Inflationary cosmology leads to the picture of a "multiverse," involving an infinite number of (spatially infinite) post-inflationary thermalized regions, called pocket universes. In the context of theories with many vacua, such as the landscape of string theory, the effective constants of Nature are randomized by quantum processes during inflation. We discuss an analytic estimate for the volume distribution of the constants within each pocket universe. This is based on the conjecture that the field distribution is approximately ergodic in the diffusion regime, when the dynamics of the fields is dominated by quantum fluctuations (rather than by the classical drift). We then propose a method for determining the relative abundances of different types of pocket universes. Both ingredients are combined into an expression for the distribution of the constants in pocket universes of all types.

Using the recently introduced method to calculate bubble abundances in an eternally inflating spacetime, we investigate the volume distribution for the cosmological constant Λ in the context of the Bousso-Polchinski landscape model. We find that the resulting distribution has a staggered appearance which is in sharp contrast to the heuristically expected flat distribution. Previous successful predictions for the observed value of Λ have hinged on the assumption of a flat volume distribution. To reconcile our staggered distribution with observations for Λ, the BP model would have to produce a huge number of vacua in the anthropic range ∆Λ A of Λ, so that the distribution could conceivably become smooth after averaging over some suitable scale δΛ ≪ ∆Λ A .

We identify instantons representing vacuum decay in a 6-dimensional toy model for string theory flux compactifications, with the two extra dimensions compactified on a sphere. We evaluate the instanton action for tunneling between different flux vacua, as well as for the decompactification decay channel. The bubbles resulting from flux tunneling have an unusual structure. They are bounded by two-dimensional branes, which are localized in the extra dimensions. This has important implications for bubble collisions.

Topology-changing transitions between vacua of different effective dimensionality are studied in the context of a 6-dimensional Einstein-Maxwell theory. The landscape of this theory includes a 6d de Sitter vacuum (dS 6 ), a number of dS 4 × S 2 and AdS 4 × S 2 vacua, and a number of AdS 2 × S 4 vacua. We find that compactification transitions dS 6 → AdS 2 × S 4 occur through the nucleation of electrically charged black hole pairs, and transitions from dS 6 to dS 4 × S 2 and AdS 4 × S 2 occur through the nucleation of magnetically charged spherical black branes. We identify the appropriate instantons and describe the spacetime structure resulting from brane nucleation.

The successful anthropic prediction of the cosmological constant depends crucially on the assumption of a flat prior distribution. However, previous calculations in simplified landscape models showed that the prior distribution is staggered, suggesting a conflict with anthropic predictions. Here we analytically calculate the full distribution, including the prior and anthropic selection effects, in a toy landscape model with a realistic number of vacua, N ∼ 10 500 . We show that it is possible for the fractal prior distribution we find to behave as an effectively flat distribution in a wide class of landscapes, depending on the regime of parameter space. Whether or not this possibility is realized depends on presently unknown details of the landscape.