1. Inflationary multiverse and string theory
Andrei Linde

2. Contents: Inflation as a theory of a harmonic oscillator
Inflation in string theory
Vacuum stabilization and the gravitino mass
Initial conditions for inflation
Does our universe looks like a sphere or like a bagel?
Eternal inflation and string theory landscape: From a bagel to a fractal

3. Inflation as a theory of a harmonic oscillator
Eternal Inflation

4. Equations of motion: Einstein: Klein-Gordon:
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. Inflation makes the universe flat, homogeneous and isotropic
In this simple model the universe typically grows times during inflation.
Now we can see just a tiny part of the universe of size ct = 1010 light yrs. That is why the universe looks homogeneous, isotropic, and flat.

7. Generation of Quantum Fluctuations

8. WMAP and the temperature of the sky

9. Name Recognition
Stephen Hawking

10. 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…

11. A solution: shift symmetry
Kawasaki, Yamaguchi, Yanagida 2000
Equally good 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:

12. The volume stabilization problem:
Inflation in String Theory
The volume stabilization problem:
A potential of the theory obtained by compactification in string theory of type IIB:
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
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.
Giddings, Kachru, Polchinski 2001
Dilaton stabilization:
Kachru, Kallosh, A.L., Trivedi 2003
Volume stabilization: KKLT construction
Burgess, Kallosh, Quevedo, 2003
Maloney, Silverstein, Strominger, in non-critical string theory

13. Volume stabilization Basic steps of the KKLT scenario:
Kachru, Kallosh, A.L., Trivedi 2003
Basic steps of the KKLT scenario:
Start with a theory with runaway potential discussed above
Bend this potential down due to (nonperturbative) quantum effects
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
AdS minimum
Metastable dS minimum

14. 3H2 = VINFLATION < VBARRIER ~ VAdS/3 = m3/2
Inflation and the gravitino mass
V =
+ D/3
VAdS = - 3 eK |W|2
m3/2 == eK |W|2 =VAdS/3 2
3H2 = VINFLATION < VBARRIER ~ VAdS/3 = m3/2 2

15. Inflationary Unplifting
Vinfl < 3Vbarrier ~ 3VAdS ~ m23/2
minflaton << H ~ m3/2

16. Perhaps 10100 - 101000 different minima
String Theory Landscape
Perhaps different minima
Lerche, Lust, Schellekens 1987
Bousso, Polchinski; Susskind; Douglas, Denef,…

17. Example: Racetrack Inflation
Vbarrier = Vinflation
waterfall from the saddle point

18. A new class of KKLT models
Kallosh, A.L. hep-th/
One can obtain a supersymmetric Minkowski vacuum without any uplifting of the potential
Inflation in the new class of KKLT models can occur at H >> m3/2
Small mass of gravitino, no correlation with the height of the barrier and with the Hubble constant during inflation

19. One of the problems with string inflation is that inflation in such models starts relatively late. A typical closed universe will collapse before inflation begins. Open or flat universes would not collapse, but they are infinite, it is hard to make them...
Can we create a finite flat universe?
Yes we can!
Take a box (a part of a flat universe) and glue its opposite sides to each other. What we obtain is a torus, which is a topologically nontrivial flat universe.

20. The size of the torus (our universe) grows as t1/2, whereas the mean free path of a relativistic particle grows much faster, as t
Therefore until the beginning of inflation the universe remains smaller that the size of the horizon t

21. If the universe initially had a Planckian size (the smallest possible size), then within the cosmological time t >> 1 (in Planck units) particles run around the torus many times and appear in all parts of the universe with equal probability, which makes the universe homogeneous and keeps it homogeneous until the beginning of inflation
Zeldovich, Starobinsky 1984; Cornish, Starkman, Spergel 1996; A.L. hep-th/

22. Closed versus compact flat universe in quantum cosmology
tunneling
Closed universe
Wave function is exponentially suppressed at large scale factor a
Compact flat universe
Wave function is not exponentially suppressed

23. Spheres are expensive, bagels are free
Creation of a closed inflationary universe, and of an infinite flat or open universe is exponentially less probable than creation of a compact topologically nontrivial flat or open universe
Spheres are expensive, bagels are free
This generalizes the standard Kaluza-Klein idea that some spatial dimensions are compactified. Now it seems likely that all spatial dimensions are compactified. Some of them remain small (KKLT mechanism), whereas some other dimensions become large due to inflation

24. This does not necessarily mean that our universe looks like a torus
This does not necessarily mean that our universe looks like a torus. Inflation in string theory is always eternal, due to large number of metastable dS vacua (string theory landscape).
The new-born universe typically looks like a bagel, but the grown-up universe looks like an eternally growing fractal.

25. What is good in 4D may be bad in 10D
Consider creation of a 3D box (torus) with a large volume modulus field  From the 4D point of view, we create a universe which may have Planck size in all directions when created at the Planck density - should be easy to do it! 3D 3D 6D
Meanwhile from the 10D perspective, this box at the Planck time has a large size in 6 directions, corresponding to the large volume modulus  This means that the universe consists of MANY independent Planck size domains which should have similar properties; difficult to create.
This is a new version of the horizon problem: The probability must be exponentially suppressed.

26. We want to start with the universe which has a Planckian (or stringy) size in all 9 space directions. But this implies starting at small , which may lead to overshooting, and decompactification: V 

27. Second uplifting in D3/D7 model

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

29. Potential of hybrid inflation with a stabilized volume modulus
Unlike in the brane-antibrane scenario, inflation in D3/D7 model does not require fine-tuning

30. Is D3/D7 inflation eternal?
Condition for a slow-roll eternal inflation:
For D3/D7 model it implies that the inflaton field S should be greater than 104 g. But S describes the distance between the branes; it must be much smaller than the CY size. Thus eternal inflation is possible only for very small g.
One may improve the situation by increasing V without changing V’. This requires an additional uplifting. The idea is that by changing fluxes one can increase V without altering the flatness of the potential, which remains protected by shift symmetry.

31. The resulting scenario:
1) The universe eternally jumps from one dS vacuum to another. Each bubble containing a new dS vacuum from the point of view of an internal observer is an infinite open dS universe. This process definitely occurs, but 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 is in dS state with large V but with the flat D3/D7 direction. Quantum fluctuations during eternal inflation in this state push the inflaton field S in all directions along the D3/D7 valley.
3) Eventually, this state decays, and a bubble is produced which is an open inflationary universe containing all possible values of the inflaton field S in its different parts. D3/D7 slow-roll inflation begins and makes this universe flat. It produces particles, galaxies, and the participants of this conference.

32. Self-reproducing Inflationary Universe

33. Landscape of eternal inflation

34. > 0 Let 10500 flowers blossom = 0 < 0 Mao Zedong
Yen'an Talks
< 0