2. by Silke Weinfurtner Victoria University of Wellington, New Zealand Stefano Liberati SISSA/INFN Trieste, Italy Constraining quantum gravity phenomenology via analogue spacetimes Fourth Meeting on Constrained Dynamics and Quantum Gravity Cala Gonone (Sardinia, Italy) September 12-16, 2005 present ed at Matt Visser Victoria University Wellington, New Zealand

3. Using the presented system for an Analogue Model for Quantum Gravity Phenomenology Extended Analogue Models for gravity in a coupled 2-component BEC Analogue Models for gravity

4. The first Analogue Models for Gravity Bill Unruh, “Experimental black hole evaporation?”, Phys. Rev. Lett., 46, (1981) 1351-1353. space-time  convergent fluid flowparticle  small excitations (sound waves)  equations of motion for irrotational fluid flow  linearizing equation about some solutions  equations for  interpretation:equation for massless scalar field in a geometry with metric

5. Bose-Einstein condensates as Analogue Models Bose-Einstein condensate ~ in experiment gas of bosons, e. g. 87 Rb (Eric Cornell) or 23 Na (Wolfgang Ketterle) extremely low densities,  10 15 atoms / cm 3 very cold temperature, T  1  K nearly all atoms occupy the ground state non condensed atoms are neglected microscopic system can be replaced by a classical mean-field, a macroscopic wave-function Bose-Einstein condensate ~ in theory L. J. Garay, J. R. Anglin, J. I. Cirac, P. Zoller, Sonic Analog of Gravitational Black Holes in Bose-Einstein Condensates, Phys. Rev. Lett. 85, 4643–4647 (2000)  interpretation in terms of Analogue Models: The kinematics for sound waves in BEC is given by the Euler and continuity equation, in the so called hydrodynamic limit the BEC is a superfluid.

6. Extending AM for massive scalar fields and use it for QGP Matt Visser, Silke Weinfurtner, Massive Klein-Gordon equation from a BEC-based analogue spacetime, Phys.Rev. D72 (2005) 044020  It is possible to extend the Analogue Models to describe massive scalar fields:  application for Quantum Gravity Phenomenology: One expected Quantum Gravity Phenomena is the violation of spacetime symmetries, e.g. Lorentz violation: Universality and naturalness problem? We would need an analogue model for different interacting (naturalness problem?) particles (universality issue?)…

7. Gross-Pitaevskii equations Macroscopic wave functions Sound waves in a 2-component BEC U AA U BB U AB

8. From the GPE to a pair of coupled wave equations Physical interpretations: this equation represents kinematics of sound waves in the 2-component BEC a small (in amplitude) perturbation in 2-component BEC results in pair of coupled sound waves coupling matrix this description holds for low and high energetic perturbations interaction matrix + quantum pressure term  contains the modified interactions due to the external coupling mass-density matrix background velocity Sound waves in a 2-component BEC

9. Klein-Gordon equation for massive phonon modes. The two decoupled wave equations can be written as two scalar fields in curved space-times: in-phase mode anti-phase mode the in-phase mode represents a massless scalar field the anti-phase mode represents a massive scalar field the two effective metrics are different, due to different speeds of sound:

10. Klein-Gordon equation for massive phonon modes. The fine tuning for the decoupling the wave equations: The two speed of sounds are: the mono-metricity condition must be which requires the fine tuning the densities and interactions within each condensate are equal Fine tuning of the interactions via the external coupling field : the external laser field modifies the interactions U AA U BB U AB ~ ~ ~ the sign of can be positive or negative ( additional trapping frequency ), e.g it is possible to make the modified XX or XY interactions zero: U AA U BB ~ ~

11. Dispersion relation for uniform condensate. Changing into momentum space leads to the dispersion relation:  interpretation in terms of Analogue Models: We recover perfect special relativity for the decoupled phonon modes in the hydrodynamic limit. Note: The change to momentum space is only exact, if the densities are uniform and the background velocity is at rest ( Minkowski space-time ).

12. Beyond the hydrodynamic limit the quantum potential has to be taken into account the quantum potential term (here in flat space-time) can be absorbed in the redefinition of the interaction matrix between the atoms (effective interaction matrix) this term gets relevant at wave length comparable to the healing length a change to momentum space shows the effective interaction is k-dependent For perturbations compareable to the healing length - high energy modes - the calculations have to be modified, by including the quantum potential. This will brake the Lorentz invariance in the dispersion relation! The hydrodynamic limit

13. Dispersion relation for high energy phonon modes. change into momentum-space coupled wave equation for phonon modes in uniform condensate beyond the hydrodynamic limit: perturbations have to fulfill the generalized Fresnel equation Calculating the dispersion relation for the 2 coupled phonon-modes..

14. Dispersion relation for high energy phonon modes. Taylor expansion of  k 2  around zero Note that with H(k 2 ) as a function of k 2 only even parameters of k appear!

15. the dimensionless coefficients - within that fine tuning - are: this is equivalent to cancel on our LIV coefficients all the terms which do not depend on the quantum pressure potential; this requires the following constraints: Predictions of Quantum Gravity: UV LIV LIV purely UV physics (only QP due terms) Note that for m A =m B it follows that  4,I =  4,II =1/2 !

16. A coupled 2-component Bose-Einstein condensate can be used as an Analogue Model for a massive and massless scalar field in curved-spacetime This is a typical situation studied in QG phenomenology with purely higher order LIV characterized by different  coefficients of LIV are particle dependent (no universality) Conclusions At low energies one recovers perfect special relativity  LI At high energies the theory has to be modified about the quantum potential  LIV At both orders - k 2 and k 4 - deviations show up. Planck-suppressed! Order one!

17. Thank you for your attention.

18. Supersonic and subsonic region… horizon  fluid velocity fluid at rest Animation: Sound waves in a moving fluid