The Scale Factor in the Universe with Dark Energy
  Friedmans cosmological equations for the scale factor are analyzed for the Universe containing dark energy. The parameter of the equation of state of the dark energy is treated as an arbitrary constant whose value lies within the interval $w \in [-1.5, -0.5]$, the limits of which are set by current observations. A unified analytic solution is obtained for the scale factor as a function of physical and conformal time. We obtain approximated solutions for scale factor to an accuracy of better then 1%. This accuracy is better then measurement errors of global density parameters and therefore is suitable for the approximated models of our Universe. An analytic solution is obtained for the scale factor in $\Lambda$CDM cosmological model both in physical and conformal time, for the description of the evolution of the Universe from the epoch of matter domination up to the infinite future. 
M.V. Sazhin, O.S. Sazhina, U. Chadayammuri
astro-ph.CO gr-qc