Exact brane cosmology in 6 D warped flux

研究会： 宇宙初期における時空と物質の進化 @ 東大 Exact brane cosmology in 6 D warped flux compactifications 小林 努 (早大 理 ) with 南辻真人 (Arnold Sommerfeld Center for Theoretical Physics) Based on ar. Xiv: 0705. 3500[hep-th]

Motivation � 6 D brane models � � � may help to resolve cosmological constant problem… � 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 6 D � 2 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. 5 D, codimension 1 brane models) � � Fundamental scale of gravity ~ weak scale Arkani-Hamed, Dimopoulos, Dvali (1998) Large extra dimensions ~ micrometer length scale Aghababaie et al. (2003); Gibbons et al. (2004); Flux-stabilized extra dimensions Burgess et al. (2004); Mukohyama et al. (2005) some relation with dynamical solutions in 6 D gauged chiral supergravity

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

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) 4

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

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

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

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

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

(4+n)D solutions � Kasner-type metric: � (4+n)D Field eqs. : � Case 1: de Sitter � Case 2: Kasner-d. S � Case 3: Kasner 10 :

Cosmological dynamics on 4 D brane � Brane induced metric: � Case 1: power-law inflation � � noninflating for supergravity case Tolley et al. (2006) Case 2: nontrivial solution � Early time: with � � Late time Case 1 Power-law solution is the late-time attractor Case 3: same as early-time behavior of case 2 Maeda and Nishino (1985) for supergravity case 11 Cosmic no hair theorem in (4+n)D Wald (1983)

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

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 13

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

Summary � 6 D Einstein-Maxwell-dilaton (6+n)D pure Einstein-Maxwell � Generate 6 D 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 15 Rare case in brane models useful toy model