Evolution of the scale factor as a function of time in logarithmic scales. The early universe undergoes a phase of inflation due to the vacuum energy before entering into the radiation era. During the early inflation, the scale factor increases by orders of magnitude in less than . This is followed by the matter era and the dark energy era responsible for the accelerated expansion of the universe. The universe exhibits two types of inflation: an early inflation corresponding to the Planck density (vacuum energy) due to quantum mechanics (Planck constant) and a late inflation corresponding to the cosmological density (dark energy) due to general relativity (cosmological constant). The evolution of the early and late universe is remarkably symmetric. We have represented in dashed line the standard model leading to a primordial singularity (Big Bang). The dotted line corresponds to the model of Section 5.3 where the radiation is neglected. This analytical model provides a good description of the late universe. We have also represented the location of the present universe (bullet). It happens to be just at the transition between the matter era and the dark energy era.