1 Moduli Stabilization and Cosmology in String Gas Compactification Cosmological Landscape: Strings, Gravity, and Inflation, Seoul, Jiro Soda Kyoto University Based on the work with Sugumi Kanno S.Kanno and J.Soda, hep-th/ “Moduli Stabilization in String Gas Compactification”

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

3 Introduction to string gas cosmology

4 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.

5 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.

6 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...

7 Cosmological Landscape: Strings, Gravity, and Inflation

8 Cosmological Landscape Inflation Strings Gravity P 73% 23% 4% Flux compactification String Gas compactification Brane Inflation Braneworld cosmology RS warped compactification ?

9 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

10 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.

11 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. String Gas Cosmology Brandenberger & Vafa (1989) Picture No cosmological singularity T-duality minimal length

12 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.

13  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 Main challenges Besides to verify the validity of the B-V mechanism, we need to investigate

14 T-duality invariant 4-d effective action

15 Action for a String string scale world sheet in conformal gauge

16 graviton, 2 form, dilaton ◆ Massless modes are important at low energy. String spectrum in 10-d flat spacetime where are 10-d spacetime indices. mass spectrum level matching condition A string looks different depending on how it oscillates.

17 graviton2-form dilaton Let us consider a string in a general background. Weyl invariance Low energy effective action

18 shifted dilaton 4-d universe Dimensional reduction 6-d toroidal space We assume Brandenberger-Vafa mechanism works. Thus, the 4-d spacetime is practically non-compact while 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.

19 T-duality invariant 4-d effective action T-duality transformation Matrix notation

20 windingmomentum String spectrum in compactified spacetime 4-d universe 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

21 More ….. is the self-dual radius. momentum winding ◆ Massless modes at the self dual point are important at low energy. Because of T-duality, one can not distigush the following two different geometries.

22 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 T-duality invariant 4-d momentum Comoving number density of a string gas

23 Moduli stabilization in string gas compactification

24 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

25 Numerical analysis revisited To verify the stability for the simplest case, we performed numerical calculation. Scale factor Stabilized! 4-d is expanding Stabilized at the string scale 6-d internal space 4-d universe

26 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.

27 1 shape moduli Identify the opposite sides A model of Compactfication We can analyze each torus separately.

28 volume moduli flux moduli shape moduli Moduli for the compactification dilaton

29 4-d effective action: Einstein frame volume flux shape Shifted dilaton Mass of a string Effective potential

30 Mass of a string

31 T-duality and Self-dual point Self-dual point T-duality transformation 1 1

32 Moduli Stabilization I 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 Effective potential

33 Flat direction The second kind of string gas consisting of modes which are massless at the self dual point. Mass formula for this wrapped string gas Moduli Stabilization II

34 Volume, shape, and flux moduli get stabilized at the self dual point! 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. The dilaton potential disappears! Effective potential Hubble dampingModulation due to moduli oscillation We have thus understood our numerical result and shown that the dilaton is marginally stable. Equation for the dilaton : moduli

35 Cosmology in string gas compactification

36 Phenomenology in string gas compactification Patil & Brandenberger 2004 overclosure condition5-th force constraint phenomenological constraint on the number density of the string gas If, then this constraint can be satisfied. Moreover, under this condition, it turns out that the stabilization mechanism is effective. 4-d momentum

37 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.

38 Is the mobile dilaton so bad? Dilaton gravity T-duality + time reversal super - inflation solution

39 “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) Big-bang H Pre-big-bang H There exists a graceful exit problem in this case.

40 Summary

41 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.