We construct models of universe with a generalized equation of state \documentclass[12pt]{minimal}
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\begin{document}$ p=(\alpha \rho+k\rho^{1+1/n})c^{2}$\end{document} having a linear component and a polytropic component. Concerning the linear equation of state \documentclass[12pt]{minimal}
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\begin{document}$ p=\alpha\rho c^{2}$\end{document}, we assume \documentclass[12pt]{minimal}
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\begin{document}$ -1\le \alpha\le 1$\end{document}. This equation of state describes radiation (\documentclass[12pt]{minimal}
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\begin{document}$ \alpha=1/3$\end{document}) or pressureless matter (\documentclass[12pt]{minimal}
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\begin{document}$ \alpha=0$\end{document}). Concerning the polytropic equation of state \documentclass[12pt]{minimal}
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\begin{document}$ p=k\rho^{1+1/n} c^{2}$\end{document}, we remain very general allowing the polytropic constant k and the polytropic index n to have arbitrary values. In this paper, we consider negative indices n < 0. In this case, the polytropic component dominates the linear component in the late universe where the density is low. For \documentclass[12pt]{minimal}
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\begin{document}$ \alpha=0$\end{document}, \documentclass[12pt]{minimal}
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\begin{document}$ n=-1$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$ k=-\rho_{\Lambda}$\end{document}, where \documentclass[12pt]{minimal}
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\begin{document}$ \rho_{\Lambda}=7.02 10^{-24}$\end{document} g/m3 is the cosmological density, we obtain a model of late universe describing the transition from the matter era to the dark energy era. The universe exists at any time in the future and there is no singularity. It undergoes an inflationary expansion (exponential acceleration) with the cosmological density \documentclass[12pt]{minimal}
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\begin{document}$ \rho_{\Lambda}=7.02 10^{-24}$\end{document} g/m3 (dark energy) on a timescale \documentclass[12pt]{minimal}
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\begin{document}$ t_{\Lambda}=1.46 10^{18}$\end{document} s. Coincidentally, we live close to the transition between the matter era and the dark energy era, corresponding to a size \documentclass[12pt]{minimal}
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\begin{document}$ a_{2}=8.95 10^{25}$\end{document} m and a time \documentclass[12pt]{minimal}
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\begin{document}$ t_{2}=2.97 10^{17}$\end{document} s. For \documentclass[12pt]{minimal}
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\begin{document}$ \alpha=0$\end{document}, \documentclass[12pt]{minimal}
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\begin{document}$ n=-1$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$ k=\rho_{\Lambda}$\end{document}, we obtain a model of cyclic universe appearing and disappearing periodically. If we were living in this universe, it would disappear in about 2.38 billion years. We make the connection between the early and the late universe and propose a simple equation describing the whole evolution of the universe. This leads to a model of universe that is eternal in past and future without singularity (aioniotic universe). It refines the standard \documentclass[12pt]{minimal}
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\begin{document}$ \Lambda$\end{document} CDM model by incorporating naturally a phase of early inflation and removing the primordial singularity (Big Bang). This model exhibits a nice “symmetry” between the early and the late universe, the cosmological constant in the late universe playing the same role as the Planck constant in the early universe. The pressure is successively negative (early inflation), positive (radiation and matter eras), and negative again (late inflation). We interpret the cosmological constant as a fundamental constant of nature describing the “cosmophysics” just like the Planck constant describes the “microphysics”. The Planck density and the cosmological density represent fundamental upper and lower bounds differing by 122 orders of magnitude. The cosmological constant “problem” may be a false problem. We determine the potential of the scalar field (quintessence, tachyon field) corresponding to the generalized equation of state \documentclass[12pt]{minimal}
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\begin{document}$ p=(\alpha \rho+k\rho^{1+1/n})c^{2}$\end{document}. We also propose a unification of vacuum energy, radiation, and dark energy through the quadratic equation of state \documentclass[12pt]{minimal}
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\begin{document}$ p/c^{2}=-4\rho^{2}/3\rho_{P}+\rho/3 - 4\rho_{\Lambda}/3$\end{document}.
