Quantum gravity phenomenology in an emergent spacetime concepts constraints and speculations
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Quantum gravity phenomenology in an emergent spacetime: concepts, constraints and speculations. Fourth Meeting on Constrained Dynamics and Quantum Gravity Cala Gonone (Sardinia, Italy). September 12-16, 2005. Stefano Liberati SISSA/INFN Trieste.

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Quantum gravity phenomenology in an emergent spacetime: concepts, constraints and speculations

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Quantum gravity phenomenology in an emergent spacetime: concepts, constraints and speculations

Fourth Meeting on

Constrained Dynamics and Quantum Gravity

Cala Gonone (Sardinia, Italy). September 12-16, 2005

Stefano Liberati



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) 021101

T. Jacobson, SL, D. Mattingly: astro-ph/0505267, Annals of Phys. Special Issue Jan 2006

Lorentz violation: first evidence of QG?

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

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

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

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

(no violation of rotational symmetry)

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…

See e.g. D. Mattingly, “Modern tests of Lorentz invariance,”

Liv. Rev. Rel. [arXiv:gr-qc/0502097].

Theoretical Frameworks for LV

Real LIV with a preferred frame

Apparent 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


Renormalizable, or higher dimension operators

Non-commutative spacetime

EFT, non-renormalizable ops,

(all op. of mass dimension> 4)

Mainly astrophysical constraints

Extended Standard Model

Renormalizible ops. (lab constraints)

E.g. QED, dim 3,4 operators

E.g. QED, dim 5 operators

Astrophysical tests of Lorentz violation

  • Cumulative effects: times of flight & birefringence:

    Time of fight: time delay in arrival of different colors

    Birefringence: linear polarization direction is rotated

    through a frequency dependent angle due to

    different phase 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):

Amelino-Camelia, et al. Nature 393, 763 (1998)

LI synchrotron critical frequency:

T. Jacobson, SL, D. Mattingly

Nature 424, 1019 (2003)

There is now a maximum achievable synchrotron frequency max for ALL electrons!

Hence one gets a constraints by askingmax≥ (max)observed

  • Also: LV induced decays not characterized by a threshold (e.g. decay of a particle

  • from one helicity to the other or photon splitting), Sidereal variation of LV couplings (as the lab moves with respect to a preferred frame or directions), Cosmological variation of couplings…. (see e.g. Mattingly review)

Novelties in threshold reactions: why

  • Upper thresholds:

    The range of available energies of the incoming particles for which the reactions happens is changed.

    Lower threshold can be shifted and upper thresholds can be introduced

  • Asymmetric configurations:

    Pair production can happen with asymmetric distribution

    of the final momenta

If LI holds there is never an upper threshold

However the presence of different coefficients for different particles allows Ei to intersect two or more times Ef switching on and off the reaction!

Sufficient condition for asymmetric Threshold.

Applications: the GZK cut-off

Since the sixties it is well-known that the universe is opaque to protons (and other nuclei) on cosmological distances via the interactions

In this way, the initial proton energy is degraded with an attenuation length of about 50 Mpc.

Since plausible astrophysical sources for UHE particles (like AGNs) are located at distances larger than 50-100 Mpc, one expects the so-called Greisen-Zatsepin-Kuzmin (GZK) cutoff in the cosmic ray flux at the energy given by

  • HiRes collaboration claim that they see the expected event reduction

  • A recent reevaluation of AGASA data seems to confirm the violation of the GZK cutoff.

  • Several explanations proposed (e.g. Z-burst, Wimpzillas), remarkably LIV appears as one one of the less exotic…

  • Everybody is waiting for the Auger experiment to give a definitive answers…

Possible constraints from GZK

Constraint from photon-pion production

p+ CMB  p+0if GZK confirmed

The range of p,for n = 3 dispersion modifications where the GZK cutoff is between 21019 eV and 71019 eV:

constraints of order 10-11 on both p,

Constraint from absence of proton vacuum Cherenkovp+  p+ 

pand are in multiples of 10-10

(Jacobson, SL, Mattingly: PRD 2003)

In 2004 Gagnon and Moore performed analysis taking into account the partonic structure founding same orders of magnitude for the constraints

Applications: QED with LIV at O(E/M)

Let’s consider all the Lorentz-violating dimension 5 terms (n=3 LIV in dispersion relation) that are quadratic in fields, gauge & rotation invariant, not reducible to lower order terms (Myers-Pospelov, 2003). For E»m

Warning: All these LIV terms also violate CPT

electron helicities have independent LIV coefficients

photon helicities have opposite LIV coefficients

u= unit timelike 4-vector that fix the preferred system of reference

Moreover electron and positron have inverted and opposite positive and negatives helicities LIV coefficients (JLMS, 2003).

The EM spectrum of the Crab nebula

Crab nebula (and other SNR) well explained by synchrotron self-Compton (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 synchrotron ambient photons


Inverse Compton

From Aharonian and Atoyan, astro-ph/9803091

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

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

  • 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 Emax100 MeV. B at most 0.6 mG

T. Jacobson, SL, D. Mattingly: Annals of Phys. Special Issue Jan 2006

Constraints for EFT with O(E/M) LIV

TOF: ||0(100) from MeV emission GRB

Birefringence:||10-4 from UV light of radio galaxies (Gleiser and Kozameh, 2002)

Using the Crab nebula we infer:

-decay: for ||10-4 implies || 0.2

from 50 TeV gamma rays from Crab nebula

Inverse 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

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

Alternatively one can see that even if one postulates classically a dispersion relation with only terms (n)pnMn-2 with n3 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 (Collins et al. 2004).

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

Idea: SUSY protect dim<5 operators but SUSY is broken…

Can we get some hint of how things might work using some toy model?

Dilute BEC analogue gravity

The propagations of quantum excitations in a BEC system simulates that of a scalar field on a curved spacetime with an energy dependent metric. In particular adopting the eikonal approximation the dispersion relation for the BEC quasi-particles is

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)=px

with x healing length=1/(8a)1/2

  • For »Cone gets the standard phonon dispersion relation ≈c|k|.

  • For «Cone gets instead the dispersion relation for an individual gas particle (breakdown of the continuous medium approximation)≈(h2k2)/(2m).

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

The need for a more complex model: 2-BEC

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

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

So what we need is an analogue model which has at least two kind of quasiparticles which “feel” the same effective geometry at low energies and show LIV in the dispersion relations at high energies

Fortunately we have such a model:

a system of two coupled BEC

2-BEC: teaser

This system reproduces a QFT in a spacetime with two scalar fields,

one massive the other massless.

As such this is an ideal system for reproducing the salient features of the situations studied in quantum gravity phenomenology…

Separately each BEC as a dispersion relation of the form


When one switched the laser coupling  on, at low frequencies

one can tune the system so to have a common speed of light and gets

12k12, 12k12+m2(cs=1) [Note that the mass is generated via the coupling]

What happens if one looks at the behaviour at high energies?

Will the Lorentz violation remain at order k4 or a naturalness problem will arise?

See Silke Weinfurtner’s talk later…

  • Definitively rule out n=3 LV, O(E/M), EFT including chirality effects

    • Strengthen the n=3 bounds. E.g. via possible role of positrons in Crab nebula emission.

    • naturalness problem: better understanding via an analog model?

  • Constraint on n=4 (favored if CPT also for QG):

  • From Auger, Euso, OWL

    • No GZK protons Cherenkov: ≤O(10-5)

    • If GZK cutoff seen: ≈≥O(-10-2)

  • From Amanda, IceCube, Euso, OWL

    • Neutrinos: 100 TeV neutrinos give order unity constraint by absence of vacuum Cherenkov but rate of energy loss too low. Recent calculations shows one need 1015- 1020 eV UHE cosmological neutrinos. Possibly to be seen via EUSO and/or OWL satellites

  • From AGILE or GLAST we shall hardly get constraints in n=4.

    For GRB we need anyway

    • better measures of energy, timing, polarization from distant -ray sources. O(1) constraint on || requires polarization detection of at 100 MeV

    • AGILE/GLAST could see TOF n=3 LIV, unfortunately no polarization

The future?

Probably new advances will require better understanding of the astrophysics

Better understanding of the naturalness problem could tell us something important about EFT in an emergent spacetime…

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