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Wrapped brane gas as a

candidate for Dark Matter

Masakazu Sano

Hokkaido University

KEK Mini Workshop on String Theory, 09 Nov , 2009

arXiv:0907.2495

in collaboration with Hisao Suzuki

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• 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

Table of Contents

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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.

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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?

(Sano , Suzuki(2008))

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.

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•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.

Dark matter candidate of wrapped brane?

•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. 6

Shiu, Wang(2003)： A world volume of D-brane wrapping over 6-torus has

been considered in the string frame.

•Analysis of interactions (scalar and RR flux) has not been done

in brane gas models

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Gubser, Peebles(2004)：•A toy model of density perturbation of string gas

wrapping over 6-torus was considered in 4-im.

Einstein frame.

•The density perturbation differs from CDM

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.

Purpose

ＣＤＭ ＳＧ

•Dilaton is ignored

•Scalar interactions of moduli fields generate

attractive forces

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4-dim. Einstein frame

• Dimensional reduction

• Time-independent Newton constant requires 4-dim.

Einstein frame after the dimensional reduction.

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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.

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• 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.

Mass and charge of a wrapped brane

• For simplicity we ignore a dependence of B-field.

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Effective action

homogeneous

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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.

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In the 4-dim.Einstein frame the mass does not

depend on the dilaton. For p<3 , and

give a light mass.

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Obers, Pioline(1998)

The dual relation between masses

time independent

time dependent

When the Dp-brane becomes light, the D(6-p)-brane

has a heavy mass, although each mass depends on

a time-dependent function.

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•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.

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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:0907.2495

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Mass of a D0-brane

The mass of the D0-brane is lighter than

the Planck mass.

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RR charge

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Scalar interaction through fluctuations

of the moduli field

This is equivalent to the RR charge

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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

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Calabi-Yau compactification (IIA,B=0)

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RR flux (IIA, CY3)

When the volume of CY is

large, RR charge of D0

and D2 becomes small.

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D3-brane

AssumptionBecker, Becker, Strominger (1995)

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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.

• It is interesting to compare CDM with the

density perturbation of wrapped branes .

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Future directions

• 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.

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appendix

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4-dim. Einstein frame

• action

• 4-dim. Einstein frame

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.

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 .

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

Dual relation of masses

• F1-NS5

• Momentum-KK5

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RR flux (IIB, CY3)

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• Effective potential （Newton approximation）

4-dim. Einstein frame

String frame

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