We construct a simple model of universe with a generalized equation of state p = ( \alpha + k \rho ^ { 1 / n } ) \rho c ^ { 2 } 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 } = 5.16 10 ^ { 99 } { g } / { m } ^ { 3 } is the Planck density , this equation of state provides a model of the early universe without singularity describing the transition between the pre-radiation era and the radiation era . The universe starts from t = - \infty but , when t < 0 , its size is less than the Planck length l _ { P } = 1.62 10 ^ { -35 } { m } . The universe undergoes an inflationary expansion that brings it to a size a _ { 1 } = 2.61 10 ^ { -6 } { m } on a timescale of a few Planck times t _ { P } = 5.39 10 ^ { -44 } { s } . When t \gg t _ { P } , the universe decelerates and enters in the radiation era . For \alpha = 0 , n = -1 and k = - \rho _ { \Lambda } , where \rho _ { \Lambda } = 7.02 10 ^ { -24 } { g } / { m } ^ { 3 } is the cosmological density , this equation of state describes the transition from a decelerating universe dominated by baryonic and dark matter to an accelerating universe dominated by dark energy ( second inflation ) . The transition takes place at a size a _ { 2 } = 8.95 10 ^ { 25 } { m } corresponding to a time of the order of the cosmological time t _ { \Lambda } = 1.46 10 ^ { 18 } { s } . The present universe turns out to be just at the transition ( t _ { 0 } \sim t _ { \Lambda } ) . This polytropic model reveals a nice “ symmetry ” between the early and late evolution of the universe , the cosmological constant \Lambda in the late universe playing a role similar to the Planck constant \hbar 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 .