  We obtain the exact analytic form of the growth index at present epoch
($a=1$) in a flat universe with the cosmological constant ({\it i.e.} the dark
energy with its equation of state $\omega_{de} = -1$). For the cosmological
constant, we obtain the exact value of the current growth index parameter
$\gamma = 0.5547$, which is very close to the well known value 6/11. We also
obtain the exact analytic solution of the growth factor for $\omega_{de}$ =
-1/3 or -1. We investigate the growth index and its parameter at any epoch with
this exact solution. In addition to this, we are able to find the exact
$\Omega_{m}^{0}$ dependence of those observable quantities. The growth index is
quite sensitive to $\Omega_{m}^{0}$ at $z = 0.15$, where we are able to use 2dF
observation. If we adopt 2dF value of growth index, then we obtain the
constrain $0.11 \leq \Omega_{m}^{0} \leq 0.37$ for the cosmological constant
model. Especially, the growth index is quite sensitive to $\Omega_{m}^{0}$
around $z \leq 1$. We might be able to obtain interesting observations around
this epoch. Thus, the analytic solution for this growth factor provides the
very useful tools for future observations to constrain the exact values of
observational quantities at any epoch related to growth factor for $\omega_{de}
= -1$ or -1/3.
