Stefano LiberatiSISSA/INFN

Trieste

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Analogue gravity work made in collaboration with Matt Visser and Silke WeinfurtnerQG phenomenology work made in collaboration with Ted Jacobson and David Mattingly

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The Quantum Gravity problem…The Quantum Gravity problem…

To eventually understand QG, we will need to

observe phenomena that depend on QG

extract reliable predictions from candidate theories & compare with observations

To eventually understand QG, we will need to

observe phenomena that depend on QG

extract reliable predictions from candidate theories & compare with observations

Why we need a QG theory?

Philosophy of unification QM-GR (reductionism in physics)

Lack of predictions by current theories (e.g. GR singularities, time machines, DE?)

Old “dogma” we cannot access any quantum gravity effect…

Lorentz violation: first evidence of QG?Lorentz violation: first evidence of QG?In recent years several ideas about sub-Planckian consequences of QG physics have been

explored:

e.g. Extra dimensions effects on gravity at sub millimeter scales, TeV BH at LHC, violations of spacetime symmetries…

Idea: LI linked to scale-free spacetime -> unbounded boosts expose ultra-short distances…

Suggestions for Lorentz violation come from: need to cut off UV divergences of QFT & BH entropy

transplanckian problem in BH evaporation end Inflation

tentative calculations in various QG scenarios, e.g.

semiclassical spin-network calculations in Loop QG

string theory tensor VEVs

spacetime foam

non-commutative geometry

some brane-world backgrounds

Very different approaches but common prediction of modified dispersion relations for elementary particles

QG phenomenologyvia modified dispersion relations

QG phenomenologyvia modified dispersion relations

€

E 2 = p2 + m2 + Δ( p,M)

€

M ≡ spacetime structure scale, generally assumed ≈ MPlanck =1019 GeV

If we presume that any Lorentz violation is associated with quantum gravity and we violate only boost symmetry

(no violation of rotational symmetry)

Almost all of the above cited framework do lead to modified dispersion relations that can be cast in this form

Note: (1,2) are dimensionless coefficients which must be necessarily small -> standard conjecture

1(/M)+1, 2(/M) with 1 and with <<M where is some particle physics mass scale

This assures that at low p the LIV are small and that at high p the highest order ones pn with n 3 are dominant…

Theoretical Frameworks for LVTheoretical Frameworks for LV

QFT+LV Renormalizable, or higher

dimension operators

Real LIV with a preferred frameApparent LIV with an extended SR

(i.e. possibly a new special relativity with two invariant scales: c and lp)

Spacetime foam leading to stochastic Lorentz violations

EFT, non-renormalizable ops,(all op. of mass dimension> 4)

Non-commutative spacetime

Extended Standard ModelRenormalizible ops.E.g. QED, dim 3,4 operators E.g. QED, dim 5 operators

Astrophysical tests of Lorentz violationAstrophysical tests of Lorentz violation Cumulative effects: times of flight & birefringence:Time of fight: time delay in arrival of different colorsBirefringence: linear polarization direction is rotated through a frequency dependent angle due to differentphase velocities of photons polarizations

Anomalous threshold reactions (usually forbidden, e.g. gamma decay, Vacuum Cherekov):

Shift of standard thresholds reactions (e.g. gamma absorption or GZK):

Reactions affected by “speeds limits” (e.g. synchrotron radiation):

€

m2

p2≈

pn−2

M n−2⇒ pcrit ≈ m2M n−2n

€

E 2 = c 2 p2 1+m2c 2

p2+ η

pn−2

M n−2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

LI synchrotron critical frequency:

€

ωcLI =

3

2

eBγ 2

m

The key point is that for negative , is now a bounded function of E! There is now a maximum achievable synchrotron frequency ωmax for ALL electrons!

€

=(1− v2)−1/ 2 ≈m2

E 2− 2η

E

MQG

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

−1/ 2

So one gets a constraints from asking ωmax≥ (ωmax)observed

The EM spectrum of the Crab nebulaThe EM spectrum of the Crab nebula

Crab alone provides three of the best constraints.We use:

From

Aharonian and Atoyan,

astro-ph/9803091

Crab nebula (and other SNR) well explained by synchrotron self-Compton model.

SSC Model:1. Electrons are accelerated to very high energies at pulsar2. High energy electrons emit synchrotron radiation3. High energy electrons undergo inverse Compton with ambient photons

synchrotron Inverse Compton

We shall assume SSC correct and use Crab observation to constrain LV.

Observations from CrabObservations from Crab

Gamma rays up to 50 TeV reach us from Crab: no photon annihilation up to 50 TeV.

By energy conservation during the IC process we can infer that electrons of at least 50 TeV propagate in the nebula: no vacuum Cherenkov up to 50 TeV

The synchrotron emission extends up to 100 MeV (corresponding to ~1500 teV electrons if LI is preserved):LIV for electrons (with negative ) should allow an Emax100 MeV.B at most 0.6 mG

Constraints for EFT with O(E/M) LIV

Constraints for EFT with O(E/M) LIV

TOF: ||0(100) from MeV emission GRBBirefringence: ||10-4 from UV light of radio

galaxies (Gleiser and Kozameh, 2002)-decay: for ||10-4 implies || 0.2

from 50 TeV gamma rays from Crab nebulaInverse Compton Cherenkov: at least one of 10-2 from inferred presence of 50 TeV electrons

Synchrotron: at least one of -10-8 Synch-Cherenkov: for any particle with satisfying

synchrotron bound the energy should not be so high to radiate vacuum Cherenkov

T. Jacobson, SL, D. Mattingly: PRD 66, 081302 (2002); PRD 67, 124011-12 (2003)T. Jacobson, SL, D. Mattingly: Nature 424, 1019 (2003)

T. Jacobson, SL, D. Mattingly, F. Stecker: PRL 93 (2004) 021101T. Jacobson, SL, D. Mattingly: Annals of Phys. Special Issue Jan 2006

An open problem: un-naturalness of small LV.An open problem: un-naturalness of small LV.

Renormalization group arguments might suggest that lower powers of momentum in

will be suppressed by lower powers of M so that n≥3 terms will be further suppressed w.r.t. n≤2 ones. I.e. one could have that

This need not be the case if a symmetry or other mechanism protects the lower dimensions operators from violations of Lorentz symmetry.

Ideas: SUSY protect dim<5 operators or brane-world scenarios or analog models…

€

E 2 = p2 + m2 + ˜ η 1μ 2

M 2Mp1 + ˜ η 2

μ

Mp2 + ˜ η 3

μ

M 2p3 + ˜ η 4

μ

M 3p4 + ...+ ˜ η n

μ

M n−1pn

Alternatively one can see that even if one postulates classically a dispersion relation with only terms (n)pnMn-2 with n3 and (n)O(1) then radiative (loop) corrections involving this term will generate

terms of the form (n)p2+(n)p M which are unacceptable observationally.

Analogue gravity as a test fieldAnalogue gravity as a test field

Idea: Study in condensed matter systems typical phenomena of semiclassical gravity.

At least three possible motivations for this study:

I. Experimental issues: try to gain an indirect observational confirmations of SG predictions (e.g. laboratory BH evaporation).

II. Theoretical issues for GR: use the analogy to guess behavior in real semiclassical gravity

III. Theoretical issues for CM: use analogy to have new interpretation of known phenomena or prediction of new ones.

LIV in BEC analogue gravityLIV in BEC analogue gravityThe quasiparticles propagate as on a curved spacetime with an energy dependent metric. This metric is the standard acoustic spacetimes metric when the quantum potential term is negligible, and shows at high frequencies a violation of the Lorentz “acoustic” invariance. In particular adopting the eikonal approximation

€

ω =voi ki ± csk

2 +h

2mk 2 ⎛

⎝ ⎜

⎞

⎠ ⎟2

This dispersion relation (already found by Bogoliubov in 1947) actually interpolates between two different regimes depending on the value of the fluctuations wavelength |k| with

respect to the “acoustic Planck wavelength” C=h/(2mc)= with healing length=1/(8a)1/2

For »C one gets the standard phonon dispersion relation ω≈c|k|. For «C one gets instead the dispersion relation for an individual gas particle (breakdown

of the continuous medium approximation) ω≈(h2k2)/(2m) .

So we see that analogue gravity via BEC reproduces that kind of LIV that people has conjectured in quantum gravity phenomenology…

€

θ1 t,x( ) = Re Αθ exp −iφ( ){ }

n1 t,x( ) = Re Αn exp −iφ( ){ }

€

ω=∂φ∂t

; ki =∇ iφ

So the dispersion relation for the BEC quasi-particles is

€

ˆ Ψ t,x( ) =ψ t,x( ) + ˆ ϕ t,x( )

The need for a more complex model: 2-BEC

The need for a more complex model: 2-BEC

Unfortunately the 1-BEC system just discussed is not enough complex to discuss the most interesting issues in quantum gravity phenomenology

In fact

Naturalness: in order to “see” a modification of the coefficient of k2 in the dispersion relation one needs at least two particles

Universality: the possibility or not of having different LIV coefficients for different particles in the dispersion relation

requires at least two species of particles

So it would be nice to have an analogue model which has at least two kind of quasiparticles which “feel” the same effective geometry at low

energies and show in the dispersion relations at high energies LIV of the sort studied in QG phenomenology

An example of such a model is a 2-BEC system

2-BEC analogue system2-BEC analogue system

We are interested in deviations from SR then:• Homogeneous (position-independent) and time independent background

• BEC at rest (vA0=vB0=0)• Equal background phases, θA0=θB0=0 (simplification)

As usual one analyzes the excitation spectrum by introducing the Madelung representation

Note: with the above notation the scattering length and U are proportional

2-BEC wave equation2-BEC wave equation

Mass Density Matrix Coupling Matrix

Interaction Matrix Quantum Potential Matrix

Linearized Quantum Potential

2-BEC: dispersion relation 2-BEC: dispersion relation

The dispersion relation can now be easily seen by considering the eikonal approximation. In this case

and

It is now convenient to introduce the new set of variables

So that the wave equation can be cast in the form

“All” we need now is to make explicit the content of this dispersion relation

2-BEC: masses2-BEC: masses

Let’s define

Then we can read the dispersion relation as

That is

Masses

We can read the mass matrix by looking at the constant factor left for H(0)

One gets

Note: in order to get quantities with dimension of masses we need to define a speed of sound. This requires some choices…

2-BEC: LIV2-BEC: LIVWe now expect to get LIV with p4 (as in standard 1-BEC) but will we get also

p2 terms due to the particle interactions? If yes will we have the latter dominant over the p4?

So we want to see if our dispersion relation takes the form

A simple way to determine the coefficients consist in taking a suitable number of derivatives in k2 of ω2

and then look at the limit for k0.

One gets:

Introducing the following 2x2 matrices

Note: X=Y=0 and c2=tr[C02]/2 if one neglects the quantum potential

Note: there are LIV terms which do not disappear for X0

2-BEC LIV: Case 1, universality and naturalness

2-BEC LIV: Case 1, universality and naturalness

Let’s see what happens if fine tune our system so that all the Lorentz violations which are purely interaction-dependent vanish.

This is equivalent to cancel in our LIV coefficients all the terms which do not depend on X and Y

This request implies the following conditions to hold

These conditions assure that all the the violations of Lorentz invariance which are not due to the UV physics of the analogue

spacetime are removed. These are also the same conditions one has to impose in order to get “mono-metricity” in the hydrodynamical limit

Few computations show in this case that

2-BEC: the naturalness problem2-BEC: the naturalness problem

Does this mean that the k4 LIV is always more suppressed of the k2 one?This would be the case if both 2 and 4 are found to be proportional to

some small ratio (/M)n given that the k4 term has a further 1/M2 suppression

A careful analysis shows that

As long we are in a regime where mII<<Meff (I.e. UAB<< UAA and/or UBB) this is indeed a small ratio

But the 4 coefficients do not show any similar “Planck” suppression

This implies that at momenta k» the quartic order LIV is dominant on the

quadratic one!

We see that the coefficients of LIV are particle dependent (no universality) but we might have a naturalness problem because 2

Note also that for mA=mB 4,I= 4,II=1/2

2-BEC LIV: Case 2, mono-metricity at low momenta

2-BEC LIV: Case 2, mono-metricity at low momenta

Let’s see what happens instead if fine tune our system so that all the Lorentz violations at order k2 vanish.

This request implies the following condition to hold

Note that this condition alone is not enough for removing in 4 all the QP-indipendent terms. In particular it is not cancelled the term

This is not strictly speaking a QGP analogue but something new where the LIV are not just a UV effect… but let’s see if it would be anyway

compatible with approximate SR at low energies.

The above condition is a quadratic in . In order to get a simple solution let’s specialize to the case mA= mB=m, A= B= 0, UAA=UBB=U0

In this limits the solutions read

2-BEC LIV: Case 22-BEC LIV: Case 2

The solution with 2 is pathological as in this case mII=0 and tr[Y2]=2eff

The solution with 1=-0UAB is instead interesting. In fact in this case

Note that in spite of the presence of the QP-independent term the LIV coefficients are still O(1) and identical in the limit considered here. Again

a small mII requires very different UAB and U0.

The qp-indep term in d2ω2/d(k2)2|k=0 is

Meanwhile the other qp-dep terms are at leading orderThis tells us that the qp-independent term is always subdominat. One

might think that it would still be visible in the hydrodynamic regime (no qp terms) but in order to see a term like this one would requires

frequencies beyond the hydro limit

ConclusionsConclusions• Potential violations of Lorentz invariance seems to be generic in QG

models• These violations typically lead to modified dispersion relations for the

elementary particles suppressed by the QG scale• This kind of Planck suppressed modifications have been put to the test

via laboratory and astrophysical observations• EFT with LIV is the most explored avenue but has also some conceptual

problems• Analogue models of gravity provide a natural environment where to look

for answers to these problems

Lessons from 2-BEC as an analogue QG systemLessons from 2-BEC as an analogue QG system

• Interplay between interaction-dependent LIV and microstructure (Quantum potential)-dependent LIV

• The system can be fine tuned in order to reproduce both a situations with low energy (small) LIV and one with exact mono-metricity at low

energies• Different quasi-particles have generically different coefficients of LIV if

there is more than one microphysical scale • When both k2 and k4 deviations exists the latter are dominant. Because of the different origin? (the latter are also there in a 1-BEC the former are

induced by the interactions)

The future?The future?

• Is there an EFT interpretation for the Case1 and Case2 condensates conditions?

• What is going wrong with the naive EFT prediction?

• Could these systems tell us that LIV is not the only low energy effect of QG? Can they inspire us to look for something else?

Once again analogue models seems able to teach us about gravity and quantum gravity effects more than we expected to.

And I cherish more than anything else the Analogies, my most trustworthy masters. They know all the secrets of Nature, and they

ought least to be neglected in Geometry.Johannes Kepler