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