Models of universe with a polytropic equation of state: II. The late
  universe
  We construct models of universe with a generalized equation of state
$p=(\alpha \rho+k\rho^{1+1/n})c^2$ having a linear component and a polytropic
component. In this paper, we consider negative indices $n<0$. In that case, the
polytropic component dominates in the late universe where the density is low.
For $\alpha=0$, $n=-1$ and $k=-\rho_{\Lambda}$, we obtain a model of late
universe describing the transition from the matter era to the dark energy era.
The universe exists eternally in the future and undergoes an inflationary
expansion with the cosmological density $\rho_{\Lambda}=7.02 10^{-24} g/m^3$ on
a timescale $t_{\Lambda}=1.46 10^{18} s$. For $\alpha=0$, $n=-1$ and
$k=\rho_{\Lambda}$, we obtain a model of cyclic universe appearing and
disappearing periodically. If we were living in this universe, it would
disappear in about 2.38 billion years. We make the connection between the early
and the late universe and propose a simple equation describing the whole
evolution of the universe. This leads to a model of universe that is eternal in
past and future without singularity (aioniotic universe). This model exhibits a
nice "symmetry" between an early and late phase of inflation, the cosmological
constant in the late universe playing the same role as the Planck constant in
the early universe. The Planck density and the cosmological density represent
fundamental upper and lower bounds differing by 122 orders of magnitude. The
cosmological constant "problem" may be a false problem. We determine the
potential of the scalar field (quintessence, tachyon field) corresponding to
the generalized equation of state $p=(\alpha \rho+k\rho^{1+1/n})c^2$. We also
propose a unification of pre-radiation, radiation and dark energy through the
quadratic equation of state $p/c^2=-4\rho^2/3\rho_P+\rho/3-4\rho_{\Lambda}/3$.
