0.95 1.00 1.05 1.10 1.15 1.20 1.25 10 -4 10 -3 10 -2 10 -1 10 0 (  fRG  /  CDM  ) now k / hMpc -1 n=2 =1 n=2 =3 n=2 =10 FIG. 2: The ratio of linear density perturbations δ$_{fRG}$/δ$_{ΛCDM}$ at present as a function of k for three different values of λ with n = 2. In order to make a simple comparison of our results with observations of galaxy clustering, we define an effective wavenumber, k$_{eff}$ ( r ), corresponding to each length scale r , in terms of the top-hat mass fluctuation within the same radius: Here P ( k ) is the linear matter spectrum obtained by the standard CDM transfer function[31] with the scale-invariant initial power spectrum of perturbations, i.e. with the primordial spectral index n$_{s}$ = 1, and W ( kr ) is the Fourier transform of the top-hat window function. The wavenumber of our particular interest is the scale corresponding to σ$_{8}$ normalization, for which we find k$_{eff}$ ( r = 8 h − $^{1}$Mpc) = 0 . 174 h Mpc − $^{1}$. Figure 3 depicts the redshift evolution of the ratio δ$_{fRG}$/δ$_{ΛCDM}$ for this scale for the same values of n and λ as in Fig. 2. Note that this ratio does not stop growing at the accelerated stage of the Universe expansion which begins at z ≈ 0 . 8 for λ = 1 and z ≈ 0 . 75 for two other values of λ . Since the standard ΛCDM model normalized by large-scale CMB observations explains galaxy clustering at small scales well, δ$_{fRG}$ should not be too much larger than δ$_{ΛCDM}$ at these scales. We may typically require σ 2 r = ∫ d $^{3}$k (2 π ) 3 | W ( kr ) | $^{2}$P ( k ) ≡ 4 πk 3 eff (2 π ) 3 P ( k$_{eff}$ ( r )) , W ( kr ) ≡ 3 j$_{1}$ ( kr ) kr . (28) 10