1. Inflation and String Cosmology Andrei Linde Andrei Linde

2. The Simplest Inflationary Model Eternal Inflation

3. Predictions of Inflation: 1) The universe should be homogeneous, isotropic and flat,  = 1 + O(10 -4 ) [    Observations: the universe is homogeneous, isotropic and flat,  = 1 + O(10 -2 ) 2)Inflationary perturbations should be gaussian and adiabatic, with flat spectrum, n s = 1+ O(10 -1 ) Observations: perturbations are gaussian and adiabatic, with flat spectrum, n s = 1 + O(10 -2 )

4. Any problems of principle? 1) We must introduce a small parameter to explain a small amplitude of density perturbations . To explain   one should have m =    Is it a real problem? 2)Transplanckian physics? These effects are not expected to affect basic features of inflationary scenario. In the worst case, one may expect minor corrections to the spectrum of perturbations. These effects vanish for low-scale inflation.

5. 3) Singularity problem This is NOT a problem of inflation. Moreover, inflationary predictions practically do not depend on the existence of the singularity. 4)Cosmological constant problem This is NOT a problem of inflation. Moreover, the only presently known solution of this problem requires inflation, in combination with anthropic principle and string theory landscape.

6. 5) Inflation requires initial homogeneity on scale greater than horizon In simplest models of chaotic inflation homogeneity is requires on the smallest possible scale, Planck length, which is not a problem. In low-scale inflation, such as new or hybrid inflation, this was a real problem. However, this problem was solved by considering inflation in compact topologically nontrivial flat or open universes: In this case homogeneity is required on Planck scale, as in chaotic inflation. A.L. hep-th/0408164

7. Are there any real problems of inflation? The main problem is to construct realistic models of inflation in the situation when the final theory of all fundamental interactions is still absent.

8. Inflation in String Theory The volume stabilization problem: A potential of the theory obtained by compactification in string theory of type IIB: The volume stabilization problem: A potential of the theory obtained by compactification in string theory of type IIB: The potential with respect to X and Y is very steep, these fields rapidly run down, and the potential energy V vanishes. We must stabilize these fields. Volume stabilization: KKLT construction Kachru, Kallosh, A.L., Trivedi 2003 Burgess, Kallosh, Quevedo, 2003 X and Y are canonically normalized field corresponding to the dilaton field and to the volume of the compactified space;  is the field driving inflation Dilaton stabilization: Giddings, Kachru, Polchinski 2001

9. Volume stabilization Basic steps of the KKLT scenario: AdS minimum Metastable dS minimum Kachru, Kallosh, A.L., Trivedi 2003 1) Start with a theory with runaway potential discussed above 2) Bend this potential down due to (nonperturbative) quantum effects 3) Uplift the minimum to the state with positive vacuum energy by adding a positive energy of an anti-D3 brane in warped Calabi-Yau space

10. Main conclusions after 2 years of investigation: It is possible to stabilize internal dimensions, and obtain an accelerating universe. Eventually, our part of the universe will decay and become ten-dimensional, but it will only happen in 10 10 120 years n Apparently, vacuum stabilization can be achieved in 10 100 - 10 1000 different ways. This means that the potential energy V of string theory may have 10 100 - 10 1000 minima where we (or somebody else) can enjoy life…

11. Related ideas existed long before the stringy landscape Example: Supersymmetric SU(5) V SU(5)SU(3)xSU(2)xU(1)SU(4)xU(1) Weinberg 1982: No way to tunnel from SU(5) to SU(3)xSU(2)XU(1) A.L 1983: Inflationary fluctuations bring us there

12. Self-reproducing Inflationary Universe

13. String Theory Landscape Perhaps 10 100 - 10 1000 different minima Bousso, Polchinski; Susskind; Douglas, Denef,… Lerche, Lust, Schellekens 1987

14. Stringy landscape provides us with a DISCRETE set of parameters corresponding to 10 1000 vacua of string theory. In addition, we may have many CONTINUOUS parameters, such as the amplitude of density perturbations, the ratio of dark matter to baryons, etc., which depend on cosmological dynamics.

15. The simplest curvaton model A.L., Mukhanov, astro-ph/9610212 Consider a light field  during inflation with Hubble constant H. Bunch-Davies distribution of fluctuations: The main contribution to these fluctuations is given by exponentially large wavelengths

16. Interpretation:Interpretation: Because of the fluctuations, curvaton typically takes values of the order + H 2 /m or - H 2 /m in domains separated by walls where it vanishes (shown as a shoreline in the figure below). A typical size of these domains is

17. Amplitude of the curvaton perturbations Amplitude of the curvaton perturbations The constant C shows the energy density of the curvaton particles produced during reheating. As far as we know, this contribution previously was ignored, but it can be very large. The amplitude of density perturbations depends on our position in the curvaton landscape. In the interior of the exponentially large islands the perturbations are (locally) gaussian. On a larger scale including many domains the perturbations are nongaussian. A.L., Mukhanov, astro-ph/0511736

18. Usually we assume that the amplitude of inflationary perturbations is constant,  ~ 10 -5 everywhere. However, in the curvaton scenario the value of  is different in different exponentially large parts of the universe. Curvaton Web A.L., Mukhanov, astro-ph/0511736

19. Consider inflaton and curvaton with masses M >> m Curvaton-Inflaton Transmutations Curvaton-Inflaton Transmutations  is a curvaton only near the walls   everywhere else it is an inflaton  The amplitude of perturbations is almost everywhere  but it grows near the walls, forming the curvaton web. Even if initially  >> , then during eternal inflation the fluctuations of the field  are generated, and it becomes much greater than  almost everywhere in the universe. Then the field  rolls down (its potential is more steep) and the curvaton  drives inflation, i.e. it becomes the inflaton. Bartolo, Liddle 2003, A.L., Mukhanov 2005

20. Dark Energy (Cosmological Constant) is about 73% of the cosmic pie. Why? What’s about Dark Matter, another 23% of the pie? Why there is 5 times more dark matter than ordinary matter?

21. Inflation and Cosmological Constant 1) Anthropic solutions of the CC problem using inflation and fluxes of antisymmetric tensor fields (A.L. 1984), multiplicity of KK vacua (Sakharov 1984), and slowly evolving scalar field (Banks 1984, A.L. 1986). All of these authors took for granted that we cannot live in the universe with 2) Derivation of the anthropic constraint (Weinberg 1987, Martel, Shapiro, Weinberg 1997) Three crucial steps in finding the anthropic solution of the CC problem: 3) String landscape (Bousso-Polchinski 2000, KKLT 2003, Susskind 2003, Douglas 2003,…)

22. Latest anthropic constraints on  Aguirre, Rees, Tegmark, and Wilczek, astro-ph/0511774 observed value

23. Dark matter: the axion scenario Standard lore: If the axion mass is smaller than 10 -5 eV, the amount of dark matter in the axion field contradicts observations, for a typical initial value of the axion field. Anthropic argument: Due to inflationary fluctuations, the amount of the axion dark matter is a CONTINUOUS RANDOM PARAMETER. We can live only in those parts of the universe where the initial value of the axion field was sufficiently small ( A.L. 1988).

24. Latest anthropic constraints on Dark Matter  Aguirre, Rees, Tegmark, and Wilczek, astro-ph/0511774 Anthropic predictions for Dark Matter are even better than the predictions for the cosmological constant ! observed value

25. Why do we live in a 4D space? Why do we live in a 4D space? Ehrenfest, 1917: Stable planetary and atomic systems are possible only in 4D space. Indeed, for D > 4 planetary system are unstable, whereas for D < 4 there is NO gravity forces between stars and planets. If one wants to suggest an alternative solution to a problem that is solved by anthropic principle, one is free to try. But it may be more productive to concentrate on many problems that do not have an anthropic solution. For example, there is no anthropic replacement for inflation

26. Two types of string inflation models: Moduli Inflation. Moduli Inflation. The simplest class of models. They use only the fields that are already present in the KKLT model. Brane inflation. Brane inflation. The inflaton field corresponds to the distance between branes in Calabi-Yau space. Historically, this was the first class of string inflation models. Moduli Inflation. Moduli Inflation. The simplest class of models. They use only the fields that are already present in the KKLT model. Brane inflation. Brane inflation. The inflaton field corresponds to the distance between branes in Calabi-Yau space. Historically, this was the first class of string inflation models.

27. Inflation in string theory KKLMMT brane-anti-brane inflation Racetrack modular inflation D3/D7 brane inflation Kahler modular inflation

28. D3/D7 inflation Unlike in the brane-antibrane scenario, inflation in D3/D7 model does not require fine-tuning because of the shift symmetry Herdeiro, Hirano, Kallosh, Dasgupta 2001, 2002

29. Let 10 500 flowers blossom Let 10 500 flowers blossom  < 0  = 0  > 0

30. In the beginning one has eternal inflation when the fields jumped from one de Sitter minimum to another. However, at some point the fields must stop jumping, as in old inflation, and start rolling, as in new or chaotic inflation: the last stage of inflation must be of the slow-roll type. Otherwise we would live in an empty open universe with  << 1. How can we create initial conditions for a slow-roll inflation after the tunneling?

31. V Initial Conditions for D3/D7 Inflation Slow roll inflation Eternal inflation in a valley with different fluxes The field drifts in the upper valley due to quantum fluctuations and then tunneling occurs due to change of fluxes inside a bubble H >> m H >>> m s In D3/D7 scenario flatness of the inflaton direction does not depend on fluxes

32. The resulting scenario: The resulting scenario: 1) The universe eternally jumps from one dS vacuum to another due to formation of bubbles. Each bubble contains a new dS vacuum. The bubbles contain no particles unless this process ends by a stage of a slow-roll inflation. Here is how: 2) At some stage the universe appears in dS state with a large potential but with a flat inflaton direction, as in D3/D7 model. Quantum fluctuations during eternal inflation in this state push the inflaton field S in all directions along the inflaton valley. 3) Eventually this state decays, and bubbles are produced. Each of these bubbles may contain any possible value of the inflaton field S, prepared by the previous stage. A slow-roll inflation begins and makes the universe flat. It produces particles, galaxies, and the participants of this conference:)