Moduli Stabilization and Cosmology in String Gas Compactification

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Moduli Stabilization and Cosmology in String Gas
Compactification
Cosmological Landscape Strings, Gravity, and
Inflation,
Seoul, 2005.9.23
Based on the work with Sugumi Kanno S.Kanno
and J.Soda, hep-th/0509074 Moduli
Stabilization in String Gas Compactification
Jiro Soda
Kyoto University
2
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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Standard Picture of the Universe
4-d universe described by general relativity
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
Dominant components
Standard model particles 4
Dark energy 73
Dark matter 23
Inflaton is also necessary to explain current
observations.
Although 4-d universe described by general
relativity,
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
P
Brane Inflation
Flux compactification
73
RS warped compactification
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Braneworld
String Gas compactification
4
cosmology
?
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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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A natural picture of the universe emerges
At low energy, a string gas looks like a gas of
particles from 4-d observer.
However, winding modes in the internal space
would play an important role in solving
cosmological issues.
10-d universe
Dark energy?
Dark matter?
Inflaton?
4-d observer
winding modes
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No cosmological singularity
String Gas Cosmology
Brandenberger Vafa (1989)
Picture

• Initially, all of 9 spatial dimensions are small
  and toroidally compactified.
• And, the universe is filled with a closed string
  gas.
• Strings winding around the circle prevent
  expansion.

T-duality
minimal length
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A possible solution of dimensionality problem
Brandenberger-Vafa mechanism
Pair annihilation of windings can not occur if
large spatial dimensions are more than 4.
4-d spacetime becomes large due to annihilation
of winding modes.
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Main challenges
Besides to verify the validity of the B-V
mechanism, we need to investigate

• 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.

string coupling
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T-duality invariant 4-d effective
action
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Action for a String
world sheet
string scale
in conformal gauge
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String spectrum in 10-d flat spacetime
A string looks different depending on how it
oscillates.
where
mass spectrum
level matching condition
? Massless modes are important at low energy.
graviton, 2 form , dilaton
are 10-d spacetime indices.
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Low energy effective action
Let us consider a string in a general background.
graviton
2-form
dilaton
Weyl invariance
Low energy effective action
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Dimensional reduction
We assume Brandenberger-Vafa mechanism
works. Thus, the 4-d spacetime is practically
non-compact while 6-d internal space is
toroidally compactified.
4-d universe
6-d toroidal space
By dimensional reduction, we can derive the
4-dimensional effective action which is useful
to describe the low energy dynamics.
shifted dilaton
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T-duality invariant 4-d effective action
T-duality transformation
Matrix notation
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String spectrum in compactified spacetime
Consider the toroidaly compactified spacetime
with the raius R. The internal momentum is
quantized to be pn/R, and there is a winding
mode, wmR.
mass spectrum
winding
momentum
level matching condition
4-d universe
Target space (T-) duality
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More ..
Because of T-duality, one can not distigush the
following two different geometries.
momentum
winding
is the self-dual radius.
? Massless modes at the self dual point
are important
at low energy.
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Action for a string gas
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
Comoving number density of a string gas
T-duality invariant
4-d momentum
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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 revisited
To verify the stability for the simplest case,
we performed numerical calculation.
Scale factor
4-d is expanding
4-d universe
6-d internal space
Stabilized!
Stabilized at the string scale
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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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A model of Compactfication

We can analyze each torus separately.
shape moduli
1
Identify the opposite sides
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Moduli for the compactification
volume moduli
shape moduli
flux moduli
dilaton
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4-d effective action Einstein frame
volume
flux
Shifted dilaton
shape
Mass of a string
Effective potential
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Mass of a string
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T-duality and Self-dual point
T-duality transformation
Self-dual point
1
1
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Moduli Stabilization I
The first kind of string gas consisting of modes
which are massless at the self-dual point.
Effective potential
Mass formula for this wrapped string gas
Flat direction
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Moduli Stabilization II
The second kind of string gas consisting of modes
which are massless at the self dual point.
Mass formula for this wrapped string gas
Flat direction
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Stable compactification
Let 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.
Volume, shape, and flux moduli get stabilized at
the self dual point!
Effective potential
The dilaton potential disappears!
Equation for the dilaton
moduli
Hubble damping
Modulation due to moduli oscillation
We have thus understood our numerical result and
shown that the dilaton is marginally stable.
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Cosmology in string gas compactification
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Phenomenology in string gas compactification
Patil Brandenberger 2004
phenomenological constraint on the number density
of the string gas
overclosure condition
5-th force constraint
4-d momentum
If
, then this constraint can be satisfied.
Moreover, under this condition, it turns out that
the stabilization mechanism is effective.
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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 history. 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?
Dilaton gravity
solution
T-duality time reversal
super - inflation
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Inflation in string gas compactfication

• From the T-duality point of view, it is natural
  to consider the super - inflation which is
  driven by the mobile dilaton.

Gasperini Veneziano (1993)
H
Big-bang
H
Pre-big-bang
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 the flux moduli in the string
  gas compactification.
• However, the dilaton is only marginally stable.
• The string gas cosmology is one approach to
  string cosmology which has various nice features.
• The many challenges remains to be solved. In
  particular, the structure formation problem is
  crucial for the success of this scenario.
  Although the conventional inflation seems to be
  incompatible with the string gas cosmology,
  pre-big-bang type scenario seems to be viable.