A simple model of universe with a polytropic equation of state
  We construct a simple model of universe with a generalized equation of state
$p=(\alpha +k\rho^{1/n})\rho c^2$ having a linear component $p=\alpha\rho c^2$
and a polytropic component $p=k\rho^{1+1/n}c^2$. For $\alpha=1/3$, $n=1$ and
$k=-4/(3\rho_P)$, where $\rho_P=5.16 10^{99} g/m^3$ 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 $\alpha=0$, $n=-1$ and
$k=-\rho_{\Lambda}$, where $\rho_{\Lambda}=7.02 10^{-24} g}/m^3$ 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 $t_{\Lambda}=1.46 10^{18} s$. This polytropic model
reveals a nice "symmetry" between the early and late evolution of the universe,
the cosmological constant $\Lambda$ 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.
