 0  2  4  6  8  10 2.0 2.5 3.0 3.5 4.0 n = 0.174 h Mpc -1 (  $_{fRG}$ /  $_{CDM}$ ) 2 now = 1.1 (  $_{fRG}$ /  $_{CDM}$ ) 2 now = 1.2 min FIG. 5: Constraints for parameter space. the ΛCDM model in the Einstein gravity. Figures 6 show 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 be- cause f ( R ) gravity is indistinguishable from the Einstein gravity plus a positive cosmological constant then. It gradually decreases in time, reaches a minimum, and then increase again towards the present epoch. We can understand this tendency from the evolution equation for γ ( z )[36], − (1 + z ) ln(1 − Ω$_{DE}$) dγ dz = − (1 − Ω$_{DE}$) γ − 1 2 [1 + 3(2 γ − 1) w$_{DE}$ Ω$_{DE}$] + 3 2 G$_{eff}$ G (1 − Ω$_{DE}$) $^{1}$− $^{γ}$, (31) where Ω$_{DE}$ = 1 − Ω$_{m}$ is the density parameter of dark energy based on (8). In the high- redshift era when Ω$_{DE}$ is small, the above equation may be approximated as (1 + z )Ω$_{DE}$ dγ dz = 3 2 ( G$_{eff}$ G − 1 ) + Ω$_{DE}$ [ 11 2 ( γ − 6 11 ) − 3 2 (1 − γ ) ( G$_{eff}$ G − 1 ) − 3 2 (2 γ − 1)( w$_{DE}$ + 1) ] . (32) 13