1. 1 P. Pasipoularides in Collaboration with K. Farakos (NTUA) and N. Mavromatos (Kings ‘ s College London) THESSALONIKI 2008, NEB XIII RECENT DEVELOPMENTS IN GRAVITY Title: BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES

2. 2 BRANE WORLD MODELS Theorists, in an attempt to solve the hierarchy problem, invented new phenomenological models with extra dimensions, which are known as brane world models (non stringy brane models). Brane models give new physics and predictions in: Astrophysics and cosmology Modifications to Newtons Law (r~160μm) High energy particle physics (1TeV-10TeV LHC)

3. 3 Brane Bulk z (extra dimension) Our world is trapped in a three dimensional brane which is embedded in a multidimensional space (BULK) Only gravitons can propagate in the bulk, hence only gravitons possess KK excitations. However, beyond the standard brane world scenario, there are other models where standard model particles, or some of standard model particles, can live in the bulk. BRANE WORLD MODELS

4. 4 I. LARGE EXTRA DIMENSIONS (R≤160μm) : ADD Model (I. Antoniadis, Arkani-Hammed, Dimopoulos and Dvali). Flat space-time. II. INFINITE EXTRA DIMENSIONS: SECOND RANDALL-SUNDRUM MODEL. (but there is an effective size for the extra dimension due to the curvature of the extra space) III. SMALL EXTRA DIMENSIONS (r~10^(-33)cm): FIRST RANDALL-SUNDRUM MODEL. (Bulk Gauge fields and fermions, Higgs field Localized on the negative tension brane) BRANE WORLD MODELS

5. 5 FIRST RANDALL-SUNDRUM MODEL SMVisible SM or Visible brane Planck Hidden Planck or Hidden brane

6. 6 FIRST RANDALL-SUNDRUM MODEL i. The RS metric is a solution of the Einstein equations only when the following fine-tuning is satisfied. ii. The induced metric on the brane is Minkowski. iii. The RS metric preserves 4D Lorentz invariance in the bulk.

7. 7 ASYMMETRYCALLY WARPED SPACETIMES Beyond the Randall-Sundrum metric we can assume a more general answatz: If the space and time warp factors are different the 5D spacetime is called asymmetrically warped

8. 8 ASYMMETRICALLY WARPED SPACETIMES The induced metric on the brane (z=0) is Minkowski if we assume that However, 4D Lorentz invariance is violated in the Bulk, due to the difference between the space and time warp factors.

9. 9 ASYMMETRICALLY WARPED SPACETIMES Asymmetrically warped Static solutions: 1) M. Visser, Physics Letters B159, 22-25 (1985). 2) Csaba Csaki, Joshua Erlich and Christophe Grojean, Nucl.Phys.B604:312-342,2001. 3) S.L. Dubovsky, JHEP 0201:012,2002. 4) Peter Bowcock, Christos Charmousis and Ruth Gregory, Class.Quant.Grav.17:4745-4764,2000. Extra matter in the bulk is necessary otherwise the Einstein equation is not satisfied by the assymetrically warped static solutions

10. 10 Static solutions like that of RS-Model can be used as background approximately only for a short period of time around our epoch t=t 0. For larger periods of time the complete cosmological evolution must be taken into account. The corresponding static solution is obtained if we set t=t0 In general we expect different warp factors in our epoch Daniel J.H. Chung, Edward W. Kolb and Antonio Riotto, Phys.Rev.D65:083516, 2002 ASYMMETRICALLY WARPED SPACETIMES Cosmological evolution reasons for asymmetrically warped brane models

11. 11 BRANE AND BULK FIELDS IN ASYMMETRICALLY WARPED SPACETIMES Brane fields (completely pinned on the brane) can not “see” the difference between the warp factors. The space-time for these field is Minkowski. Bulk fields are described by a wave function. Due to the extension of the wave function in the bulk the bulk field “sees” the difference between the warp factors and 4D Lorenzt symmetry is violated.

12. 12 ASYMMETRICALLY WARPED SPACETIMES Photons pinned on the brane Gravitons traveling in the bulk extra dimension z Gravitational violation of Lorentz symmetry. Direct signal from superluminous propagation of gravitational waves.

13. 13 ASYMMETRICALLY WARPED SPACETIMES AdS-Reissner Nordstrom Black Hole Solution μ and Q are the mass and charge of the five dimensional AdS Black Hole

14. 14 ASYMMETRICALLY WARPED SPACETIMES AdS-Reissner Nordstrom Black Hole Solution as a linearized perturbation around the RS-metric.

15. 15 ASYMMETRICALLY WARPED SPACETIMES Ads RN Black Hole metric as a linearized perturbation around the RS-metric. We assume that δh(z) is small everywhere in the bulk We consider Z 2 symmetry and a second brane at the position z=z c, in order to achieve the structure (S 1 /Z 2 ) of the first RS-model.

16. 16 BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES We aim to study bulk photons in an asymmetrically warped spacetime which is a linear perturbation around the RS metric.

17. 17 BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES Equation of motion in the case of 5D U(1) Gauge fields Plane wave Answatz

18. 18 BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES Boundary Condition on the brane If we use the formulation of time independent perturbation theory we obtain:

19. 19 BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES Zero Mode plus Discrete Spectrum Unperturbed Equation Photons in the case of the RS1-model

20. 20 BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES Time independent perturbation theory: Zero mode First order correction Second order correction Zero order

21. 21 BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES Energy dependent phase and group velocity of light

22. 22 BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES Final Formulas for phase and group velocity of light when β G <<1 Quadratic dependence on the energy

23. 23 Our main result is a subluminous effective refractive index for vacuum We see that, photons with different energies propagate with different velocities. Hence, we will observe a time lag of the arrival times of photons, which were emitted simultaneously by a remote astrophysical source. In particular photons with smaller energies will arrive first, and photons with larger energies will follow.

24. 24 COMPARING WITH THE EXPERIMENTAL DATA OF MAGIC Magic is an imaging atmospheric Cherenkov telescope which can detect very high energy electromagnetic particles (VHE), in particular gamma rays. VHE photons have energies between 0.1TeV-30TeV. They are photons from conversion of gravitational Energy, when a very massive rotating star is collapsing to a supermassive black hole (Blazar or AGN).

25. 25 J. Albert et al, astro- ph/070008 Magic observations during a flare (which lasts twenty minutes) of the nearby blazar (AGN) Markarian 501 ( Z=0.034), in July 9 (2005), indicates a 4±1 min time delay between the peaks of the time profile envelops for photons with energies smaller than ω 1.2 TeV 150-250 GeV 250-600 GeV 600-1200 GeV 1.2-10 TeV 4 mins-bins

26. 26 COMPARING WITH THE EXPERIMENTAL DATA OF MAGIC  It is an energy dependent effect in the source (SSC mechanism).  New physics induces an effective refractive index for vacuum.  Most of Quantum Gravity Models seem to predict a dispersion relation for vacuum.  Or may be brane models with Bulk photons and an assymetrically warped metric (especially in our model we have quadratic prediction for the refractive index). Possible interpretations of delays of more energetic photons

27. 27 COMPARING WITH THE EXPERIMENTAL DATA OF MAGIC A numerical analysis based on Magic results, which aim at the reconstruction of the original electromagnetic pulse maximizing its energy, assuming a refractive index for vacuum (two cases linear and quadratic): MAGIC Coll. & Ellis, Mavromatos, Nanopoulos, Sakharov, Sarkisyan, arXive: 07082889 [astro-ph] predicts the following values for the two mass scales

28. 28 COMPARING WITH THE EXPERIMENTAL DATA OF MAGIC Numerical analysis: Magic results fitting Theoretical analysis: Brane Models

29. 29 COMPARING WITH THE EXPERIMENTAL DATA OF MAGIC We compute numerically the parameter β G AdS-Schwarzchild Black Hole Solution AdS-Reissner Nordstrom Black Hole Solution Average deviation around the RS1-Metric

30. 30 COMPARING WITH THE MAGIC EXPERIMENT The small values we obtain for are consistent with the weak nature of the perturbation

31. 31 CONCLUSIONS We study asymmetrically warped brane models with bulk photons. We show that the standard Lorentz invariant dispersion relation for 4D photons, possesses nonlinear corrections which lead to an Energy-dependent speed of light. We compared with the experimental data of Magic and we set concrete restrictions to the specific brane models we examined. We propose further investigation for other types of particles such as, gravitons and fermions. Similar dispersion relations are expected.