1. Inflation, String Theory, Andrei Linde Andrei Linde and Origins of Symmetry

2. Contents: Inflation as a theory of a harmonic oscillator Inflation and observations Inflation in supergravity String theory and cosmology Eternal inflation and string theory landscape Origins of symmetry: moduli trapping Inflation as a theory of a harmonic oscillator Inflation and observations Inflation in supergravity String theory and cosmology Eternal inflation and string theory landscape Origins of symmetry: moduli trapping

3. Inflation as a theory of a harmonic oscillator

4. Einstein: Klein-Gordon: Einstein: Klein-Gordon: Equations of motion: Compare with equation for the harmonic oscillator with friction:

5. Logic of Inflation: Large φ large H large friction field φ moves very slowly, so that its potential energy for a long time remains nearly constant No need for false vacuum, supercooling, phase transitions, etc.

6. Add a constant to the inflationary potential - obtain two stages of inflation

7. Comparing different inflationary models: Chaotic inflation can start in the smallest domain of size 10 -33 cm with total mass ~ M p (less than a milligram) and entropy O(1) New inflation can start only in a domain with mass 6 orders of magnitude greater than M p and entropy greater than 10 9 Cyclic inflation can occur only in the domain of size greater than the size of the observable part of the universe, with mass > 10 55 g and entropy > 10 87 Chaotic inflation can start in the smallest domain of size 10 -33 cm with total mass ~ M p (less than a milligram) and entropy O(1) New inflation can start only in a domain with mass 6 orders of magnitude greater than M p and entropy greater than 10 9 Cyclic inflation can occur only in the domain of size greater than the size of the observable part of the universe, with mass > 10 55 g and entropy > 10 87 Solves flatnes, mass and entropy problem Not very good with solving flatnes, mass and entropy problem Does not solve flatnes, mass and entropy problem

8. A photographic image of quantum fluctuations blown up to the size of the universe

9. Comparing different inflationary models: Chaotic inflation can start in the smallest domain of size 10 -33 cm with total mass ~ M p (less than a milligram) and entropy O(1) New inflation can start only in a domain with mass 6 orders of magnitude greater than M p and entropy greater than 10 9 Cyclic inflation can occur only in the domain of size greater than the size of the observable part of the universe, with mass > 10 55 g and entropy > 10 87 Chaotic inflation can start in the smallest domain of size 10 -33 cm with total mass ~ M p (less than a milligram) and entropy O(1) New inflation can start only in a domain with mass 6 orders of magnitude greater than M p and entropy greater than 10 9 Cyclic inflation can occur only in the domain of size greater than the size of the observable part of the universe, with mass > 10 55 g and entropy > 10 87 Solves flatnes, mass and entropy problem Not very good with solving flatnes, mass and entropy problem Does not solve flatnes, mass and entropy problem

10. How important is the gravitational wave contribution? For these two theories the ordinary scalar perturbations coincide:

11. Is the simplest chaotic inflation natural? Often repeated (but incorrect) argument: Thus one could expect that the theory is ill-defined at However, quantum corrections are in fact proportional to and to These terms are harmless for sub-Planckian masses and densities, even if the scalar field itself is very large.

12. Chaotic inflation in supergravity Main problem: Canonical Kahler potential is.. Therefore the potential blows up at large |φ|, and slow-roll inflation is impossible: Too steep, no inflation…

13. A solution: shift symmetry Kawasaki, Yamaguchi, Yanagida 2000 Equally legitimate Kahler potential and superpotential The potential is very curved with respect to X and Re φ, so these fields vanish But Kahler potential does not depend on The potential of this field has the simplest form, without any exponential terms:

14. Inflation in String Theory The volume stabilization problem: Consider a potential of the 4d theory obtained by compactification in string theory of type IIB The volume stabilization problem: Consider a potential of the 4d theory obtained by compactification in string theory of type IIB Here is the dilaton field, and describes volume of the compactified space The potential with respect to these two fields is very steep, they run down, and V vanishes The problem of the dilaton stabilization was solved in 2001, Giddings, Kachru and Polchinski 2001 but the volume stabilization problem was most difficult and was solved only recently (KKLT construction) Kachru, Kallosh, Linde, Trivedi 2003 Burgess, Kallosh, Quevedo, 2003

15. Volume stabilization Basic steps: Warped geometry of the compactified space and nonperturbative effects AdS space (negative vacuum energy) with unbroken SUSY and stabilized volume Uplifting AdS space to a metastable dS space (positive vacuum energy) by adding anti-D3 brane (or D7 brane with fluxes) Warped geometry of the compactified space and nonperturbative effects AdS space (negative vacuum energy) with unbroken SUSY and stabilized volume Uplifting AdS space to a metastable dS space (positive vacuum energy) by adding anti-D3 brane (or D7 brane with fluxes) AdS minimumMetastable dS minimum

16. Inflation with stabilized volume Use KKLT volume stabilization Use KKLT volume stabilization Kachru, Kallosh, Linde, Maldacena, McAllister, Trivedi 2003 Introduce the inflaton field with the potential which is flat due to shift symmetry Introduce the inflaton field with the potential which is flat due to shift symmetry Break shift symmetry either due to superpotential or due to radiative corrections Break shift symmetry either due to superpotential or due to radiative corrections Hsu, Kallosh, Prokushkin 2003 Koyama, Tachikawa, Watari 2003 Firouzjahi, Tye 2003 Hsu, Kallosh 2004 Alternative approach: Modifications of kinetic terms in the strong coupling regime Silverstein and Tong, 2003 Alternative approach: Modifications of kinetic terms in the strong coupling regime Silverstein and Tong, 2003

17. String inflation and shift symmetry Hsu, Kallosh, Prokushkin 2003

18. Why shift symmetry? It is not just a requirement which is desirable for inflation model builders, but, in a certain class of string theories, it is an unavoidable consequence of the mathematical structure of the theory Hsu, Kallosh, 2004

19. The Potential of the Hybrid D3/D7 Inflation Model is a hypermultiplet is an FI triplet

20. In many F and D-term models the contribution of cosmic strings to CMB anisotropy is too large This problem disappears for very small coupling g Another solution is to add a new hypermultiplet, and a new global symmetry, which makes the strings semilocal and topologically unstable

21. Semilocal Strings are Topologically Unstable Achucarro, Borill, Liddle, 98

22. D3/D7 with two hypers Detailed brane construction - D-term inflation dictionary Dasgupta, Hsu, R.K., A. L., Zagermann, hep-th/0405247 Resolving the problem of cosmic string production: additional global symmetry, no topologically stable strings, only semilocal strings, no danger Confirmation of Urrestilla, Achucarro, Davis; Binetruy, Dvali, R. K.,Van Proeyen Resolving the problem of cosmic string production: additional global symmetry, no topologically stable strings, only semilocal strings, no danger Confirmation of Urrestilla, Achucarro, Davis; Binetruy, Dvali, R. K.,Van Proeyen Brane construction of generalized D-term inflation models with additional global or local symmetries due to extra branes and hypermultiplets.

23. Bringing it all together: Double Uplifting First uplifting: KKLT KKL, in progress

24. Second uplifting in D3/D7 model

25. Inflationary potential at as a function of S and Shift symmetry is broken only by quantum effects

26. Potential of hybrid inflation with a stabilized volume modulus

27. For two hypers: Inflaton potential: Symmetry breaking potential:

28. Can we have eternal inflation in such models? Yes, by combining these models with the ideas of string theory landscape

29. String Theory Landscape Perhaps 10 different vacua 100

30. de Sitter expansion in these vacua is eternal. It creates quantum fluctuations along all possible flat directions and provides necessary initial conditions for the low-scale inflation

31. Landscape of eternal inflation

32. Self-reproducing Inflationary Universe

33. Finding the way in the landscape Anthropic Principle : Love it or hate it but use it Vacua counting : Know where you can go Moduli trapping : Live in the most beautiful valleys Anthropic Principle : Love it or hate it but use it Vacua counting : Know where you can go Moduli trapping : Live in the most beautiful valleys

34. Two possible regimes: Resurrection: From any dS minimum one can always jump back with probability e ΔS, and experience a stage of inflation Eternal youth : A much greater fraction of the total volume is produced due to eternal jumps in dS space at large energy density, and subsequent tunneling followed by chaotic inflation Resurrection: From any dS minimum one can always jump back with probability e ΔS, and experience a stage of inflation Eternal youth : A much greater fraction of the total volume is produced due to eternal jumps in dS space at large energy density, and subsequent tunneling followed by chaotic inflation

35. Quantum effects lead to particle production which result in moduli trapping near enhanced symmetry points These effects are stronger near the points with greater symmetry, where many particles become massless This may explain why we live in a state with a large number of light particles and (spontaneously broken) symmetries Quantum effects lead to particle production which result in moduli trapping near enhanced symmetry points These effects are stronger near the points with greater symmetry, where many particles become massless This may explain why we live in a state with a large number of light particles and (spontaneously broken) symmetries Beauty is Attractive Kofman, A.L., Liu, McAllister, Maloney, Silverstein: hep-th/0403001 also Silverstein and Tong, hep-th/0310221 Kofman, A.L., Liu, McAllister, Maloney, Silverstein: hep-th/0403001 also Silverstein and Tong, hep-th/0310221

36. Basic Idea Consider two interacting moduli with potential Suppose the field φ moves to the right with velocity. Can it create particles  ? Nonadiabaticity condition: is related to the theory of preheating after inflation Kofman, A.L., Starobinsky 1997 It can be represented by two intersecting valleys

37. V φ When the field φ passes the (red) nonadiabaticity region near the point of enhanced symmetry, it created particles χ with energy density proportional to φ. Therefore the rolling field slows down and stops at the point when Then the field falls down and reaches the nonadiabaticity region again…

38. V φ When the field passes the nonadiabaticity region again, the number of particles  (approximately) doubles, and the potential becomes two times more steep. As a result, the field becomes trapped at distance that is two times smaller than before.

39. Trapping of a real scalar field

40. Thus anthropic and statistical considerations are supplemented by a dynamical selection mechanism, which may help us to understand the origin of symmetries in our world.