Phantom dark energy, cosmic doomsday, and the coincidence problem

Phantom dark-energy models, with w<-1, are characterized by a future singularity and therefore a finite lifetime for the universe. Because the future singularity is triggered by the onset of dark-energy domination, the universe spends a significant fraction of its total lifetime in a state for which the dark-energy and matter densities are roughly comparable. We calculate, as a function of w, the fraction of the total lifetime of the universe for which the dark-energy and matter densities differ by less than the ratio r₀ in either direction. For r₀=10, this fraction varies from 1/3 to 1/8 as w varies from -1.5 to -1.1; the fraction is smaller for smaller values of r₀. This result indicates that the coincidence problem is significantly ameliorated in phantom-dominated cosmologies with a future singularity.