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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:0705.3500[hep-th] 研究会： 宇宙初期における時空と物質の進化 @ 東大

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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)

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

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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)

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

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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

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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

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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

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

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

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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

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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

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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

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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)

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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

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