Infrared divergences in the inflationary brane world

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

2. Quantum fluct in cosm BW Motivation in BW cosmology

3. 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? Motivation in BW cosmology Bulk inflaton model: Bulk scalar with light mode drives inflation on the brane How do IR divergences look like in the BW?? Is the backreaction from quantum effects important?

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

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

6. Masslessscalar in de Sitter e.o.m.

7. Masslessscalar in de Sitter e.o.m. dS invariant dS

8. Masslessscalar 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 (in 3 dimensions) • breaks dS inv.

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

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

13. Special case: Massless minimally coupled

14. Special case: Massless minimally coupled Garriga Kirstenvacuum

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

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

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

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) in Rindler coords: De Sitter

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

22. Flat bulk Generic scalar field brane bulk

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

25. For , the KK contribution

26. Exactly massless bound state

27. Exactly massless bound state AF vacuum A) Bound state:

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

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

30. Regular on the light cone

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

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

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

35. Divergence at !! (even with )

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

37. Massless minimally coupled Special case:

38. Application: bulk inflaton model

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

40. Lightbound state

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

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

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

44. bound state no bound state on the brane:

45. Conclusions (and either or )

46. Conclusions (and either or ) • 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??

47. Conclusions (and either or ) • 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?? • a regular and dS inv vacuum exists

48. Conclusions (and either or ) • 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?? • when the lowest lying mode is light, the dS-invariant vacuum can generate a large if mbs fine tuned • no fine tuning of mbs • no large backreaction • in the bulk inflaton model • perturbations on the brane dominated by b.s. if • can be mimicked by a massive mode ? • a regular and dS inv vacuum exists