On likely values of the cosmological constant

We discuss models in which the smallness of the effective vacuum energy density ${\rho }_{\Lambda }$ and the coincidence of the time of its dominance $t_{\Lambda }$ with the epoch of galaxy formation $t_G$ are due to anthropic selection effects. In such models, the probability distribution for ${\rho }_{\Lambda }$ is a product of an a priori distribution ${\mathcal{P}}_{*}\left({\rho }_{\Lambda }\right)$ and of the number density of galaxies at a given ${\rho }_{\Lambda }$ (which is proportional to the number of observers who will detect that value of ${\rho }_{\Lambda }).\mathrm{}$ To determine ${\mathcal{P}}_{*},$ 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 ${\rho }_{\Lambda },$ 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 ${\rho }_{\Lambda }$ and for $t_G{/t}_{\Lambda }$ in simple models with power-law potentials. For some of these models, the agreement with the observationally suggested values of ${\rho }_{\Lambda }$ is better than with a flat a priori distribution. We also discuss a quantum-cosmological approach in which ${\rho }_{\Lambda }$ 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.