On a Relation of Vacuum Energy to the Hierarchy of Forces

Abstract: We investigate the relation between vacuum energy and the hierarchy of forces in habitable universes. The hierarchy of forces studied in \cite{fred} was bound by $\frac{\alpha_G}{\alpha} \le 10^{-34}$, using structure formation arguments which involve the fine structure constant $\alpha$ and the gravitational constant $\alpha_G$. Previously we showed that the requirement of vacuum domination epoch occuring after matter radiation equality time, places a bound on vacuum energy in terms of the density perturbation parameter Q. Here we impose a further condition: we require that at equality time the size of the gas cloud which forms a galaxy at virialization time, be smaller than the horizon size. The latter condition leads to an intriguing relation whereby vacuum energy is bound by a power law function of the hierarchy of forces. The constraint introduced by the hierarchy of forces on the amount of dark energy is suggestive of an unknown microphysical mechanism that relates vacuum energy to the other constants of nature, specifically to the gravitational and fine structure constants and to their hierarchy.