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Moduli Stabilization and

Cosmology in String Gas

Compactification

Cosmological Landscape: Strings, Gravity, and Inflation, Seoul, 2005.9.23

Jiro SodaKyoto University

Based on the work with Sugumi Kanno S.Kanno and J.Soda, 　 hep-th/0509074 　　　　 “Moduli Stabilization in String Gas Compactification”

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Plan of this talk

Introduction to string gas cosmology

T-duality invariant 4-d effective action

Moduli stabilization in string gas compactification

Cosmology in string gas compactification

Conclusion

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Introduction to string gas cosmology

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4-d universe described by general relativity

Standard Picture of the Universe

Surely, standard model particles are components of the universe.

However, WMAP and other cosmological data tells us that they are not dominant components.

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Standard Picture of the Universe

Dark energy 73%

Dark matter 23%

Inflaton is also necessary to explain current observations.

Although 4-d universe described by general relativity,

Standard model particles 4%

Dominant components

general relativity is suffering from the singularity problem.

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Problems in cosmology to be solved…

Dark energy (cosmological constant) Dark matter Inflaton

Cosmological Singularity Superstring Theory 10-dimensions Dimensionality problem Moduli stabilization and more ...

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Cosmological Landscape: Strings, Gravity, and Inflation

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Cosmological Landscape

Inflation

Strings

Gravity

P

73%

23%

4%

Flux compactification

String Gascompactification

Brane Inflation

Braneworld

cosmology

RS warped compactification

?

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A natural picture of the universe emerges

In the conventional standard cosmology, it is assumed elementaly particles occupy the universe. As the every particles can be regarded as modes of a string, it is natural to imagine the universe filled with a string gas.

10-d universe

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10-d universe

4-d observer

Dark energy?

Dark matter?

Inflaton?

winding modes

At low energy, a string gas looks like a gas of particles from 4-d observer.

A natural picture of the universe emerges

However, winding modes in the internal space would play an important role in solving cosmological issues.

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Initially, all of 9 spatial dimensions are small and toroidally compac

tified.

And, the universe is filled with a closed string gas.

Strings winding around the circle prevent expansion.

String Gas CosmologyBrandenberger & Vafa (1989)Brandenberger & Vafa (1989)

Picture

No cosmological singularity

T-duality

minimal length

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Pair annihilation of windings can not occur if large spatial dimensions are more than 4.

A possible solution of dimensionality problem Brandenberger-Vafa mechanism

4-d spacetime becomes large due to annihilation of winding modes.

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whether 6-d internal dimensions are stable or not

during the cosmological expansion.

if we can stabilize the dilaton in this string gas

compactification. ( ).

possible cosmological implications.

sg e string coupling

Main challenges

Besides to verify the validity of the B-V mechanism,we need to investigate

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T-duality invariant 4-d effective action

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Action for a String

string scale2 1s

2Polyakov

1

4ab

a bS d gg X X

( )0

0

( ) ( )2 2

inn

n

iX x e

n

( )0

0

( ) ( )2 2

inn

n

iX x e

n

2 2

2 20X

X

world sheet

in conformal gauge

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2

2 2

M p p

N N

0N N

1N N

† †1 1 | 0;a a k graviton, 2 form , dilaton

◆ Massless modes are important at low energy.

String spectrum in 10-d flat spacetime

†

1n n

n

N n a a

†

1n n

n

N n a a

where

, are 10-d spacetime indices.

mass spectrum

level matching condition

A string looks different depending on how it oscillates.

| 0; | 0;p k k k

0 0

2 2p

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(2)

22

1( ) ( ) ( )

4ab ab

d a bS d g g X i X X RG B X X

graviton 2-form dilaton

Let us consider a string in a general background.

(10) 210 2 2

210

1 14

2 12S d x Ge R H

H dB

Weyl invariance

Low energy effective action

Low energy effective action

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2 ( ) ( ) a babds g x dx dx x dy dy

( )ab abH B x

2 2e e shifted dilaton

4-d universe

1log

2

1a ay y

Dimensional reduction

6-d toroidal space

We assume Brandenberger-Vafa mechanism works.Thus, the 4-d spacetime is practically non-compactwhile 6-d internal space is toroidally compactified.

By dimensional reduction, we can derive the 4-dimensional effective action which is useful to describe the low energy dynamics.

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24 22

14

2S d x ge R

1 1

4 4ac bd ab cd

ab cd ac bdB B

11B B

11 1B B B B

abab

ababB B

T-duality invariant 4-d effective action

T-duality transformation

Matrix notationg g

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windingmomentum

2

2 22

2 2 2n

m RM N NR

N nN m

R

String spectrum in compactified spacetime

4-d universe

1R

R n m

Target space (T-) duality

Consider the toroidaly compactified spacetime with the raius R.The internal momentum is quantized to be p=n/R, and there is a winding mode, w=mR.

mass spectrum

level matching condition

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More …..

1sR is the self-dual radius.

momentum winding

1, 1 , 0n m N N

◆ Massless modes at the self dual point are important at low energy.

1, 0 , 1n m N N

R 1

R

Because of T-duality, one can not distigush the following two different geometries.

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22 2 2 2 0

nm RM N N

R

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Action for a string gas

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00gasS d g Ex 2 ( ( ), ( ))i j ab

ijabE xg M B xp p

We calculate the mass spectrum of a string with constant background fields . After the calculation we replaced them by functions of spacetime coordinates . Let be the comoving number density of the string gas. As the energy of a string is given by , we obtain

T-duality invariant

4-d momentum

Comoving number densityof a string gas

,ab abB( ), ( )ab abx B x

42 ( ( ), ( ))ij

i j ab abg p p M x B x

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Moduli stabilization in string gas compactification

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Previous works in string gas compactification

Numerical evidence of stability of the volume moduli is shown. But the dilaton is running logarithmically.

Watson & Brandenberger 2003

Using the 4-d effective action approach, it is shown that the dilaton and the radion can not be stabilized. This 4-d effective action is not manifestly T-duality invariant.

Battefeld & Watson 2004

The importance of massless modes is stressed and the stability of the volume moduli is proved analytically.

Watson 2004, Patil & Brandenberger 2004

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Numerical analysis revisitedTo verify the stability for the simplest case, we performednumerical calculation.

Scale factor

2 2 2 ( ) i jijds dt a t dx dx

2 ( ) a babb t dy dy

a

b

Stabilized!

4-d is expanding

Stabilized at the string scale

6-d internal space

4-d universe

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Shape moduli in string gas compactification

The stability of shape moduli is partially analyzed using the massless modes at the self-dual point.

　　　　　　　　　　　　　　　　　　　　　　 Brandenberger, Cheung, and Watson 2005

Here, we intend to give a complete stability analysis of all moduli for a simple compactification by using the T-duality invariant 4-d effective action.

We will also clarify why the dilaton is stabilized in our numerical result.

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1

shape moduli

Identify the opposite sides

A model of Compactfication

We can analyze each torus separately.

4M

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2 22 1 2 2 22

torusds dy d

by dy

2

2 2

1b

0

0B

volume moduli

flux moduli

b

shape moduli

Moduli for the compactification

dilaton

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4-d effective action: Einstein frame

2 2 242 4

2 2

2

1 12 2 log

2 2

1

2

ES d x g R bb

44 00

22 ( , , , )iji jd x g g p p M be

volume flux

shape

Shifted dilaton

Mass of a string

Effective potential

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21

1

11

1

22

2 2

2 22

22

22 2

221

2

1( , , , )

4( 1) 2

M bb

b

b

pp w

p w

w

w

p w

w

w p wb N

Mass of a string

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T-duality and Self-dual point

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4 2

bb

b

2 2

2 2

4 2b

1, 0, 1, 0b Self-dual point

T-duality transformation

0abB 1

1

11B B

11 1B B B B

B B

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Moduli Stabilization I

2

221 2 2

12

bM

b b

2b Flat direction

The first kind of string gas consisting of modes which are massless at the self-dual point.

Mass formula for this wrapped string gas

eff 4 002 2 ( , , , )ij

i jV g g p p M be

Effective potential

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2 2 2

22 22 2 2

11 2

bM b

b b

4 22

2

b

21 b Flat direction

The second kind of string gasconsisting of modes which are massless at the self dual point.

Mass formula for this wrapped string gas

Moduli Stabilization II

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Volume, shape, and flux moduli get stabilizedat the self dual point!

Stable compactificationLet us consider both contributions together.　 As the would-be flat directions are orthogonal to each other, the flat direction disappears at the end of the day.

4 2 24 00 ( , , , )ij

i jd x g g p p e M b The dilaton potential disappears!

Effective potential

2

2 2 22

3 6 0d d d

e H e X edt dt dt

Hubble damping Modulation due to moduli oscillation

We have thus understood our numerical result and shown thatthe dilaton is marginally stable.

Equation for the dilaton

X : moduli

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Cosmology in string gas compactification

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Phenomenology in string gas compactificationPatil & Brandenberger 2004

33 3 41 3410 GeV 10 GeV

overclosure condition5-th force constraint

phenomenological constraint on the number density of the string gas

10 eVp

If 4 , then this constraint can be satisfied.

Moreover, under this condition, it turns out that the stabilization mechanism is effective.

4-d momentum

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Cosmology in String Gas compactification

After string gas dominated stage, the radiation dominant stage commences. During this stage, moduli including the dilaton are stable.

Then, the matter dominant stage takes over. Here, we have to assume the dark matter consisting of massive string modes so that the stabilization of moduli is guaranteed.

It is difficult to incorporate the inflation in the cosmological hist

ory. The reason is apparent. If we consider the inflaton potential, it destroys the stability of the moduli. We might seek other mechanism to produce the large scale structure of the universe.

Dark energy is also difficult to explain.

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Is the mobile dilaton so bad?

24 22

14

2S d x ge R

13loga a

a

1

30, 3 1 log , 0a t t t

Dilaton gravity

T-duality

+ time reversal

1

30, 3 1 log , 0a t t t

1

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1 11 0

3 3a t

super - inflation

solution

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“Inflation” in string gas compactfication From the T-duality point of view, it is natural to consider the su

per - inflation which is driven by the mobile dilaton.

Gasperini & Veneziano (1993)Big-bangH

Pre-big-bang

H

There exists a graceful exit problem in this case.

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Summary

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Summary

We have constructed the T-duality invariant 4-d effective action.

We have shown the stability of volume moduli, shape moduli, and t

he flux moduli in the string gas compactification.

However, the dilaton is only marginally stable.

The string gas cosmology is one approach to string cosmology wh

ich has various nice features.

The many challenges remains to be solved. In particular, the struct

ure formation problem is crucial for the success of this scenario. A

lthough the conventional inflation seems to be incompatible with t

he string gas cosmology, pre-big-bang type scenario seems to be

viable.