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) Based on arXiv: [hep-th] 研究会： 東大

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…  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 Arkani-Hamed, Dimopoulos, Dvali (1998) 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)

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 in Einstein-Maxwell system (without a dilaton)

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

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

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: almost exactly solvable  Scalar perturbations…remaining issue Rare case in brane models  useful toy model