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 $p/c^2=-4\rho^2/3\rho_P+\rho/3-4\rho_{\Lambda}/3$. 
Pierre-Henri Chavanis
astro-ph.CO