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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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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Our goal 6D Einstein-Maxwell-dilaton + conical 3-branes

　　　　 : Nishino-Sezgin chiral supergravity

Look for cosmological solution

　

Conical brane

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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)

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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+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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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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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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4D FRW X 2D compact (4+n)D Kasner-type metric:

From (6+n)D to 6D

6D cosmological solution:

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(4+n)D solutions Kasner-type metric:

(4+n)D Field eqs.:

Case1: de Sitter

Case2: Kasner-dS

Case3: Kasner :

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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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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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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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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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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