Probabilities in the Bousso–Polchinski multiverse

Abstract

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 .

Anthropic prediction for a large toy landscape

“…However, the first attempt to calculate this distribution for the Bousso-Polchinski and Arkani-Hamed-Dimopolous-Kachru landscape models [29,30] revealed a staggered distribution, suggesting a conflict with anthropic predictions.…”

“…But as in Ref. [29], we can make the approximation that all upward transition rates are tiny compared to all downward transition rates from a given vacuum (see also the appendix). In that approximation, we can compute probabilities as follows.…”

“…However, to allow for numerical solution, Refs. [29,30] used models with a relatively small number of vacua and worked only in a first-order approximation.…”

Predicting the cosmological constant with the scale-factor cutoff measure

“…With scalefactor slicing, all regions would expand as a 3 = e 3t if it were not for the continuous loss of volume to terminal vacua with negative or zero Λ. Because of this loss, the value of γ is slightly smaller than 3, and the difference is determined mostly by the rate of decay of the slowestdecaying (dominant) vacuum in the landscape [44],…”

“…The conditions of validity of this heuristic argument have been studied in simple landscape models [44,50,51], with the conclusion that it does in fact apply to a wide class of models. Here, we shall assume that Eq.…”