Growth index with the exact analytic solution of sub-horizon scale
  linear perturbation for dark energy models with constant equation of state
  Three decades ago Heath found the integral form of the exact analytic growing
mode solution of the linear density perturbation $\delta$ on sub-horizon scales
including the cosmological constant or the curvature term. Recently, we
obtained the exact analytic form of this solution in our previous work
\cite{SK}. Interestingly, we are able to extend this solution for general dark
energy models with the constant equation of state $\omega_{de}$ in a flat
universe. This analytic solution provides the accurate and efficient tools for
probing the properties of dark energy models such as the behavior of the growth
factor and the growth index. We investigate the growth index and its parameter
at any epoch with this exact solution for different dark energy models and find
that the growth index is quite model dependent in the redshift space, $0.25
\leq z \leq 1.5$, so observations of the structure growth around this epoch
would be very interesting. Also one may be able to rule out some dark energy
models by using the analysis from this exact solution. Thus, the analytic
solution for the growth factor provides the very useful tools for future
observations to constrain the exact values of observational quantities at any
epoch related to the growth factor in the dark energy models.
