1. Brane cosmological solutions in 6D warped flux compactifications Tsutomu Kobayashi JCAP07(2007)016 [arXiv:0705.3500] In collaboration with M. Minamitsuji (ASC) Waseda University Cosmo 07

2. 2 Motivation  Why braneworlds with 2 extra dimensions are interesting?  Fundamental scale of gravity ~ weak scale  Large extra dimensions ~ micrometer length scale  Flux-stabilized compactifications – Motivation from string theory  Keep the setup as simple as possible  May help to resolve cosmological constant problem Chen, Luty, Ponton (2000); Carroll, Guica (2003);Navarro (2003); Aghababaie et al. (2004); Nilles et al. (2004); Lee (2004); Vinet, Cline (2004); Garriga, Porrati (2004);…… Aghababaie et al. (2003); Gibbons et al. (2004); Burgess et al. (2004); Mukohyama et al. (2005);…  Time-dependent dynamics in 6D (super)gravity models  Implication for cosmology Tolley, Burgess, de Rham, Hoover (2006); Copeland, Seto (2007) Arkani-Hamed, Dimopoulos, Dvali (1998);……

3. 3 Our goal  6D Einstein-Maxwell-dilaton + conical 3-branes  is a parameter,  : Nishino-Sezgin chiral supergravity  Look for cosmological solutions  -  Assume axial symmetry Conical branes

4. 4 Strategy  We will not solve the 6D field equations directly  Systematically construct the desired 6D solutions by dimensionally reducing known solutions in (6 + n)D Einstein-Maxwell system  Basic idea: 6D Einstein-Maxwell-dilaton system can be equivalently described by (6+ n)D pure Einstein-Maxwell theory

5. 5 Dimensional reduction approach  (6+n)D Einstein-Maxwell system  Ansatz: TK and Tanaka (2004) Dimensional reduction  6D Einstein-Maxwell-dilaton system  Redefinition:

6. 6  (6+ n)D generalization of Mukohyama el al. (2005) ~double Wick rotated Reissner-Nordstrom solution  (4+n)D metric solves  Field strength (6+n)D solution in Einstein-Maxwell Conical deficit where

7. 7 Useful reparameterization  Warping parameter:  Rugby-ball (or football):  Reparameterized metric: Parameters of the solution are: – warping parameter – cosmological const. on (4+n)D brane – controls brane tensions

8. 8 Demonstration: 4D Minkowski X 2D compact  Seed: (4+n)D Minkowski For supergravity model, Salam and Sezgin (1984) Aghababaie et al. (2003) Gibbons, Guven and Pope (2004) Burgess et al. (2004)  6D solution: From (6+n)D to 6D

9. 9 Dynamical solutions: 4D FRW X 2D compact  Seed: (4+n)D Kasner-type metric From (6+n)D to 6D  6D cosmological solution:

10. 10 (4+n)D Kasner-type metric, explicitly  Kasner-type metric:  Solves (4+n)D field eqs.:  Case1: de Sitter  Case2: Kasner-dS  Case3: Kasner :

11. 11 Cosmological dynamics on 4D brane  Case1: power-law inflation (Seed: de Sitter)  noninflating for supergravity case Tolley et al. (2006) with Maeda and Nishino (1985) for supergravity case Power-law inflationary solution is the late-time attractor Cosmic no hair theorem in (4+n)D Wald (1983)  Brane induced metric:  Case3: (Seed: Kasner)  same as early-time behavior of case2  Case2: nontrivial solution (Seed: Kasner-dS)  Early time:  Late time  Case1

12. 12 Perturbations  Perturbation dynamics in 6D models can be studied using (6 + n)D description  The power-law inflation model in 6D is equivalent to the (6 + n)D Einstein-Maxwell model with de Sitter branes; Much simpler background! Kinoshita et al. (2007)  (In)stability? – Remaining issue  6D Einstein-Maxwell model with de Sitter branes is unstable under scalar perturbations for large Hubble rate  Implies: instability of (6 + n)D Einstein-Maxwell model and of 6D Einstein-Maxwell-dilaton model for a certain parameter region

13. 13 Summary  Present a systematic method to construct brane-world solutions in 6D Einstein-Maxwell-dilaton system  Construct cosmological solutions by dimensionally reducing known solutions in (6 + n)D Einstein-Maxwell system  Power-law inflationary solution for a general dilatonic coupling, which is the late-time attractor  (6 + n)D description will simplify the analysis of perturbations