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Stefano Liberati SISSA/INFN Trieste

Quantum gravity phenomenology via Lorentz violations: concepts, constraints and lessons from analogue models. Stefano Liberati SISSA/INFN Trieste. Analogue gravity work made in collaboration with Matt Visser and Silke Weinfurtner

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Stefano Liberati SISSA/INFN Trieste

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  1. Quantum gravity phenomenology via Lorentz violations: concepts, constraints and lessons from analogue models Stefano Liberati SISSA/INFN Trieste Analogue gravity work made in collaboration with Matt Visser and Silke Weinfurtner QG phenomenology work made in collaboration with Ted Jacobson and David Mattingly

  2. The Quantum Gravity problem… • 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?) • To eventually understand QG, we will need to • observe phenomena that depend on QG • extract reliable predictions from candidate theories & compare with observations Old “dogma” we cannot access any quantum gravity effect…

  3. 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

  4. 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…

  5. Theoretical Frameworks for LV Apparent LIV with an extended SR Real LIV with a preferred frame (i.e. possibly a new special relativity with two invariant scales: c and lp) Spacetime foam leading to stochastic Lorentz violations QFT+LV Renormalizable, or higher dimension operators Non-commutative spacetime EFT, non-renormalizable ops, (all op. of mass dimension> 4) Extended Standard Model Renormalizible ops. E.g. QED, dim 3,4 operators E.g. QED, dim 5 operators

  6. 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): LI synchrotron critical frequency: The key point is that for negative ,  is now a bounded function of E! There is now a maximum achievable synchrotron frequency max for ALL electrons! So one gets a constraints from asking max≥ (max)observed

  7. The EM spectrum of the Crab nebula From Aharonian and Atoyan, astro-ph/9803091 Crab alone provides three of the best constraints. We use: synchrotron Inverse Compton Crab nebula (and other SNR) well explained by synchrotron self-Compton model. SSC Model: 1. Electrons are accelerated to very high energies at pulsar 2. High energy electrons emit synchrotron radiation 3. High energy electrons undergo inverse Compton with ambient photons We shall assume SSC correct and use Crab observation to constrain LV.

  8. Observations 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 Emax100 MeV. B at most 0.6 mG

  9. 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: 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) -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

  10. 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. 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…

  11. Analogue gravity as a test field Idea: Study in condensed matter systems typical phenomena of semiclassical gravity. At least three possible motivations for this study: Experimental issues: try to gain an indirect observational confirmations of SG predictions (e.g. laboratory BH evaporation). Theoretical issues for GR: use the analogy to guess behavior in real semiclassical gravity Theoretical issues for CM: use analogy to have new interpretation of known phenomena or prediction of new ones.

  12. The 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 LIV in BEC analogue gravity So 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…

  13. 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

  14. 2-BEC analogue system Note: with the above notation the scattering length and U are proportional • We are interested in deviations from SR then: • Homogeneous (position-independent) and time independent background • BEC at rest (vA0=vB0=0) • Equal background phases, A0=B0=0 (simplification) As usual one analyzes the excitation spectrum by introducing the Madelung representation

  15. 2-BEC wave equation Mass Density Matrix Coupling Matrix Interaction Matrix Quantum Potential Matrix Linearized Quantum Potential

  16. 2-BEC: dispersion relation It is now convenient to introduce the new set of variables So that the wave equation can be cast in the form The dispersion relation can now be easily seen by considering the eikonal approximation. In this case and “All” we need now is to make explicit the content of this dispersion relation

  17. 2-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…

  18. 2-BEC: LIV We 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 2 and then look at the limit for k0. Introducing the following 2x2 matrices One gets: Note:X=Y=0andc2=tr[C02]/2if one neglects the quantum potential Note: there are LIV terms which do not disappear for X0

  19. 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

  20. 2-BEC: the naturalness problem We see that the coefficients of LIV are particle dependent (no universality) but we might have a naturalness problem because 2 Does this mean that the k4 LIV is always more suppressed of the k2 one? This would be the case if both 2and 4 are found to be proportional to some small ratio (/M)n given that the k4 term has a further 1/M2suppression 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! Note also that for mA=mB4,I= 4,II=1/2

  21. 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

  22. 2-BEC LIV: Case 2 The solution with 2 is pathological as in this case mII=0 and tr[Y2]=2eff 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 d22/d(k2)2|k=0is Meanwhile the other qp-dep terms are at leading order This 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

  23. Conclusions • 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 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)

  24. 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

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