Exact brane cosmology in 6D warped flux compactifications

1. Exact brane cosmology in 6D warped flux compactifications 小林 努 (早大 理工) with 南辻真人 (Arnold Sommerfeld Center for Theoretical Physics) 研究会： 宇宙初期における時空と物質の進化 @ 東大 Based on arXiv:0705.3500[hep-th]

2. Motivation 6D brane models Fundamental scale of gravity ~ weak scale Large extra dimensions ~ micrometer length scale Flux-stabilized extra dimensions may help to resolve cosmological constant problem… Arkani-Hamed, Dimopoulos, Dvali (1998) Aghababaie et al. (2003); Gibbons et al. (2004); Burgess et al. (2004); Mukohyama et al. (2005) 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) • Codimension 2 brane (c.f. 5D, codimension 1 brane models) • cannot accommodate matter fields other than pure tension • ??? 3-branes with Friedmann-Robertson-Walker geometry ??? • Bulk matter fields can support cosmic expansion on the brane • Cosmological solutions in the presence of a scalar field, flux, and conical 3-branes in 6D • some relation with dynamical solutions in 6D gauged chiral supergravity

3. Our goal 6D Einstein-Maxwell-dilaton + conical 3-branes : Nishino-Sezgin chiral supergravity Look for cosmological solution Conical brane

4. Our strategy Dependent on time and internal coordinates Difficult to solve Einstein eqs. + Maxwell eqs. + dilaton EOM Generate desired solutions from familiar solutions inEinstein-Maxwell system (without a dilaton)

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

6. (6+n)D solution in Einstein-Maxwell ~double Wick rotated Reissner-Nordstrom solution where (4+n)D metric: Field strength 6D case: Mukohyama et al. (2005) Conical deficit

7. Reparameterization Warping parameter: Rugby-ball (football): Reparameterized metric: • Parameters of solutions are: • – warping parameter • – cosmological const. on (4+n)D brane • – controls brane tensions

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

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

10. (4+n)D solutions Kasner-type metric: (4+n)D Field eqs.: Case1: de Sitter Case2: Kasner-dS Case3: Kasner :

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

12. Cosmological perturbations Axisymmetric tensor perturbations, for simplicity (6+n)D Einstein eqs. – separable perturbation eq. • General solution: • Boundary conditions at two poles: Separation eigenvalue

13. Cosmological perturbations t direction: Exactly solvable for inflationary attractor background • Extra direction: • Zero mode • No tachyonic modes • Kaluza-Klein modes • Exact solutions for • given numerically for general

14. KK mass spectrum For small , KK modes are “heavy” Small is likely from the stability consideration Larger makes flux smaller Unstable mode in scalar perturbations; expected for large Kinoshita, Sendouda, Mukohyama (2007)

15. Summary 6D Einstein-Maxwell-dilaton  (6+n)D pure Einstein-Maxwell Generate 6D brane cosmological solutions from (6+n)D Einstein-Maxwell Power-law inflationary solutions and two nontrivial ones Power-law solution is the late-time attractor Noninflating for supergravity case… Cosmological perturbations Tensor perturbations: almostexactly solvable Scalar perturbations…remaining issue Rare case in brane models  useful toy model