1.00 1.05 1.10 1.15 1.20  0  1  2  3  4  5  6 fRG / CDM z = 0.174 h Mpc -1 n=2 =1 n=2 =3 n=2 =10 FIG. 3: The ratio of linear density perturbations δ$_{fRG}$/δ$_{ΛCDM}$ ( k = 0 . 174 h Mpc − $^{1}$) as a function of redshift for three different values of λ with n = 2. ( δ$_{fRG}$/δ$_{ΛCDM}$ ) $^{2}$( k = 0 . 174 h Mpc − $^{1}$) ≲ 1 . 1. Although we neglect non-linear effects here, the difference between linear calculation and non-linear N-body simulation remained smaller than 5% at the wavenumber 0 . 174 h Mpc − $^{1}$[32]. Figures 4 represent ( δ$_{fRG}$/δ$_{ΛCDM}$ ) $^{2}$( k = 0 . 174 h Mpc − $^{1}$) as a function of λ for n = 2 , 3 , and 4. From the analytic formula (19), this λ dependence would have the form ( δ$_{fRG}$/δ$_{ΛCDM}$ ) 2 ∝ C $^{2}$( k ) ∝ λ − (2 n − p $_{$_{n}$+1)(}$√$_{33}$ − 5) 4(3 n +2) which is depicted by a broken line in each figure. This curve, however, does not match the asymptotic behaviour ( δ$_{fRG}$/δ$_{ΛCDM}$ ) 2 −→ 1 for large λ . We find that an exponential function fits the numerical calculation very well with ( n, b$_{n}$, q$_{n}$ ) = (2 , 0 . 47 , 0 . 19) , (3 , 0 . 43 , 0 . 49) , and (4 , 0 . 39 , 0 . 70) , respectively. From these figures, in order to keep deviation from ΛCDM model smaller than 10% at k = 0 . 174 h Mpc − $^{1}$, we find λ should be larger than 8.2, 3.0, and 1.9 for n = 2 , 3 , and 4, respectively. ( δ$_{fRG}$/δ$_{ΛCDM}$ ) 2 = 1 + b$_{n}$e − q$_{n}$λ (29) From these analysis, we can constrain the parameter space as Fig. 5. The region which 11