A simple model of universe with a polytropic equation of state

TL;DR

This paper introduces a simple universe model with a generalized polytropic equation of state that describes both early inflationary and late accelerating phases, highlighting a symmetry between them and proposing a new perspective on the cosmological constant problem.

Contribution

It presents a unified polytropic model of the universe that captures early and late cosmic evolution, linking the cosmological constant to fundamental physics constants.

Findings

Models early universe without singularity and describes transition to radiation era.

Explains late universe transition to dark energy dominated acceleration.

Highlights symmetry between early and late universe evolution.

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 $\rho_P=5.16 10^{99} g/m^3$ 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.