Properties of the exact analytic solution of the growth factor and its applications

There have been the approximate analytic solution \cite{Silveria} and several approximate analytic forms \cite{0508156,Carroll,0303112} of the growth factor $D_g$ for the general dark energy models with the constant values of its equation of state $\text{omde}$ after Heath found the exact integral form of the solution of $D_g$ for the Universe including the cosmological constant or the curvature term. Recently, we obtained the exact analytic solutions of the growth factor for both $\text{omde}=-1$ or $-\text{fr}13$ \cite{SK1} and the general dark energy models with the constant equation of state $\text{omde}$ \cite{SK3} independently. We compare the exact analytic solution of $D_g$ with the other well known approximate solutions. We also prove that the analytic solutions for $\text{omde}=-1$ or $-\text{fr}13$ in Ref. \cite{SK1} are the specific solutions of the exact solutions of the growth factor for general $\text{omde}$ models in Ref. \cite{SK3} even though they look quite different. We scrutinize the issue of using the well known parameterizations of the growth index and its parameter given in Ref. \cite{WS} to obtain the growth factor for general dark energy models. We also investigate the possible extensions of the exact solution of $D_g$ to the time-varying $\text{omde}$ for the comparison with observations.