1. Brane Gravity and Cosmological Constant Tetsuya Shiromizu Tokyo Institute of Technology Tokyo Institute of Technology 白水 White Water

2. Plan 1. Warped extra dimension - Randall-Sundrum braneworld - 2. Effective theory 3. D braneworld 4. Summary

3. 1. Warped Extra Dimension - Randall-Sundrum Braneworld -

4. Warped extra dimension y flat brane Randall&Sundrum(1999) Extra dimension 5-dimensions

5. The Einstein equation on brane Shiromizu, Maeda, Sasaki, 2000 4dim Einstein equation Correction terms Cosmological constant on the brane Bulk cosmological constant Brane tension For flat brane, we assume Randall-Sundrum tuning

6. Two cosmological constants Bulk cosmological constant Cosmological constant on brane Do not confused !

7. The Einstein equation on brane Shiromizu, Maeda, Sasaki, 2000

8. Cosmology - FRW model - If homogeneous and Isotropic spatial derivative fluid 4-velocity

9. Cosmology II For the homogeneous and isotropic universe, it becomes the energy-momentum of “radiation”. (Equation of state) (Equation of motion) (Dark radiation) Equation is closed! Mass parameter of Schwarzshild-antideSitter Black Hole Braneworld adS/CFT (Witten,Gubser,Garriga&Sasaki,Shiromizu&Ida,Tanaka,…) Scale factor (~radius of universe )

10. Cosmology III Friedmann equation on the brane: Bulk spacetimes: 5-dim. Schwarzshild-anti deSitter spacetime (unit 3-dim. Sphere, plane or hyerboloid(K=1,0, or -1) Langlois et al, Garriga & Sasaki, Ida, Kraus, Mukohyama et al, …

11. Problems How to solve? ✓ Linear perturbation ( Garriga & Tanaka, Sasaki, Shiromizu & Maeda,…) ✓ Cosmological Perturbation ( Kodama,Ishibashi&Seto, Mukohyama, Koyama&Soda, Langlois,…) ✓ Numerical analysis ( Shiromizu&Shibata, Kudoh,Tanaka&Nakamura,…) ✓ Gradient expansion ( Wiseman, Kanno & Soda, Shiromizu & Koyama,…) ✓ Close limit of two branes (Shiromizu,Koyama,Takahashi, de Rham&Webseter,…) ✓ Lower dimension (Emparan,Horowitz&Myers)

12. 2. Effective theory

13. Model Randall-Sundrum I type - brane+ brane Bulk action + Brane action -Brane action

14. Induced metric & extrinsic curvature paralell transport Induced metric hypersurface Extrinsic curvature

15. (1+4)-Decomposition (Bulk) (On Branes) We solve the evolution equations along the extra dimension y Junction conditions

16. 2-1. Low energy effective theory Wiseman 2002, Kanno & Soda 2002, Shiromizu & Koyama 2003, de Rham 2004

17. Small parameter We are interested in long wave scale. Then small parameter is the square of ratio of the bulk curvature scale to the brane intrinsic curvature scale:

18. Sketch The junction condition at the brane Gradient expansion or close limit

19. 0 th order

20. “Integration constant” constraints Junction conditions

21. 0 th order Summary

22. 1 st order

23. ・ The traceless part of evolutional equation becomes solution Integration constant ・ From the Hamiltonian constraint, we obtain

24. 1 st order Effective gravitational equation Junction conditions imply

25. Comment on RSII model undetermined In principle, it is determined by boundary condition at “y=∞” It may corresponds to the dark radiation, but we have to confirm that.

26. 2-2. Close limit Shiromizu, Koyama and Takahashi, 2003 (primitive version) de Rham and Webster 2005 (elegant version)

27. Assumption and motivation Brane collisions is fundamental like particle collisions Brane collision may give us a new picture of big-bang The brane distance is much smaller than the bulk curvature scale

28. What we have to do actually can be written in terms of energy- momentum tensor on branes (Junction condition) We must evaluate this

29. de Rham & Webster’s way

30. Effective equation

31. Tensor perturbation Equation for tensor perturbation is same with 4-dim. one except for gravitational constant

32. 3. D Braneworld Shiromizu, Koyama, Onda & Torii, 2003 Shiromizu, Koyama & Torii, 2003 Onda, Shiromizu, Koyama & Hayakawa, 2004 Shiromizu, Himemoto, Takahashi, 2004 Iwashita, Shiromizu, Takahashi, Fujii,2005

33. D brane cosmology D brane Brane world is motivated by D brane deSitter/Inflation model in warped flux compactification Kachru,Kallosh,Linde,Trivedi, 2003 Kachru, Kallosh, Linde, Maldacena, McAlister, Trivedi (KKLMMT), 2003

34. Gravity on D brane D brane compactification Tension T = Charge Q Bulk: IIB supergravity compactified on S^5 (~ 5-dim. theory) Z_2 symmetry H_3=dB_2, F_3=dC_2, G_5=dD_4 B_2 D_4 Brane: Born Infeld action + Chern Simons

35. From Ten to Five IIB Supergravity

36. Theory

37. Way to look at gravity on brane The junction condition at the brane Replaced by stress tensor on the brane using the junction condition for bulk form fields cancel Gradient expansion or close limit 5-dim Einstein equation Anti-de Sitter curvature radius

38. Gravity on brane The gravitational theory at low energy Z_2 symmetry H_3=dB_2, F_3=dC_2, G_5=dD_4 B_2 D_4 B_2 on the brane is not the source of the brane gravity Tension T = Charge Q

39. Breaks “supersymmetry” T≠Q Introduction of anti-D brane

40. T ≠ Q Tension T ≠ Charge Q Z_2 symmetry H_3=dB_2, F_3=dC_2, G_5=dD_4 B_2 D_4 No cancellation Cosmological constant appears at the same time as B_2 is a source of the gravity Shiromizu, Koyama, Torii, 2003, Iwashita, Shiromizu, Takahashi, Fujii, 2005

41. 4. Summary

42. Summary Braneworld effective theory Braneworld effective theory Gradient expansion Gradient expansion Close limit Close limit Braneworld based on D-brane Braneworld based on D-brane The gauge fields on BPS D brane is not source of gravity on brane. The gauge fields on BPS D brane is not source of gravity on brane. The gauge fields on the deSitter (anti-) D brane is source of gravity on brane. The gauge fields on the deSitter (anti-) D brane is source of gravity on brane.

43. Remaining Issues Higher co-dimension, Warped flux compactification( KKLT, KKLMMT model) Higher co-dimension, Warped flux compactification( KKLT, KKLMMT model) other fields other fields stabilisation stabilisation Basics of higher dimensional spacetimes Basics of higher dimensional spacetimes Mukohyama et al, hep-th/0506050

44. Fin Thank you very much

45. Anti-D brane Randall-Sundrum type model T_+=Q_+T_-=Q_-=-T_+= D brane with positive tension D brane with negative tension

46. Inflating anti-D brane in warped extra dimension T_+=Q_+T_-=Q_-≠-T_+ D brane with positive tension D brane with negative tension anti-D brane T_0=-Q_0 Koyama & Koyama, hep-th/0505256 cosmological constant is induced on anti-D brane Note: D branes is not deSitter brane If the brane distances are stabilised, D branes would be inflating

47. Gravity on anti-D brane? Koyama, Koyama, Shiromizu, Iwashita, to appear, 2005 Koyama & Koyama Koyama & Koyama: “anti-D brane is inflating” Shiromizu, Koyama, Torii 2003/ Iwashita, Shiromizu, Takahashi, Fujii 2005 Shiromizu, Koyama, Torii 2003/ Iwashita, Shiromizu, Takahashi, Fujii 2005: “non-BPS brane is inflating at the same time as gauge field on the brane is a source of the gravity on the brane” The gauge field on the anti-D brane would be a source of the gravity on the anti-D brane.

48. Gravity on anti-D brane Koyama, Koyama, Shiromizu, Iwashita, to appear, 2005

50. Set-up tenson=charge

51. Toy Model Background spacetime can be pure anti-deSitter+ flat branes assumption

52. Equations For simplicity, we assume (Evolutional equations) (Constraint equations)

53. Junction conditions

54. Long wave approximation

55. 0 th order

56. 1 st order

57. Traceless part Trace part Junction conditions

58. Effective Equation at 1 st order

59. Effective Equation at 2 nd order