On likely values of the cosmological constant

We discuss models in which the smallness of the effective vacuum energy density ρ_Λ and the coincidence of the time of its dominance t_Λ with the epoch of galaxy formation tG are due to anthropic selection effects. In such models, the probability distribution for ρ_Λ is a product of an a priori distribution P*(ρ_Λ) and of the number density of galaxies at a given ρ_Λ (which is proportional to the number of observers who will detect that value of ρ_Λ). To determine P*, we consider inflationary models in which the role of the vacuum energy is played by a slowly varying potential of some scalar field. We show that the resulting distribution depends on the shape of the potential and generally has a non-trivial dependence on ρ_Λ, even in the narrow anthropically allowed range. This is contrary to Weinberg's earlier conjecture that the a priori distribution should be nearly flat in the range of interest. We calculate the (final) probability distributions for ρ_Λ and for tG/t_Λ in simple models with power-law potentials. For some of these models, the agreement with the observationally suggested values of ρ_Λ is better than with a flat a priori distribution. We also discuss a quantum-cosmological approach in which ρ_Λ takes different values in different disconnected universes and argue that Weinberg's conjecture is not valid in this case as well. Finally, we extend our analysis to models of quintessence, with similar conclusions.