Density perturbations and the cosmological constant from inflationary landscapes

An anthropic understanding of the cosmological constant requires that the vacuum energy at late time scans from one patch of the universe to another. If the vacuum energy during inflation also scans, the various patches of the universe acquire exponentially differing volumes. In a generic landscape with slow-roll inflation, we find that this gives a steeply varying probability distribution for the normalization of the primordial density perturbations, resulting in an exponentially small fraction of observers measuring the Cosmic Background Explorer value of ${10}^{-5}$. Inflationary landscapes should avoid this “$\sigma$ problem,” and we explore features that can allow them to do that. One possibility is that, prior to slow-roll inflation, the probability distribution for vacua is extremely sharply peaked, selecting essentially a single anthropically allowed vacuum. Such a selection could occur in theories of eternal inflation. A second possibility is that the inflationary landscape has a special property: although scanning leads to patches with volumes that differ exponentially, the value of the density perturbation does not vary under this scanning. This second case is preferred over the first, partly because a flat inflaton potential can result from anthropic selection, and partly because the anthropic selection of a small cosmological constant is more successful.