1. Cosmological evolution of regularized branes in 6D warped flux compactifications Masato Minamitsuji (ASC, LMU, Munich) With David Langlois (APC, Paris) arXiv: 0707.1426 [hep-th] COSMO 07 @University of Sussex, August 2007 See also Papantonopoulos, et.al. arXiv: 0707.1396 [hep-th]

2. Introduction 6D braneworlds :Gravity on codimension 2 brane might help the resolution of the cosmological constant problem. : The simplest case of higher-codimensional branes :Phenomenological interests : “ Large extra dimensions ” resolution of hierarchy problem and extra-dimensions would be detected Arkani-Hamed, et.al (1998) Chen, et.al (2003); Carroll & Guica (2003); Navarro (2003); Aghababaie, et.al (2004) Navarro (2003); Nilles, et. al (2003): Vinet & Cline (2004, 2005); Garriga & Porrati (2004) It is difficult to put ordinary matter on a 3-brane due to the stronger self-gravity One way to circumvent this problem is to regularize a brane. 3-brane4-brane with one compactified dimension Cosmology on a regularized brane = Brane motion in the warped bulk

3. 6D Einstein-Maxwell model Braneworld solution with a warped bulk (Mukohyama, et.al (2005-2007)) Codimension 2 (conical) branes are located at Brane @ “ North pole ” Brane @ “ Sorth pole ” (+)-brane (-)-brane

4. Bulk configuration Northern pole (3-brane) Southern pole (3-brane)

5. Regularization of codimension 2 branes Codimension 2 3-brane Codimension 1 4-brane 4-brane Cap region Geometries in the cap regions :Continuity across the branes Cap regions are supported by different bulk CCs. :Regularities in the caps (Peloso, Sorbo & Tasinato, PRD (2006) ; Papantonopoulos, et. al, JHEP(2007))

6. Static brane Static brane Braneworld action Complex scalar field coupled to the bulk gauge field The radial mode is stabilized at a local minimum. Linearized brane gravity Solution for the brane Goldstone mode (Peloso, et. al PRD (2006) ; Kobayashi & Minamitsuji, PRD (2007)) 4D GR Scalar mode associated with the compact dimension Large distances scalesRecovery of 4D GR Dirac quantization number

7. Brane cosmological evolutions Brane motion toward the northern pole Moving 4-brane :Expansion of the ordinary 3-space :Contraction of a KK direction Cosmology

8. Junction conditions Brane induced metric Brane trajectory : Brane localized matter Israel junction conditions

9. Friedmann equation Low energy brane cosmology (Papantonopoulos, et.al., arXiv: 0707.1396, Minamitsuji & Langlois, arXiv: 0707.1426) near the northern pole ! Maxwell junction tells that the energy density vanishes at this point and an “ expansion ” loses its meaning. strong time-dependence asymmetry (Ida, JHEP (2000) for 5D case)

10. Cosmological evolution 1 Cosmological evolution 1 Brane matter = Scalar field coupled to bulk gauge field : Solution for the brane Goldstone mode These are the same as the case of a static brane Solving coupled Israel and Maxwell junction conditions : Evolution of the radial mode and cosmic scale factor We now allow an evolution of radial mode.

11. Cosmological evolution 2 Cosmological evolution 2 :Evolution of a radial mode in terms of cosmic scale factor :Scalar potential is determined by the geometry. : Time scale of the brane evolution.

12. Cosmological evolution 3 Cosmological evolution 3 : A contracting brane cannot cross the point :In the vicinity of the northern pole Brane reaches the northern pole for larger gauge coupling within a finite time scale An example potential total density The total energy density is vanishing at the point that “ “

13. Summary and discussions We have investigated cosmology on a regularized brane in a warped static bulk in 6D Einstein-Maxwell model. : Even in the low energy limits, the effective gravitational coupling becomes time-dependent and negative in the vicinity of the northern pole. : An expanding brane reaches the northern pole within a finite time scale. 5D Randall-Sundrum cosmology : The evolution of a contracting brane is terminated at some finite scale factor with an infinite Hubble rate, although the bulk geometry is regular there. : Similar results are obtained for 6D Nishino-Sezgin supergravity. It is hard to get an ever expanding brane. Bulk time-dependence would be able to help to get more realistic cosmologies. (de Rham, et. al. New J. Phys(2006) ; Kobayashi & Minamitsuji, JCAP(2007); Copeland & Seto, JHEP (2007))