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 = (α + kρ1/n)ρc2 having a linear component p = αρc2 and a polytropic component p = kρ1+1/nc2. For α = 1/3, n = 1 and k = –4/(3ρP), where ρp = 5.16 × 1099 g/m3 is the Planck density, the equation of state provides a model of early universe without singularity describing the transition between the vacuum energy era (Planck era) and the radiation era. The universe starts from t = −∞ but, when t < 0, its size is less than the Planck length lp = 1.62 × 10−35 m, requiring a quantum gravity treatment. For t ≥ 0, the universe undergoes an accelerated expansion (early inflation) that brings it from the Planck size lP = 1.62 × 10−35 m to a size a1 = 2.61 × 10−6 m on a timescale of about 23.3 Planck times tP = 5.39 × 10−44 s. When t > t1 = 23.3tP = 1.25 × 10−42 s, the universe decelerates and enters in the radiation era. For α = 0, n = −1 and k = −ρΛ, where ρΛ = 7.02 × 10−24g/m3 is the cosmological density, the equation of state p = −ρΛc2 describes the transition from a decelerating universe dominated by dark matter to an accelerating universe (late inflation) dominated by dark energy (de Sitter era). This transition takes place at a size a2 = 0.204lΛ = 8.95 × 1025m corresponding to a time t2 = 0.203tΛ = 2.97 × 1017s where lΛ = 4.38 × 1026m is the cosmological length and tΛ = 1.46 × 1018 s the cosmological time. The present universe turns out to be just at the transition (t0 ∼ t2). Our model generalizes the standard ACDM model by incorporating naturally a phase of early inflation that avoids the primordial singularity. Furthermore, it reveals a nice “symmetry” between the early universe (vacuum energy + radiation) and the late universe (dark matter + dark energy). They are described by two polytropic equations of state with index n = +1 and n = −1 respectively. Furthermore, the cosmological constant Λ in the late universe plays a role similar to the Planck constant ℏ in the early universe. The mathematical formulae in the early and in the late universe are strikingly symmetric. 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 123 orders of magnitude. They are responsible for a phase of inflation in the early and late universe. The cosmological constant “problem” may be a false problem. Our model does not present any singularity and exists eternally in the past and in the future (aioniotic universe). On the other hand, it admits a scalar field interpretation based on an inflaton, quintessence, or tachyonic field. This correspondence is interesting because the early inflation and the late acceleration of the universe are usually described in terms of a scalar field. We determine the potential and the mass of this scalar field both in the early and late universe.