Growth index with the cosmological constant

Abstract

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 $ω_{de} = -1$). For the cosmological constant, we obtain the exact value of the current growth index parameter $γ= 0.5547$, which is very close to the well known value 6/11. We also obtain the exact analytic solution of the growth factor for $ω_{de}$ = -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 $Ω_{m}^{0}$ dependence of those observable quantities. The growth index is quite sensitive to $Ω_{m}^{0}$ 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 $0.11 \leq Ω_{m}^{0} \leq 0.37$ for the cosmological constant model. Especially, the growth index is quite sensitive to $Ω_{m}^{0}$ 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 $ω_{de} = -1$ or -1/3.