content_image/346965.jpg content_image/346966.jpg content_image/346973.jpg
<img src='content_image/346965.jpg'>

is forbidden from instability of de Sitter regime. 

<img src='content_image/346966.jpg'>

Next we turn to another important quantity used to distinguish different theories of gravity, namely, the gravitational growth index, γ ( z ), of density fluctuations. It is defined through 

<img src='content_image/346973.jpg'>

It takes a practically constant value γ ∼ = 0 . 55 in the standard ΛCDM model, but it evolves with time in modified gravity theories in general. We also note that γ ( z ) has a nontrivial k -dependence in f ( R ) gravity since density fluctuations with different wavenumbers evolve differently. Therefore, this quantity is a useful measure to distinguish modified gravity from ΛCDM model in the Einstein gravity. 

Figures 3 show the evolution of γ ( z ) together with that of G$_{eff}$/G for different values of k . In the early high-redshift regime, γ ( z ) takes a constant value identical to the ΛCDM model. It gradually decreases in time, reaches a minimum which may be even negative, and then increase again towards the present epoch. We note that recently Narikawa and Yamamoto[9] numerically calculated time evolution of γ ( z ) in a simplified model which we had used in the previous paper and also obtained some analytic expansion, which behaves qualitatively the same as the numerical result but with much more exaggerated amplitudes. Our results, which satisfy all viability conditions, exhibit milder deviation from ΛCDM model than they found. At present, the constraints for the growth index is not so strict to distinguish the deviation from the ΛCDM model[10], but observations may reveal its time and wave number 

