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Brane Gravity and Cosmological Constant

Brane Gravity and Cosmological Constant. 白水 White Water. Tetsuya Shiromizu Tokyo Institute of Technology. Plan. Warped extra dimension - Randall-Sundrum braneworld - Effective theory D braneworld Summary. 1. Warped Extra Dimension.

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Brane Gravity and Cosmological Constant

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

  2. Plan • Warped extra dimension - Randall-Sundrum braneworld - • Effective theory • D braneworld • Summary

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

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

  5. The Einstein equation on brane Shiromizu, Maeda, Sasaki, 2000 Correction terms 4dim Einstein equation 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 - fluid 4-velocity spatial derivative If homogeneous and Isotropic

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

  10. Cosmology III Langlois et al, Garriga & Sasaki, Ida, Kraus, Mukohyama et al, … 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)

  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 + Brane action -Brane action Bulk action

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

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

  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. 0th order

  20. 0th order “Integration constant” constraints Junction conditions

  21. 0th order Summary

  22. 1st order

  23. 1st order ・The traceless part of evolutional equation becomes solution ・From the Hamiltonian constraint, we obtain Integration constant

  24. 1st order Junction conditions imply Effective gravitational equation

  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 We must evaluate this can be written in terms of energy-momentum tensor on branes (Junction condition)

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

  35. From Tento Five IIB Supergravity

  36. Theory

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

  38. Gravity on brane The gravitational theory at low energy Tension T = Charge Q 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 Z_2 symmetry

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

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

  41. 4. Summary

  42. Summary • Braneworld effective theory Gradient expansion Close limit • Braneworld based on D-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.

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

  44. Fin Thank you very much

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

  46. Inflating anti-D brane in warped extra dimension Koyama & Koyama, hep-th/0505256 anti-D brane T_0=-Q_0 D brane with positive tension D brane with negative tension cosmological constant is induced on anti-D brane T_+=Q_+ T_-=Q_-≠-T_+ 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: “anti-D brane is inflating” 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

  49. Set-up tenson=charge tenson=charge

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