On likely values of the cosmological constant - art. no. 083502

We discuss models in which the smallness of the effective vacuum energy den sity 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 eff ects. In such models. the probability distribution for rho(Lambda) is a pro duct of an a priori distribution P-*(rho(Lambda)) 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)). To determine P-*, we consider inflationary models in which the role of the vacuum energy is pla yed by a slowly varying potential of some scalar field. We show that the re sulting distribution depends on the shape of the potential and generally ha s 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 ca lculate the (final) probability distributions for rho(Lambda) and for t(G)/ t(Lambda) in simple models with power-law potentials. For some of these mod els, the agreement with the observationally suggested values of rho(Lambda) is better than with a Rat a priori distribution. We also discuss a quantum -cosmological approach in which rho(Lambda) takes different values in diffe rent disconnected universes and argue that Weinberg's conjecture is not val id in this case as well. Finally, we extend our analysis to models of quint essence, with similar conclusions.