Moduli stabilization in brane gas cosmology

1
Moduli stabilization in brane gas cosmology

• Jin Young Kim
• 11th IK Meeting
• Kunsan National University
• arXiv0908.4314hep-th

2
Moduli and string phenomenology

• moduli
• Parameters labeling the geometry of the internal
  space (volume moduli, shape moduli, dilaton,
  axion)
• Stabilization of moduli VEV of all moduli should
  be fixed if string theory is not to contradict
  with the observed phenomena.

• Particle phenomenology
• Coupling constant of elementary particles are
  determined by the VEV of moduli.

• Cosmology
• Scale factor of the extra dimension (volume
  moduliradion) should be fixed.
• VEV of dilaton determines the gravitational
  coupling constant.

Cosmology based on string theory D-brane
inflation, Moduli inflation, cyclic and ekpyrotic
scenarios, mirage cosmology, string/brane gas
cosmology,
3
String Gas Cosmology- BV scenario Brandenberger
and Vafa, NPB 316, 391 (1989)

• Generate dynamically the spatial dimensionality
  and explain the problem of initial singularity.
• The universe was small and hot at early times
  with a gas of strings.
• Target space duality (T-duality, winding?
  momentum) implies the minimum scale. gt can avoid
  the singularity
• Winding mode and anti-winding mode can annihilate
  if their world sheets intersects (pair
  annihilation).
• Spatial dimensions without winding modes will
  expand.
• Winding strings can intersect efficiently in
  three or lower spatial dimensions. gt
  (31)-dimensional spacetime

Brane Gas Cosmology BV scenario with D-branes

• D-branes in string theory string theory is not a
  theory of only strings and has richer structure
  of branes.
• String cosmology with D-brane gas.
• Branes with opposite winding numbers can
  annihilate if their
• world volume intersects.
• Hierarchy of scales between the wrapped and
  unwrapped dimensions.

4
Brane gas formalism with flux

• Start from the late stage of BV scenario where
  the radii and curvature scales are grown enough.
• Include the running of dilaton.
• After the thermal equilibrium is broken,
  
• 3 dimensions unwrapped, expand
• (D-3) dimensions wrapped by gas of branes
  whose dimensions are less than or equal to (D-3)
• Assumption
• - each type of brane gas makes a comparable
  contribution to energy momentum tensor
• To prevent collapse, introduce the gauge fields
  representing RR flux in the transverse directions

5
Action in string frame
Bulk action
6
Action in the Einstein frame frame
7
One-dimensional effective action
8
Equations of motion
9
Equations of motion
10
Equations of motion for SO(3)xSO(D-3) subspace
Assume 3-dimensional space and
(D-3)-dimensional space are isotropic
11
Equations of motion for volume factors
volume of large space
volume of extra space
12
dilaton potential
Motion of a particle under potential

13
Volume factors for fixed dilaton
For fixed dilaton
Combining the second derivative equations
14
Expanding three-volume for fixed dilaton
Three dimensions expand monotonically
15
Radion stabilization for fixed dilaton
Radion is stabilized for fixed dilaton
16
Dilaton stabilization for fixed radion
dilaton perturbation disappears exponentially
17
Stabilizing both moduli simultaneously
Dilaton and radion are stabilized separately if
one is fixed.
18
Eigenvalues of coupled harmonic oscillation
Radion and dilaton can be stabilized
simultaneously
19
Discussion
Sources of moduli potential terms in string
theory curvature, D-branes, orientifold, flux,
nonperturbative effects (instanton, gaugino
condensation), tachyon, orbifold, etc
Find completely explicit examples for stabilizing
moduli.
For coupled system, we can check the stability
perturbatively from their critical value.
20
Summary

• The anisotropic evolution of the spatial
  dimensions and the stability of the extra
  dimensions and dilaton is possible with brane gas
  and RR-flux.
• The effective potentials of three-dimensional
  volume is runaway type so that they can expand
  indefinitely.
• The effective potentials of radion and dilaton
  show global minima that can provide stabilizing
  forces.
• Dilaton and radion perturbations around their
  minima are stable.

21
Motivation running dilaton in radion
stabilization

• In the point of string theory, cosmology is
  described by the coupled system of the metric and
  dilaton.
• Studies on string/brane gas cosmology were
  concentrated on the stabilization of radion
  assuming that the dilaton can be stabilized by
  some other mechanism.
• Running dilaton can cause serious problems at
  late-time evolution (unfixed dilaton ? unfixed
  Newton constant)
• Dilaton stabilization and radion stabilization
  are compatible (string gas and gaugino
  condensation) Danos, Frey, Brandenberger, arXiv
  0802.1577, PRD 77, 126009 (2008)
• Stabilize dynamically both the dilaton and the
  radion in the Einstein frame with brane gas and
  flux