Models of universe with a polytropic equation of state: I. The early universe

Abstract.

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. Concerning the linear equation of state $p=\alpha\rho c^{2}$, we assume $-1\le\alpha\le 1$. This equation of state describes radiation ( $\alpha=1/3$ or pressureless matter ($\alpha = 0$. Concerning the polytropic equation of state $p=k\rho^{1+1/n}c^{2}$, we remain very general allowing the polytropic constant k and the polytropic index n to have arbitrary values. In this paper, we consider positive indices n > 0 . In that case, the polytropic component dominates the linear component in the early universe where the density is high. For $\alpha = 1/3$, n = 1 and $k=-4/(3\rho_{P})$, where $\rho_{P}=5.16 10^{99}$ g/m³ is the Planck density, we obtain a model of early universe describing the transition from the vacuum energy era to the radiation era. The universe exists at any time in the past and there is no primordial 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/m³ (vacuum energy) that brings it from the Planck size $l_{P}=1.62 10^{-35}$ m at t = 0 to a size $a_{1}=2.61 10^{-6}$ m at $t_{1}=1.25 10^{-42}$ s (corresponding to about 23.3 Planck times $t_{P}=5.39 10^{-44}$ s). 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 enters in the radiation era and evolves like in the standard model. We describe the transition from the vacuum energy era to the radiation era by analogy with a second-order phase transition where the Planck constant ℏ plays the role of finite-size effects (the standard Big Bang theory is recovered for ℏ = 0 .