Brane cosmological solutions in6D warped flux compactifications

Tsutomu Kobayashi

JCAP07(2007)016 [arXiv:0705.3500]In collaboration with M. Minamitsuji (ASC)

Waseda University

Cosmo 07

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Motivation Why braneworlds with 2 extra dimensions are

interesting? Fundamental scale of gravity ~ weak scale Large extra dimensions ~ micrometer length scale

Flux-stabilized compactifications – Motivation from string theory Keep the setup as simple as possible

May help to resolve cosmological constant problemChen, 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);…

Time-dependent dynamics in 6D (super)gravity models Implication for cosmologyTolley, Burgess, de Rham, Hoover (2006); Copeland, Seto (2007)

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

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

is a parameter, 　　　　 : Nishino-Sezgin chiral supergravity

Look for cosmological solutions

- Assume axial symmetry 　

Conical branes

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Strategy

We will not solve the 6D field equations directly

Systematically construct the desired 6D solutions by dimensionally reducing known solutions in (6 + n)D Einstein-Maxwell system

Basic idea: 6D Einstein-Maxwell-dilaton system can be equivalently described by (6+ n)D pure Einstein-Maxwell theory

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Dimensional reduction approach (6+n)D Einstein-Maxwell system

Ansatz:

TK and Tanaka (2004)

Dimensional reduction 6D Einstein-Maxwell-dilaton system

Redefinition:

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(6+ n)D generalization of Mukohyama el al. (2005)~double Wick rotated Reissner-Nordstrom solution

(4+n)D metric solves

Field strength

(6+n)D solution in Einstein-Maxwell

Conical deficit

where

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Useful reparameterization Warping parameter:

Rugby-ball (or football):

Reparameterized metric:

Parameters of the solution are: – warping parameter – cosmological const. on (4+n)D brane – controls brane tensions

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Demonstration: 4D Minkowski X 2D compact Seed: (4+n)D Minkowski

For supergravity model, 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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Dynamical solutions: 4D FRW X 2D compact Seed: (4+n)D Kasner-type metric

From (6+n)D to 6D

6D cosmological solution:

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(4+n)D Kasner-type metric, explicitly Kasner-type metric:

Solves (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(Seed: de Sitter) noninflating for supergravity case Tolley et al. (2006)

with

Maeda and Nishino (1985) for supergravity case

Power-law inflationary solution is the late-time attractor

Cosmic no hair theorem in (4+n)D Wald (1983)

Brane induced metric:

Case3: (Seed: Kasner) same as early-time behavior of case2

Case2: nontrivial solution (Seed: Kasner-dS) Early time:

Late time Case1

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Perturbations Perturbation dynamics in 6D models can be studied

using (6 + n)D description The power-law inflation model in 6D is equivalent to

the (6 + n)D Einstein-Maxwell model with de Sitter branes; Much simpler background!

Kinoshita et al. (2007)

(In)stability? – Remaining issue 6D Einstein-Maxwell model with de Sitter branes is unstable under

scalar perturbations for large Hubble rate Implies: instability of (6 + n)D Einstein-Maxwell model and of 6D

Einstein-Maxwell-dilaton model for a certain parameter region

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Summary

Present a systematic method to construct brane-world solutions in 6D Einstein-Maxwell-dilaton system

Construct cosmological solutions by dimensionally reducingknown solutions in (6 + n)D Einstein-Maxwell system

Power-law inflationary solution for a general dilatonic coupling, which is the late-time attractor

(6 + n)D description will simplify the analysis of perturbations