We develop a cosmological model based on a quadratic equation of state p / 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 generalized Chaplygin gas model .
For \rho \rightarrow \rho _ { P } , it reduces to p = - \rho c ^ { 2 } leading to a phase of early accelerated expansion ( early inflation ) with a constant density equal to the Planck density \rho _ { P } = 5.16 10 ^ { 99 } { g } / { m } ^ { 3 } ( vacuum energy ) .
For \rho _ { \Lambda } \ll \rho \ll \rho _ { P } , we recover the standard linear equation of state p = \alpha \rho c ^ { 2 } describing radiation ( \alpha = 1 / 3 ) or pressureless matter ( \alpha = 0 ) and leading to an intermediate phase of decelerating expansion .
For \rho \rightarrow \rho _ { \Lambda } , we get p = - \rho c ^ { 2 } leading to a phase of late accelerated expansion ( late inflation ) with a constant density equal to the cosmological density \rho _ { \Lambda } = 7.02 10 ^ { -24 } { g } / { m } ^ { 3 } ( dark energy ) .
We show a nice “ symmetry ” between the early universe ( vacuum energy + \alpha -fluid ) and the late universe ( \alpha -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 \rho _ { P } in the early universe plays a role similar to the cosmological density \rho _ { \Lambda } in the late universe .
They represent fundamental upper and lower density bounds differing by 122 orders of magnitude .
The cosmological constant “ problem ” may be a false problem .
We study the evolution of the scale factor , density , and pressure .
Interestingly , 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 ( \alpha -era ) .
This model does not present any singularity at t = 0 and exists eternally in the past ( although it may be incorrect to extrapolate the solution to the infinite past ) .
On the other hand , it admits a scalar field interpretation based on a quintessence field or a tachyon field .