Stabilizing moduli with flux in brane gas cosmology

1. Stabilizing moduli with flux in brane gas cosmology Jin Young Kim (Kunsan National Univ.) CosPA 2009, Melbourne Based on arXiv:0908.4314[hep-th]; PRD 78, 066003 (arXiv:0804.0073[hep-th]); PLB 652, 43 (hep-th/0608131).

2. Moduli and string phenomenology moduli • Parameters labeling the geometry of the internal space (volume moduli, shape moduli, dilaton, axion) • VEV of 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 should be finite. • 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 picture of string cosmology- BV scenario [Brandenberger and Vafa, NPB 316, 391 (1989)] Initial singularity: • Minimum scale by T-duality (winding momentum) Spacetime dimension: • Early universe - hot and small with a gas of strings • All directions can fluctuate about the self-dual radius. • Directions without winding modes can expand. • Strings can be annihilated efficiently in three or lower spatial dimensions. 2(1+1)=> (3+1) Brane gas cosmology: BV scenario with D-branes • String cosmology with D-brane gas • Hierarchy in sizes of the extra (wrapped) dimensions

4. Motivation: running dilaton in radion stabilization • String cosmology is described by the coupled system of the metric and dilaton (Gravitational constant is fixed by VEV of dilaton) • Studies on BGC were concentrated on the stabilization of radion (volume modulus) assuming that the dilaton can be stabilized. • Are dilaton and radion stabilizations compatible? Yes, with string gas and gaugino condensation. [Danos et al, arXiv: 0802.1577, PRD 77, 126009 (2008)] • Stabilize dynamically both the dilaton and the radion in the Einstein frame with brane gas and flux

5. Brane gas formalism with flux Late stage of BV scenario • After the thermal equilibrium is broken, 3 dimensions: unwrapped, expand (D-3) dimensions: wrapped by gas of branes of dimensions less than or equal to (D-3) Assumption: each type of brane gas makes a comparable contribution to EM tensor => effectively (D-3) brane gas. • Include the running of dilaton. • To prevent collapse, introduce RR flux in the transverse directions

6. RR field dilaton Brane action(DBI): induced metric gauge field antisymmetric tensor negligible when the temperature is low enough Total action: bulk + 6 brane Action in string frame Bulk action:

7. Brane action acts not only as a source term for gravity but also provides potential for dilaton and radion. F^2 term give potential term for radion and dilaton. Singularity at the position of the brane along the transverse direction can be smoothed out by uniform gas approximation. Consider the simple case where it is a constant to incorporate the cosmological constant. Action in the Einstein frame frame

8. Metric ansatz: Induced metric: Static brane : brane time = bulk time To satisfy the Bianchi identity, take the RR field One-dimensional effective action

9. solution: integration constant Equations of motion

10. Equations of motion

11. : redefinition of parameters Equations of motion for isotropic 3 and (D-3) subspace Assume 3-dimensional space and (D-3)-dimensional space are isotropic

12. Equations of motion in terms of volume factors volume of large space volume of extra space

13. dilaton potential Motion of a particle under potential

14. Motion of particles under potential: Volume factors for fixed dilaton For fixed dilaton: Second derivative equations for volume factors

15. Condition for a monotonic expansion of : Expanding three-volume for fixed dilaton Three dimensions expand monotonically

16. Condition for a confining potential for : for Perturbation around Radion stabilization for fixed dilaton Radion can be stabilized for fixed dilaton

17. Dilaton perturbatiuon: Solution: i) : are complex with their real parts negative Damped oscillation ii) : are negative Exponentially decreasing Dilaton stabilization for fixed radion dilaton perturbation disappears exponentially

18. Perturb both moduli around their critical value. Ignore the terms proportional to due to the inflation in three dimensions. Ignore the damping terms since they always contribute to the stabilization positively. Stabilizing both moduli simultaneously Dilaton and radion are stabilized separately if one is fixed.

19. stability condition : are real and positive satisfied for Eigenvalues of coupled harmonic oscillation Radion and dilaton can be stabilized simultaneously

20. : existence of a critical point Criteria for the stabilization of a single scalar field : well-type, confining force 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.

21. 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.