Anthropic Prediction for Λ and theQCatastrophe

We discuss probability distributions for the cosmological constant Lambda andthe amplitude of primordial density fluctuations Q in models where they bothare anthropic variables. With mild assumptions about the prior probabilities,the distribution P(Lambda,Q) factorizes into two independent distributions forthe variables Q and $y \propto \Lambda/Q^3$. The distribution for y is largelymodel-independent and is in a good agreement with the observed value of y. Theform of P(Q) depends on the origin of density perturbations. If theperturbations are due to quantum fluctuations of the inflaton, then P(Q) tendsto have an exponential dependence on Q, due to the fact that in such models Qis correlated with the amount of inflationary expansion. For simple models witha power-law potential, P(Q) is peaked at very small values of Q, far smallerthan the observed value of 10^{-5}. This problem does not arise incurvaton-type models, where the inflationary expansion factor is not correlatedwith Q.Comment: 20 pages, 1 figur