1. Infrared divergences in the inflationary brane world Oriol Pujolàs Yukawa Institute for Theoretical Physics, Kyoto University In collaboration with Takahiro Tanaka & Misao Sasaki gr-qc/0407085 to appear in JCAP Trobades de Nadal 2004 Universitat de Barcelona, 21/12/04

2. Motivation Quantum fluct in cosm BW in BW cosmology

3. Motivation How do IR divergences look like in the BW?? Is the backreaction from quantum effects important? Bulk inflaton model: Bulk scalar with light mode drives inflation on the brane in BW cosmology Describe bulk inflaton: modific of RS to include period of infl: inflaton in brane or in bulk It’s well known that in dS the BD vac suffers from IR divergences Do the kk modes modify the fluctuations?

4. IR divergences in de Sitter IR divergences in de Sitter Brane World Application: Bulk inflaton model Conclusions PLAN

5. IR divergence in de Sitter Light scalars in de Sitter in Bunch Davies vacuum Broadening of the homogeneous mode for

6. Massless scalar in de Sitter e.o.m.

7. Massless scalar in de Sitter e.o.m. dS invariant dS

8. Massless scalar in de Sitter e.o.m. dS invariant dS But KG norm dS invariant vacuum

9. Allen Follaci vacuum is a free parameter breaks dS inv.

10. Allen Follaci vacuum breaks dS inv. (in 3 dimensions)

11. Allen Follaci vacuum Vilenkin Ford ’82 Linde ’82 breaks dS inv. (in 3 dimensions)

12. Allen Follaci vacuum Vilenkin Ford ’82 Linde ’82 breaks dS inv. is finite

13. Massless minimally coupled Special case:

14. Massless minimally coupled Garriga Kirsten vacuum Special case:

15. Massless minimally coupled Garriga Kirsten vacuum is finite and dS-invariant Special case: but Shift symmetry why?

16. In summary, in de Sitter space: large and (massless minimal coupling) some regular dS invariant vacuum exists ( effectively massive but not minimal c.) but is regular

17. does it mean that in the brane world there are light cone divergences?… but … if, even in the massive case, the wave function of the bound state diverges on the light cone … ??

18. IR divergences in the Brane World Minimal Non-minimal

19. Model: one de Sitter brane in a flat bulk n+2 dimensions (Vilenkin-Ipser-Sikivie ’83)

20. Model: one de Sitter brane in a flat bulk n+2 dimensions (Vilenkin-Ipser-Sikivie ’83) De Sitter in Rindler coords:

21. Model: one de Sitter brane in a flat bulk n+2 dimensions (Vilenkin-Ipser-Sikivie ’83) De Sitter in Rindler coords: ‘light cone’

22. Generic scalar field bulk brane Flat bulk

23. Spectrum Continuum of KK modes m One bound state, with mass

25. For, the KK contribution

26. Exactly massless bound state

27. AF vacuum A) Bound state:

28. B) KK modes: simple poles: regular double pole: Exactly massless bound state AF vacuum

29. A) Bound state: B) KK modes: simple poles: regular double pole: Exactly massless bound state AF vacuum light cone div.

30. Regular on the light cone

31. = In fact, Regular on the light cone but its derivatives are NOT (4 dim)

32. = In fact, Regular on the light cone but its derivatives are NOT diverges on the LC in 4 and 6 dimensions if

34. Divergence at !! Continuation of decaying mode grows!!

35. (even with ) Divergence at !!

36. Massless minimally coupled Special case: is finite and dS-invariant again, because of the shift symmetry is constant so, again Note: Garriga Kirsten vacuum ?? Comment on the graviton

37. Massless minimally coupled Special case:

38. Application: bulk inflaton model

39. a bulk scalar field in ‘almost’-Randall-Sundrum II model has a light bound state in the spectrum, and a potential that drives inflation bulk brane Scales: bound state dominates higher dimensional effects are important Bound state dominates for ?? Bulk inflaton model Backreaction?

40. Light bound state

41. Regular on the light cone (thanks to the KK modes) (in the bulk)

42. Bound state wave function corresponding to Regular on the light cone (thanks to the KK modes) Light bound state

43. There are 2 possibilities for m small: cancellation; or everybody small Light bound state two possibilities for cancellation (fine tuning) No fine tuning No large backreaction

44. on the brane: no bound statebound state

45. Conclusions (and either or )

46. The analog of the Allen Follaci vacuum in the Brane World scenario does not generate IR divergences on the light cone but it can not avoid an IR divergence within the bulk is it possible to avoid this divergence by modifying vacua of KK modes?? (and either or ) Conclusions

47. The analog of the Allen Follaci vacuum in the Brane World scenario does not generate IR divergences on the light cone but it can not avoid an IR divergence within the bulk is it possible to avoid this divergence by modifying vacua of KK modes?? (and either or ) Conclusions a regular and dS inv vacuum exists

48. The analog of the Allen Follaci vacuum in the Brane World scenario does not generate IR divergences on the light cone but it can not avoid an IR divergence within the bulk is it possible to avoid this divergence by modifying vacua of KK modes?? (and either or ) when the lowest lying mode is light, the dS-invariant vacuum can generate a large if m bs fine tuned no fine tuning of m bs no large backreaction in the bulk inflaton model perturbations on the brane dominated by b.s. if can be mimicked by a massive mode ? Conclusions a regular and dS inv vacuum exists