Models of universe with a polytropic equation of state: I. The early
  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 positive indices $n>0$. In that case, the
polytropic component dominates in the early universe where the density is high.
For $\alpha=1/3$, $n=1$ and $k=-4/(3\rho_P)$, we obtain a model of early
universe describing the transition from a pre-radiation era to the radiation
era. The universe exists at any time in the past and there is no singularity.
However, for $t<0$, its size is less than the Planck length $l_P=1.62 10^{-35}
m$. In this model, the universe undergoes an inflationary expansion with the
Planck density $\rho_P=5.16 10^{99} g/m^3$ that brings it to a size $a_1=2.61
10^{-6} m$ at $t_1=1.25 10^{-42} s$ (about 20 Planck times $t_P$). For
$\alpha=1/3$, $n=1$ and $k=4/(3\rho_P)$, we obtain a model of early universe
with a new form of primordial singularity: The universe starts at t=0 with an
infinite density and a finite radius $a=a_1$. Actually, this universe becomes
physical at a time $t_i=8.32 10^{-45} s$ from which the velocity of sound is
less than the speed of light. When $a\gg a_1$, the universe evolves like in the
standard model. We describe the transition from the pre-radiation era to the
radiation era by analogy with a second order phase transition where the Planck
constant $\hbar$ plays the role of finite size effects (the standard Big Bang
theory is recovered for $\hbar=0$).
