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 . The linear equation of state p = \alpha \rho c ^ { 2 } describes radiation ( \alpha = 1 / 3 ) , pressureless matter ( \alpha = 0 ) , stiff matter ( \alpha = 1 ) , and vacuum energy ( \alpha = -1 ) . The polytropic equation of state p = k \rho ^ { 1 + 1 / n } c ^ { 2 } may be due to Bose-Einstein condensates with repulsive ( k > 0 ) or attractive ( k < 0 ) self-interaction , or have another origin . In this paper , we consider negative indices n < 0 . In that case , the polytropic component dominates in the late universe where the density is low . For \alpha = 0 , n = -1 and k = - \rho _ { \Lambda } , we obtain a model of late universe describing the transition from the matter era to the dark energy era . Coincidentally , we live close to the transition between these two periods , corresponding to a _ { 2 } = 8.95 10 ^ { 25 } { m } and t _ { 2 } = 2.97 10 ^ { 17 } { s } . The universe exists eternally in the future and undergoes an inflationary expansion with the cosmological density \rho _ { \Lambda } = 7.02 10 ^ { -24 } { g } / { m } ^ { 3 } on a timescale t _ { \Lambda } = 1.46 10 ^ { 18 } { s } . For \alpha = 0 , n = -1 and k = \rho _ { \Lambda } , we obtain a model of cyclic universe appearing and disappearing periodically . If we were living in this universe , it would disappear in about 2.38 billion years . We make the connection between the early and the late universe and propose a simple equation describing the whole evolution of the universe . This leads to a model of universe that is eternal in past and future without singularity ( aioniotic universe ) . It generalizes the \Lambda CDM model by removing the primordial singularity ( Big Bang ) . This model exhibits a nice “ symmetry ” between an early and late phase of inflation , the cosmological constant in the late universe playing the same role as the Planck constant in the early universe . 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 . We determine the potential of the scalar field ( quintessence , tachyon field ) corresponding to the generalized equation of state p = ( \alpha \rho + k \rho ^ { 1 + 1 / n } ) c ^ { 2 } . We also propose a unification of pre-radiation , radiation and dark energy through the quadratic equation of state p / c ^ { 2 } = -4 \rho ^ { 2 } / 3 \rho _ { P } + \rho / 3 - 4 \rho _ { \Lambda } / 3 .