Anisotropic Expansion and Stability of Extra Dimensions in Brane Gas Cosmology

1. Anisotropic Expansion and Stability of Extra Dimensions in Brane Gas Cosmology I. Introduction II. Anisotropic expansion of brane universe III. Stabilization with bulk RR flux IV. Conclusion/Discussion J. Y. Kim (Kunsan National University), hep-th/0608131; hep-th/403096[Phys. Rev. D70, 104024 (2004)]

2. I. Introduction • In GR, dimensionality of the universe is an assumption, not fundamental. • Unifying gravity with other forces of nature strongly suggests that there may be more than three spatial dimensions in the unifying scale. • Some of these are hidden from the low energy observers: cosmological problem of anisotropic expansion

3. Cosmology based on string theory BV mechanism [Brandenberger and Vafa,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. • If there are no winding modes around some spatial dimensions, they will expand. • Winding strings can intersect efficiently in three or lower spatial dimensions.

4. Brane gas cosmology • Discovery of D-branes in string theory: string theory is not a theory of only strings and has richer structure of branes. • String cosmology with D-branes. [Maggiore,Riotto, NPB(99); Park, Sin, Lee, PRD(2000); Brandenberger et al, PRD62 (2000)] • Branes with opposite winding numbers can annihilate if their world volume interacts. • Hierarchy of scales can be achieved between the wrapped and unwrapped dimensions.

5. Motivation • Extra dimensions should be wrapped with three or higher dimensional branes for the observed three large spatial dimensions [JYK, hep-th/403096 (PRD,2004)]. • However, it is not guaranteed that the compact dimensions can be stabilized with these branes. • Possible stabilization of the internal dimensions with bulk RR field. • There are many numerical studies. Study the problem analytically

6. II. Anisotropic expansion of Brane Universe • Start from the late stage of BV scenario where the radii and curvature scales are grown enough. • After the thermal equilibrium is broken, (p)-dims: unwrapped, (D-p)-dims: wrapped by gas of branes whose dims are less than or equal to (D-p). • Assume each type of brane gases makes a comparable contribution to energy momentum tensor. • Asymmetric evolution of a universe with metric

7. effective action • Metric • Categorize the effective potential as

8. One dimensional effective action • Equations of motion - take the variation for n and • Set n=1 at the end.

9. Equations of motion

10. Consider the case when the global rotational symmetry is broken down to • Ignore the the stabilizing potential. • Define the volumes of wrapped and unwrapped subspace to decouple the scale factors

11. Decoupling • Since LHS is a function of and RHS is a function of , each side should be a constant. Take the constant as a parameter E.

12. solution • To make expand forever, E>0

13. solution- three possibilities depending on Since wrapped subvolume expands faster than unwrapped subvolume . ii) • expands, but slower than .

14. iii) • Oscillating, but the size of is not bound from below. • Internal subvolume can shrink to zero, which violates the assumptions of brane gas approximation. • It is not possible to stabilize the internal dimensions with only branes

15. III. Stabilization with bulk RR flux Consider a model where the potential U comes from bulk flux. i) bulk effective action: type II string theory with RR field ii) matter contribution: brane (DBI action)

16. Action in the string frame Action in the Einstein frame

17. Static gauge with simple embedding • Induced metric • Static brane • Ansatz: (Bianchi identity is automatically satisfied )

18. One-dimensional effective action for constant potential Equation of motion for

20. Introduce two kinds of bulk RR potential ( ) and branes ( ) to break the global rotational symmetry down to Ignore the running of the dilaton. The contribution of constant dilaton can be absorbed in the redefinition of charge and tension Equations of motion

21. Equations of motion in terms of subvolumes Analyze the behavior of the two subvolumes by considering the effective potential

23. simplest non-trivial case: and are nonzero with p=3. • expands for E>0.

24. We need the condition to have the confining behavior for large . • To have a bounce for small , we need (automatically satisfied)

25. IV. Conclusion/Discussion • It is not possible to stabilize the internal dimensions with only branes in BGC. • Bulk RR field can give an effective potential which prevent the internal volume from collapsing. • RR flux gives a logarithmic bounce for small . • Analysis including the dilaton equation ? • Other combinations of brane gas and RR flux ? (negative tension branes)