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

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

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

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

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))

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

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

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

Friedmann equation Low energy brane cosmology (Papantonopoulos, et.al., arXiv: , Minamitsuji & Langlois, arXiv: ) 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)

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.

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.

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 “ “

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))