-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 0 1 2 3 4  (z) z n = 2,  = 1 k=10 -2 hMpc -1 k=10 -1 hMpc -1 k=1 hMpc -1 CDM (a)Evolution of γ ( z ) for n = 2 λ = 1. 1.0 1.1 1.2 1.3 1.4 0 1 2 3 4 G eff  / G z n = 2,  1 k=10 -2 hMpc -1 k=10 -1 hMpc -1 k=1 hMpc -1 (b)Evolution of G eff /G for n = 2 λ = 1. -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0 1 2 3 4  (z) z n = 2,  = 3 k=10 -2 hMpc -1 k=10 -1 hMpc -1 k=1 hMpc -1 CDM (c)Evolution of γ ( z ) for n = 2 λ = 3. 1.0 1.1 1.2 1.3 1.4 0 1 2 3 4 G eff  / G z n = 2,   = 3 k=10 -2 hMpc -1 k=10 -1 hMpc -1 k=1 hMpc -1 (d)Evolution of G eff /G for n = 2 λ = 3. FIG. 6: Evolution of γ ( z ) and G$_{eff}$/G . In the earlier stage, the first term in the right-hand side is more important. That explains why γ ( z ) starts to decrease when G$_{eff}$/G starts to increase. As time goes by towards lower redshifts, the second term becomes more important to make γ ( z ) increase again. We note that recently Narikawa and Yamamoto[38] calculated time evolution of γ ( z ) in a simplified model (15) numerically and also obtained some analytic expansion, which behaves qualita- tively the same as our numerical results but with much more exaggerated amplitudes. Our results, which satisfy all the viability conditions, exhibit milder deviation from the ΛCDM model than those they found. Existing constraints on the growth index[39] are not strong enough to detect any deviation from the ΛCDM model and/or to obtain new bounds on f ( R ) DE models, but future observations may reveal its time and wavenumber dependence. Another quantity which can characterize the evolution of density perturbations more directly is the ratio δ$_{fRG}$ ( z = 0 . 5) /δ$_{fRG}$ ( z = 0). However, it varies only from 0.75 to 0.78 for 14