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We develop a cosmological model based on a quadratic equation of statep/c^2=-(\alpha+1){\rho^2}/{\rho_P}+\alpha\rho-(\alpha+1)\rho_{\Lambda} (where\rho_P is the Planck density and \rho_{\Lambda} the cosmological density)"unifying" vacuum energy and dark energy in the spirit of a generalizedChaplygin gas model. For $\rho\rightarrow \rho_P$, it reduces to p=-\rho c^2leading to a phase of early accelerated expansion (early inflation) with aconstant density equal to the Planck density \rho_P (vacuum energy). For$\rho_{\Lambda}\ll\rho\ll \rho_P$, we recover the standard linear equation ofstate p=\alpha \rho c^2 describing radiation (\alpha=1/3) or pressurelessmatter (\alpha=0) and leading to an intermediate phase of deceleratingexpansion. For $\rho\rightarrow \rho_{\Lambda}$, we get p=-\rho c^2 leading toa phase of late accelerated expansion (late inflation) with a constant densityequal to the cosmological density \rho_{\Lambda} (dark energy). We show a nicesymmetry between the early universe (vacuum energy + \alpha-fluid) and the lateuniverse (\alpha-fluid + dark energy). In our model, they are described by twopolytropic equations of state with index n=+1 and n=-1 respectively.Furthermore, the Planck density \rho_P in the early universe plays a rolesimilar to the cosmological density \rho_{\Lambda} in the late universe. Theyrepresent fundamental upper and lower density bounds differing by 122 orders ofmagnitude. This quadratic equation of state leads to a fully analytical modeldescribing the evolution of the universe from the early inflation (Planck era)to the late accelerated expansion (de Sitter era). These two phases are bridgedby a decelerating algebraic expansion (\alpha-era). This model does not presentany singularity at t=0 and exists eternally in the past. It admits a scalarfield interpretation based on a quintessence field or a tachyon field.

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