1. C AN WE CONSTRAIN MODELS FROM THE STRING THEORY BY N ON - GAUSSIANITY ? APCTP-IEU Focus Program Cosmology and Fundamental Physics June 11, 2011 Kyung Kiu Kim (IEU) With Chanju Kim and Frederico Arroja

2. O UTLINE Motivation Large volume scenario Non-gaussianity in Multi-field Inflation (C. Peterson and M. Tegmark) Possible way to produce large Non-gaussianity in some model from Large volume scenario Evolution of non-gaussianity with the adiabaticity assumption. Summary

3. M OTIVATION Im in the IEU. I would like to know whether my main tool, string theory, can explain our universe or not. The string theory is a consistent theory and has good properties mathematically. So many string theorists hope that string theory plays a role of TOE. But the string theory looks so far from observations or experiments, because the energy scale is so high (gravity scale).

4. M OTIVATION In cosmology, the inflation model was proposed and gives good agreement or fitting to observations. It provides many nice explanations for our universe. If it is the right model for our universe and the sting theory is the theory of the universe, then the string theory should contain the inflation model. Recently, Human beings paid really big money for taking photos and we are waiting for the results.

5. M OTIVATION One of the results is the non-gaussianity of the universe. This could give very important information for our early universe. If we assume the string theory explains the inflation, we need to calculate the non- gaussianity in string theory models and compare it to the observation.

6. T HE MODELS FROM THE STRING THEORY Since we dont want inconsistency in string theory, there is many obstacles and difficulties in construction of models in string theory. The models for non-gaussianity in string theory - The large volume scenarios from flux compactification(eta problem) - DBI inflation models(movement of D brane, Freds talk) - Axion monodromy models(eta problem) - …. We are devoted to the large volume scenarios.

7. T HE LARGE VOLUME S CENARIO String theory is defined in 10 dimensional spacetime. One of way to obtain 4 dimensional model is compactifiying 6 dimension in type IIB string theory. This gives a 4 dimensional N=1 Supergravity action.

8. T HE LARGE VOLUME S CENARIO However, the supergravity has a fine tuning problem, eta problem. Without fine-tuning, we cannot produce small eta during the inflation. In order to solve the problem, we may take some assumption. The resulting scenario is the large volume scenarios.

9. T HE LARGE VOLUME S CENARIO Details of the construction can be found in the Prof. Nams talk. I gives a very short introduction here. N=1 SUGRA action given by the Kahler potential K and the holomorphic superpotential W. V_uplift is effect of the supersymmetry-breaking from other sectors of the theory.( We know physical origin.)

10. T HE LARGE VOLUME S CENARIO From flux compactification of Type IIB string theory, These K and W are given by

11. T HE LARGE VOLUME S CENARIO 4 cycle volume + i (axionic partner) Origin was known(instanton or gaugino condensation,…).

12. T HE LARGE VOLUME S CENARIO Dimensionless classical volume D_i is a harmonic 2 form in M. t^i is an area of 2 cycle in M

13. T HE LARGE VOLUME S CENARIO Because of G_3, complex structure moduli and axion have string scale masses and they are decoupled. The low energy theory Taking large volume scenario, Alpha and lambda are from the intersection number.

14. T HE LARGE VOLUME S CENARIO The model is up to with

15. T HE LARGE VOLUME S CENARIO In order to make the metric canonical, one can introduce then the metric on field space becomes canonical type.

16. T HE LARGE VOLUME S CENARIO One may introduce a simple model (an example in 1010.3261) Two light fields play role of inflatons. The model is boiled down to

17. N ON - GAUSSIANITY IN M ULTI FIELD INFLATION Single field slow roll inflation model which has canonical kinetic term gives small non- gaussianity. Easiest way to avoid small non-gaussianity is introducing multi-field which could produce large Non-gaussianity. In string theory, many fields situations are very common because there are many scalar fields. As we explained, after flux compactification, it is very natural to obtain many scalar field with canonical type of kinetic term.

18. N ON - GAUSSIANITY IN M ULTI FIELD INFLATION The local type non-gaussianity has been given by WMAP data. The Planck will give more exact value of NG. For simplicity and insight, we first consider two field case with delta N formalism(Ki-youngs talk). Starting with action

19. N ON - GAUSSIANITY IN M ULTI FIELD INFLATION (C PETERSON AND M T EGMARK ) The number of e-folds N is given by We can express time derivative with e-folds numbers The background eom becomes

20. N ON - GAUSSIANITY IN M ULTI FIELD INFLATION With the slow-roll parameters The field velocity Define some vectors

21. N ON - GAUSSIANITY IN M ULTI FIELD INFLATION Slow-roll approximation in this convention and This means low field-speed and slowly changing speed.

22. N ON - GAUSSIANITY IN M ULTI FIELD INFLATION We have many fields, one may choose another direction and take slowly changing limit Slow turn limit In the SRST limit, the evolution equation is The speed-up rate and turn-rate are approximated by Give by potential V

23. N ON - GAUSSIANITY IN M ULTI FIELD INFLATION The hessian The perturbation equation in Fourier space. In SRST limit

24. N ON - GAUSSIANITY IN M ULTI FIELD INFLATION The evolution of is approximated by The curvature mode and the iso-curvature mode This evolution is expressed by a transfer matrix

25. N ON - GAUSSIANITY IN M ULTI FIELD INFLATION Alpha and beta are given by in SRST limit Related to two point function Three kinds of spectrum (curvature, cross and iso-curvature )

26. N ON - GAUSSIANITY IN M ULTI FIELD INFLATION The curvature spectral index where, tensor spectral index was used The gradient of N Correlation angle

27. N ON - GAUSSIANITY IN M ULTI FIELD INFLATION Curvature iso-curvature correlation Tensor to scalar ratio f_NL and power spectrum in the delta N formalism then

28. N ON - GAUSSIANITY IN M ULTI FIELD INFLATION One can find f_NL becomes simpler form With a little algebra

29. N ON - GAUSSIANITY IN M ULTI FIELD INFLATION Most important term is Condition for large f_NL 1. The total amount of sourcing of curvature modes by iso- curvature modes (TRS) must be extremely sensitive to a change in the initial conditions perpendicular to the inflaton trajectory. In other words, two neighboring trajectories must experience dramatically different amounts of sourcing. 2. The total amount of sourcing must be non-zero. Usually, the amount of sourcing must also be moderate, to avoid having

30. N ON - GAUSSIANITY IN M ULTI FIELD INFLATION Fine tuning for Non-gaussianity 1 vs 100

31. P OSSIBLE WAY TO PRODUCE LARGE N ON - GAUSSIANITY IN SOME MODEL FROM L ARGE VOLUME SCENARIO One model from string flux compactification.

32. E VOLUTION OF NON - GAUSSIANITY WITH THE ADIABATICITY ASSUMPTION 1011.4934 (J. Meyers and N. Sivanandam) The adiabaticity assumption The non-gaussianity is decreasing exponentially.

33. E VOLUTION OF NON - GAUSSIANITY WITH THE ADIABATICITY ASSUMPTION If the adiabaticity assumption is considered, we cannot expect local type large non-gaussianity. So it could be difficult to produce large NG in all the multi field case. However, this assumption is not so strong.

34. S UMMARY Flux compactification of type IIB string theory can give multi-field inflation model(Large volume scenario). In the multi- field case, the large NG requires fine tuning in the field trajectories. In the string theory, we can generate the model which can produce large NG. With the adiabaticity assumption, large NG is very difficult produce. In order to constrain string theory models, we have to understand how the fine-tuning constrains the parameter space of the models.