A simple model of universe describing the early inflation and the late accelerated expansion in a symmetric manner

We construct a simple model of universe which unifies vacuum energy and radiation on the one hand, and matter and dark energy on the other hand in the spirit of a generalized Chaplygin gas model. Specifically, the phases of early inflation and late accelerated expansion are described by a generalized equation of state p/c(2) = alpha rho + k rho(1+1/n) 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) = 5.1610(99) g/m(3) is the Planck density, this equation of state describes the transition between the vacuum energy era and the radiation era. For t >= 0, the universe undergoes an inflationary expansion that brings it from the Planck size l(P) = 1.6210(-35) m to a size a(1) = 2.61 10(-6) m on a timescale of about 23.3 Planck times t(P) = 5.3910(-44) s (early inflation). When t > t(1) = 23.3t(P), the universe decelerates and enters in the radiation era. We interpret the transition from the vacuum energy era to the radiation era as a second order phase transition where the Planck constant (h) over bar plays the role of finite size effects (the standard Big Bang theory is recovered for (h) over bar = 0). For alpha = 0, n = -1 and k = -rho(Lambda), where rho(Lambda) = 7.0210(-24) g/m(3) is the cosmological density, the equation of state p/c(2) = alpha rho + k rho(1+1/n) describes the transition from a decelerating universe dominated by pressureless matter (baryonic and dark matter) to an accelerating universe dominated by dark energy (late inflation). This transition takes place at a size a(2) = 0.204l(Lambda) corresponding to a time t(2) = 0.203t(Lambda) where l(Lambda) = 4.3810(26)m is the cosmological length and t(Lambda) = 1.4610(18) s the cosmological time. The present universe turns out to be just at the transition between these two periods (t(0) similar to t(2)). Our model gives the same results as the standard Lambda CDM model for t >> t(P) and completes it by incorporating a phase of early inflation for t < 23.3t(P) in a very natural manner. Furthermore, it reveals a nice symmetry between the early and the late evolution of the universe. The early universe is modeled by a polytrope n = +1 and the late universe by a polytrope n = -1. Furthermore, the cosmological constant. in the late universe plays a role similar to the Planck constant <(h)over bar> in the early universe. The mathematical formulae in the early and in the late universe are then 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 122 orders of magnitude. The cosmological constant problem may be a false problem. Finally, we show that our model admits a scalar field interpretation based on a quintessence field or a tachyon field.