A simple model of universe with a polytropic equation of state

Abstract

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 ${\rho }_P=5.1610^{99}g/m^3$ 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 $l_P=1.6210^{-35}m$. The universe undergoes an inflationary expansion that brings it to a size $a_1=2.6110^{-6}m$ on a timescale of a few Planck times $t_P=5.3910^{-44}s$. When $t\gg t_P$, the universe decelerates and enters in the radiation era. For $\alpha =0$, $n=-1$ and $k=-{\rho }_{\Lambda }$, where $ρ_Λ=7.02 10^{-24} g}/m^3$ 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 $a_2=8.9510^{25}m$ corresponding to a time of the order of the cosmological time $t_{\Lambda }=1.4610^{18}s$. 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 $\hslash$ 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.

arXiv admin note: text overlap with arXiv:1208.0797,arXiv:1208.0801