brane cosmological solutions in 6d warped flux compactifications tsutomu kobayashi jcap07(2007)016...
TRANSCRIPT
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
2
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);……
3
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
4
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
5
Dimensional reduction approach (6+n)D Einstein-Maxwell system
Ansatz:
TK and Tanaka (2004)
Dimensional reduction 6D Einstein-Maxwell-dilaton system
Redefinition:
6
(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
7
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
8
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
9
Dynamical solutions: 4D FRW X 2D compact Seed: (4+n)D Kasner-type metric
From (6+n)D to 6D
6D cosmological solution:
10
(4+n)D Kasner-type metric, explicitly Kasner-type metric:
Solves (4+n)D field eqs.:
Case1: de Sitter
Case2: Kasner-dS
Case3: Kasner :
11
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
12
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
13
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