Coincidence Problem in Cyclic Phantom Models of the Universe
  We examine cyclic phantom models for the universe, in which the universe is
dominated sequentially by radiation, matter, and a phantom dark energy field,
followed by a standard inflationary phase. Since this cycle repeats endlessly,
the Universe spends a substantial portion of its lifetime in a state for which
the matter and dark energy densities have comparable magnitudes, thus
ameliorating the coincidence problem. We calculate the fraction of time that
the universe spends in such a coincidental state and find that it is nearly the
same as in the case of a phantom model with a future big rip. In the limit
where the dark energy equation of state parameter, w, is close to -1, we show
that the fraction of time, f, for which the ratio of the dark energy density to
the matter density lies between r_1 and r_2, is f = -(1+w) ln [(\sqrt{r_2} +
\sqrt{1+r_2})/(\sqrt{r_1} + \sqrt{1+r_1})].
