1. PROBABILITIES IN THE LANDSCAPE Alex Vilenkin Tufts Institute of Cosmology

2. STRING THEORY PREDICTS MULTIPLE VACUA WITH DIFFERENT CONSTANTS OF NATURE Bousso & Polchinski (2000) Susskind (2003) Douglas (2003) Different compactifications, branes, fluxes, etc.

3. “THE LANDSCAPE”

4. Eternal inflation the entire landscape will be explored. Transitions between vacua through bubble nucleation 1 1 2 2 2 High-energy vacua inflation

5. The multiverse You are here

6. We will not be able to calculate (all) constants from first principles. KEY PROBLEM: find the probability distribution for the constants (the measure problem). P j ~ P j (V) n j (obs) We cannot calculate, but in some interesting cases fraction of baryons clustered in galaxies. -- problems with infinities. THIS TALK

7. Spacetime structure i+i+ Structure of a bubble (pocket universe) Bubble wall Open FRW

8. Measure based on a global time coordinate t = const Calculate P j (V) on a hypersurface t = const. BUT: the result depends on what we use as t. Linde, Linde & Mezhlumian (1994) A substantial contribution to P j (V) comes from new bubbles depends on how the surface is drawn.

9. A NEW PROPOSAL FOR P j : A pocket-based measure. Introduce step by step, starting from a simple model. Based on: Garriga, Schwartz-Perlov, A.V. & Winitzki (2005) Easther, Lim & Martin (2005)

10. Only one type of bubbles A scalar field X X is randomized by quantum fluctuations during inflation varies within the bubbles. Problem: find P (V) (X). X(r,t) STEP 1: A simple model

11. Proposed solution: Calculate P (V) (X) in one pocket. It does not matter which one: all pockets are statistically equivalent. Infinite spacelike hypersurface Garriga, Tanaka & A.V. (1999) Vanchurin, A.V. & Winitzki (2000) If there is some inflation inside bubbles, with an X-dependent expansion factor Z(X), then

12. Suppose there are different types of bubbles. To compare the numbers of observers, we want to sample “equal” comoving volumes in all types of bubbles. But what exactly does this mean? STEP 2:

13. Comoving reference scale R j : All bubbles are identical at small t. Choose R j (t) to be the same in all pockets at some Then, up to a constant, Expansion factor since t ~ H -1. Note: large inflation inside bubbles is again rewarded!

14. STEP 3: Some bubbles may be more abundant than others need to introduce the bubble abundance factor p j. Definition of p j is tricky, since the total number of bubbles is infinite. What fraction of all integers are odd? 1,2,3,4, … vs. 1, 2,4, 3, 6,8, 5, …

15. A PROPOSAL FOR p j Garriga, Schwartz-Perlov, A.V. & Winitzki (2005) Include only bubbles bigger than some small comoving size. Then let. independent of initial conditions (dominated by bubbles formed at late times) does not depend on the choice of.

16. An equivalent proposal A randomly selected set of N comoving worldlines 22 122 3 4 Easther, Lim & Martin (2005) Count only bubbles that intersect at least one of the worldlines. Equivalent in the limit of large N.

17. -- fraction of co-moving volume in vacuum of type i Gained from other vacua Lost to other vacua Nucleation rate of bubbles of type “i” in vacuum “j”. Highest nonvanishing eigenvalue of (it can be shown to be negative). Corresponding eigenvector (it is non-degenerate). E.g., if all vacua emanate from a single false vacuum, then It can be shown that (as intuitively expected) Calculation of p j Garriga, Schwartz-Perlov, A.V. & Winitzki (2005)

18. The full distribution is Z j is mostly determined by the amount of slow-roll inflationary expansion inside the bubbles. P j (V) is proportional to the volume expansion factor during the slow roll -- as intuitively expected.

19. AN APPLICATION: Q Feldstein, Hall & Watari (2005) Garriga & A.V. (2005) Hall, Watari & Yanagida (2006)

20. V0V0 A simple model: is variable is fixed Amplitude of density perturbations: Small Q are exponentially favored.

21. Anthropic range: 10 -6 < Q < 10 -4 Tegmark & Rees (1998) Q obs ~ 10 -5 -- very hard to explain. THE Q CATASTROPHE The catastrophe is likely to persist in more general models: Z depends exponentially on the inflaton potential, and therefore on Q. A POSSIBLE RESOLUTION: Q may be due to some field other than inflaton (e.g., curvaton). Then Q is not correlated with Z. Linde & Mukhanov (97), Enquist & Sloth (02), Lyth & Wands (02), Dvali, Gruzinov & Zaldarriaga (04), Kofman (03).

22. CONCLUSIONS We discussed a proposal for the volume distribution of the constants in the multiverse: P j are independent of gauge & of initial conditions. How uniquely is P j specified by these requirements? Ultimate test – comparison with observations. Some attempts in this direction: -- Pogosian, Tegmark & A.V. (2004) -- Tegmark, Aguirre, Rees & Wilczek (2005) Q -- Feldstein, Hall & Watari (2005), Garriga & A.V. (2005), -- Schwartz-Perlov & A.V. (2006)

24. If in addition there is a continuous variable X, Normalized distribution at in pockets of type j