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A simple model of universe with a polytropic equation of state

We construct a simple model of universe with a generalized equation of state $p=(α+kρ^{1/n})ρc^2$ having a linear component $p=αρc^2$ and a polytropic component $p=kρ^{1+1/n}c^2$. For $α=1/3$, $n=1$ and $k=-4/(3ρ_P)$, where $ρ_P=5.16 10^{99} g/m^3$ is the Planck density, this equation of state provides a model of the early universe without singularity describing the transition between the pre-radiation era and the radiation era. The universe starts from $t=-\infty$ but, when $t<0$, its size is less than the Planck length $l_P=1.62 10^{-35} m$. The universe undergoes an inflationary expansion that brings it to a size $a_1=2.61 10^{-6} m$ on a timescale of a few Planck times $t_P=5.39 10^{-44} s$. When $t\gg t_P$, the universe decelerates and enters in the radiation era. For $α=0$, $n=-1$ and $k=-ρ_Λ$, where $ρ_Λ=7.02 10^{-24} g}/m^3$ is the cosmological density, this equation of state describes the transition from a decelerating universe dominated by baryonic and dark matter to an accelerating universe dominated by dark energy (second inflation). The transition takes place at a size $a_2=8.95 10^{25} m$ corresponding to a time of the order of the cosmological time $t_Λ=1.46 10^{18} s$. This polytropic model reveals a nice "symmetry" between the early and late evolution of the universe, the cosmological constant $Λ$ in the late universe playing a role similar to the Planck constant $\hbar$ in the early universe. 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.