1.0 1.1 1.2 1.3 1.4 1.5  0  2  4  6  8  10 ( $^{δ}$ fRG  /  CDM  ) 2 now n = 2 = 0.174 h Mpc -1 numerical fit analytical fit (a) n = 2. 1.0 1.1 1.2 1.3 1.4 1.5  0  2  4  6  8  10 ( $^{δ}$ fRG  /  CDM  ) 2 now n = 3 = 0.174 h Mpc -1 numerical fit analytical fit (b) n = 3. 1.0 1.1 1.2 1.3 1.4 1.5  0  2  4  6  8  10 ( $^{δ}$ fRG  /  CDM  ) 2 now n = 4 = 0.174 h Mpc -1 numerical fit analytical fit (c) n = 4. FIG. 4: The present ratio ( δ$_{fRG}$/δ$_{ΛCDM}$ ) $^{2}$( k = 0 . 174 h Mpc − $^{1}$) as a function of λ together with two fitting functions. satisfy ( δ$_{fRG}$/δ$_{ΛCDM}$ ) $^{2}$( k = 0 . 174 h Mpc − $^{1}$) < 1 . 1 corresponds to above the solid line. We also show the 20% boundary by the broken line. The region below the dotted line is forbidden because of instability of the de Sitter regime. Next we turn to another important quantity used to distinguish different theories of gravity, namely, the gravitational growth index, γ ( z ), of density fluctuations[33–38]. It is defined through It takes a practically constant value γ ∼ = 0 . 55 in the standard ΛCDM model[34], but it evolves in 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 d ln δ d ln a = Ω$_{m}$( z ) γ ( z $^{)}$, or γ ( z ) = log ( ˙ δ Hδ ) log Ω$_{m}$ . (30) 12