1. Masakazu Sano Hokkaido University
Wrapped brane gas as a candidate for Dark Matter
Masakazu Sano
Hokkaido University　
arXiv:
in collaboration with Hisao Suzuki
KEK　Mini Workshop on String Theory, Nov , 2009

2. Table of Contents Introduction Dark matter candidate of wrapped brane?
4-dimensional Einstein frame
Mass and charge of a wrapped brane
Electric-Magnetic duality
Summary and future directions

3. Introduction
We know from recent observations that dark matter and dark energy fills the universe.
If string theory describes the universe, the theory should explain the dark components.
Brane gas cosmology (Alexander, Brandenberger, Easson(2000) ) is one of string cosmologies.
Brane gas cosmology has considered various roles of gases of branes. For example, interesting models of the moduli stabilization (dilaton+scales) have been proposed (Danos, Frey, Brandenberger (2008), Sano and Suzuki(2008)).
It is also interesting to consider a dark matter candidate from the wrapped brane gas.

4. Introduction – idea --
We observe gases of branes wrapping over cycles of a compactified space as a dust of particles.
If the dust of the wrapped branes remains by unknown mechanism, is it possible the wrapped brane gas becomes a dark matter candidate?
A brane gas model constructed by D1-KK5 gives the moduli stabilization. After the moduli stabilization, the model gives rise to energy density of dust in the 4-dim. Einstein frame.
(Sano , Suzuki(2008))

5. Dark matter candidate of wrapped brane?
We should estimate mass of the wrapped brane. If the mass is too heavy, it is difficult to observe the wraped branes.
We have to analyze charges of RR-flux and scalar interactions generated by fluctuations of moduli fields.

6. String coupling is of order one String length is of order TeV
Shiu, Wang(2003)：
A world volume of D-brane wrapping over 6-torus has been considered in the string frame.
String coupling is of order one
String length is of order TeV
In general, a weak string coupling and a large volume gives a very heavy mass in string frame, if the string length is of order the Planck length. One may expect that DM of the wrapped brane is difficult.

7. Gubser, Peebles(2004)：
A toy model of density perturbation of string gas wrapping over 6-torus was considered in 4-im. Einstein frame.
Dilaton is ignored
The density perturbation differs from CDM
Scalar interactions of moduli fields generate attractive forces
ＣＤＭ ＳＧ
Analysis of interactions (scalar and RR flux) has not been done in brane gas models
Purpose
We will estimate mass and charge of interactions in 4dim. Einstein frame, taking weak string coupling and string length which is of order Planck length. We would like to find a light brane with weak interactions.

8. 4-dim. Einstein frame Dimensional reduction
Time-independent Newton constant requires 4-dim. Einstein frame after the dimensional reduction.

9. 4-dim. Einstein frame
World volume (single D-brane, single wrapping)
In the String frame　　　　　 gives rise to a heavy brane. However in the 4-dim. Einstein frame, one may expect light branes by the non-trivial factor.

10. Mass and charge of a wrapped brane
For simplicity we ignore a dependence of B-field.
We consider 6-dim, torus and take the 4-dim. Einstein frame.
Consider a brane wrapping only over 6-torus and .
estimate the mass.
Estimate RR charge.
Estimate a scalar interaction through fluctuations of moduli fields.

11. Effective action
homogeneous

12. A D-brane wrapping over a p-cycle
The proper time of string frame is not equivalent to the proper time defined by 4-dim. Einstein frame.

13. In the 4-dim. Einstein frame the mass does not depend on the dilaton
In the 4-dim.Einstein frame the mass does not depend on the dilaton. For p<3 , and
give a light mass.

14. The dual relation between masses
Obers, Pioline(1998)
time dependent
time independent
When the Dp-brane becomes light, the D(6-p)-brane has a heavy mass, although each mass depends on a time-dependent function.

15. For 10=(d+1)+(10-(d+1)) dimensions we consider (d+1)-dim
For 10=(d+1)+(10-(d+1)) dimensions we consider (d+1)-dim. Einstein frame.
Mass of a Dp-brane wrapping over a p-cycle of a 10-(d+1)-dim. torus.
The time-independent dual relation is realized by 4-dim. Einstein frame.

16. Comment: Moduli stabilization
Mass of a Dp-brane wrapping over a (1・・・p)-cycle
In the string frame the tension binds all cycles.
Brane binds cycles and transverse directions are stretched like a rubber tube.
Physical intuition well behaves
arXiv:

17. The mass of the D0-brane is lighter than the Planck mass.
Mass of a D0-brane
The mass of the D0-brane is lighter than the Planck mass.

18. RR charge

19. Scalar interaction through fluctuations of the moduli field
This is equivalent to the RR charge

20. D0-brane (RR charge and moduli fluctuations)
Square of the charge :
If a D-brane has a light mass and weak interactions, the D-brane has a possibility of the dark matter candidate in the 4-dim. Einstein frame

21. Calabi-Yau compactification (IIA,B=0)

22. When the volume of CY is large, RR charge of D0 and D2 becomes small.
RR flux (IIA, CY3)
When the volume of CY is large, RR charge of D0 and D2 becomes small.

23. D3-brane
Assumption
Becker, Becker, Strominger (1995)

24. Summary
In the 4-dim. Einstein frame, gives rise to some light D-branes with weak interation when the string length is of order the Plank length
Time-independent dual relation of masses and charges is realized only in the 4-dim. Einstein frame, while each mass and charge is a function of time.

25. Wrapped brane gas can be embedded in the flux compactification.
Future directions
It is interesting to compare CDM with the density perturbation of wrapped branes .
Wrapped brane gas can be embedded in the flux compactification.
various interactions arise from couplings between wrapped branes and Dark energy fields or Inflatons. Density perturbation is interesting.

26. appendix

27. 4-dim. Einstein frame
action
4-dim. Einstein frame

28. DBI action of Dp-brane wrapping over (m1…mp)-cycle
The vector potential on the D-brane and the velocity are functions of time in the homogeneous background.

29. Using equations of motion for the vector potential and velocity, the potential energy generated from DBI action is as follows in the 4-dimensional Einstein frame.
In the 4-dimensional Einstein frame, this DBI action cannot fix the dilaton field. We require other sources which include a term proportional to

30. D1-brane(+gauge field), KK5-monopole（ＩＩＢ）
Effective potential in 4-dimensional Einstein frame
D1-branes wrap around each cycle of
KK5-monopoles wrap over (45678)-cycle and its cyclic permutations

31. Dual relation of masses
F1-NS5
Momentum-KK5

32. RR flux (IIB, CY3)

33. 4-dim. Einstein frame Effective potential （Newton approximation）
String frame