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A cosmological model describing the early inflation, the intermediate decelerating expansion, and the late accelerating expansion by a quadratic equation of state

We develop a cosmological model based on a quadratic equation of state p/c^2=-(α+1){ρ^2}/{ρ_P}+αρ-(α+1)ρ_Λ (where ρ_P is the Planck density and ρ_Λ the cosmological density) "unifying" vacuum energy and dark energy in the spirit of a generalized Chaplygin gas model. For $ρ\rightarrow ρ_P$, it reduces to p=-ρc^2 leading to a phase of early accelerated expansion (early inflation) with a constant density equal to the Planck density ρ_P (vacuum energy). For $ρ_Λ\llρ\ll ρ_P$, we recover the standard linear equation of state p=αρc^2 describing radiation (α=1/3) or pressureless matter (α=0) and leading to an intermediate phase of decelerating expansion. For $ρ\rightarrow ρ_Λ$, we get p=-ρc^2 leading to a phase of late accelerated expansion (late inflation) with a constant density equal to the cosmological density ρ_Λ (dark energy). We show a nice symmetry between the early universe (vacuum energy + α-fluid) and the late universe (α-fluid + dark energy). In our model, they are described by two polytropic equations of state with index n=+1 and n=-1 respectively. Furthermore, the Planck density ρ_P in the early universe plays a role similar to the cosmological density ρ_Λ in the late universe. They represent fundamental upper and lower density bounds differing by 122 orders of magnitude. This quadratic equation of state leads to a fully analytical model describing the evolution of the universe from the early inflation (Planck era) to the late accelerated expansion (de Sitter era). These two phases are bridged by a decelerating algebraic expansion (α-era). This model does not present any singularity at t=0 and exists eternally in the past. It admits a scalar field interpretation based on a quintessence field or a tachyon field.