Growth of matter perturbations for realistic $F(R)$ models

Two different realistic $F(R)$ modified gravity models are considered in the framework of the Friedmann-Lemetre-Robertson-Walker universe. The parameters of these two models are adjusted to reach coherence with the most recent and accurate observations of the current Universe. A study of the growth of matter density perturbations is done, and several parametrizations of the growth index are developed for both models. The ansatz for the growth index given by $\gamma ={\gamma }_0+{\gamma }_{1}z/(1+z)$ seems to be the best parametrization for the two models considered. Finally, the values obtained for ${\gamma }_0$ and ${\gamma }_1$ can be used in order to characterize these two models and to differentiate them from others such as the Hu-Sawicki model.

Figure 1

Contour plot of the effective gravitational constant $G_{\mathrm{eff}}/G$ as a function of $z$ and $\log k({\mathrm{Mpc}}^{-1})$ for model I (a) and model II (b).

Figure 2

Cosmological evolution of $G_{\mathrm{eff}}/G$ as a function of $z$ for $k=0.1{\mathrm{Mpc}}^{-1}$ (blue, upper line), $k=0.01{\mathrm{Mpc}}^{-1}$ (red, middle line) and $k=0.001{\mathrm{Mpc}}^{-1}$ (green, lower line) for model I (a) and model II (b).

Figure 3

Contour plot of the growth rate $f_{\mathrm{g}}$ as a function of $z$ and $\log k({\mathrm{Mpc}}^{-1})$ for model I (a) and model II (b).

Figure 4

Cosmological evolution of the growth rate $f_{\mathrm{g}}$ as a function of $z$ for $k=0.1{\mathrm{Mpc}}^{-1}$ (blue, upper line), $k=0.01{\mathrm{Mpc}}^{-1}$ (red, middle line) and $k=0.001{\mathrm{Mpc}}^{-1}$ (green, lower line) for model I (a) and model II (b).

Figure 5

Constant growth index as a function of $\log k({\mathrm{Mpc}}^{-1})$ for model I (a) and for model II (b). The bars express the 68% CL.

Figure 6

Cosmological evolutions of the growth rate $f_{\mathrm{g}}$ (red, upper ending line) and ${\Omega }_{\mathrm{m}}^{\gamma }$ (blue, lower ending line) with $\gamma ={\gamma }_0$ as functions of the redshift $z$ in model I for $k=0.1{\mathrm{Mpc}}^{-1}$ (a), $k=0.01{\mathrm{Mpc}}^{-1}$ (c) and $k=0.001{\mathrm{Mpc}}^{-1}$ (e), and those in model II for $k=0.1{\mathrm{Mpc}}^{-1}$ (b), $k=0.01{\mathrm{Mpc}}^{-1}$ (d) and $k=0.001{\mathrm{Mpc}}^{-1}$ (f).

Figure 7

Cosmological evolution of the relative difference ${ϵ}_r=\frac{|f_{\mathrm{g}}-{\Omega }_{\mathrm{m}}^{\gamma }|}{f_{\mathrm{g}}}$ with $\gamma ={\gamma }_0$ for $k=0.1{\mathrm{Mpc}}^{-1}$ (blue, lower ending line), $k=0.01{\mathrm{Mpc}}^{-1}$ (red, middle ending line) and $k=0.001{\mathrm{Mpc}}^{-1}$ (green, upper ending line) in model I (a) and model II (b).

Figure 8

Growth index fitting parameters in the case $\gamma ={\gamma }_0+{\gamma }_1·z$ as a function of $\log k({\mathrm{Mpc}}^{-1})$ for model I [(a) and (c)] and model II [(b) and (d)]. The legend is the same as Fig. 5.

Figure 9

Cosmological evolutions of the growth rate $f_{\mathrm{g}}$ (red, upper ending line for cases (c)–(f) and lower ending line for (a) and (b)) and ${\Omega }_{\mathrm{m}}^{\gamma }$ (blue, lower ending line for cases (c)–(f) and upper ending line for (a) and (b)) with $\gamma ={\gamma }_0+{\gamma }_1·z$ as functions of the redshift $z$ in model I for $k=0.1{\mathrm{Mpc}}^{-1}$ (a), $k=0.01{\mathrm{Mpc}}^{-1}$ (c) and $k=0.001{\mathrm{Mpc}}^{-1}$ (e), and those in model II for $k=0.1{\mathrm{Mpc}}^{-1}$ (b), $k=0.01{\mathrm{Mpc}}^{-1}$ (d) and $k=0.001{\mathrm{Mpc}}^{-1}$ (f).

Figure 10

Cosmological evolution of the relative difference ${ϵ}_r=\frac{|f_{\mathrm{g}}-{\Omega }_{\mathrm{m}}^{\gamma }|}{f_{\mathrm{g}}}$ with $\gamma ={\gamma }_0+{\gamma }_1·z$ for $k=0.1{\mathrm{Mpc}}^{-1}$ (blue, lower ending line), $k=0.01{\mathrm{Mpc}}^{-1}$ (red, middle ending line) and $k=0.001{\mathrm{Mpc}}^{-1}$ (green, upper ending line) in model I (a) and model II (b).

Figure 11

Growth index fitting parameters in the case $\gamma ={\gamma }_0+{\gamma }_1·\frac{z}{1+z}$ as a function of $\log k({\mathrm{Mpc}}^{-1})$ for model I [(a) and (c)] and model II [(b) and (d)]. The legend is the same as Fig. 5.

Figure 12

Cosmological evolutions of the growth rate $f_{\mathrm{g}}$ (red, upper ending line) and ${\Omega }_{\mathrm{m}}^{\gamma }$ (blue, lower ending line) with $\gamma ={\gamma }_0+{\gamma }_1·\frac{z}{1+z}$ as functions of the redshift $z$ in model I for $k=0.1{\mathrm{Mpc}}^{-1}$ (a), $k=0.01{\mathrm{Mpc}}^{-1}$ (c) and $k=0.001{\mathrm{Mpc}}^{-1}$ (e), and those in model II for $k=0.1{\mathrm{Mpc}}^{-1}$ (b), $k=0.01{\mathrm{Mpc}}^{-1}$ (d) and $k=0.001{\mathrm{Mpc}}^{-1}$ (f).

Figure 13

Cosmological evolution of the relative difference ${ϵ}_r=\frac{|f_{\mathrm{g}}-{\Omega }_{\mathrm{m}}^{\gamma }|}{f_{\mathrm{g}}}$ with $\gamma ={\gamma }_0+{\gamma }_1·\frac{z}{1+z}$ for $k=0.1{\mathrm{Mpc}}^{-1}$ (blue, lower ending line), $k=0.01{\mathrm{Mpc}}^{-1}$ (red, middle ending line) and $k=0.001{\mathrm{Mpc}}^{-1}$ (green, upper ending line) in model I (a) and model II (b).

Figure 14

Growth index fitting parameters in the case $\gamma ={\gamma }_0+{\gamma }_1·z$ as a function of $\log k({\mathrm{Mpc}}^{-1})$. The legend is the same as Fig. 5.