Evolution of density perturbations in $f(R)$ theories of gravity

In the context of $f(R)$ theories of gravity, we study the evolution of scalar cosmological perturbations in the metric formalism. Using a completely general procedure, we find the exact fourth-order differential equation for the matter density perturbations in the longitudinal gauge. In the case of sub-Hubble modes, the expression reduces to a second-order equation which is compared with the standard (quasistatic) equation used in the literature. We show that for general $f(R)$ functions the quasistatic approximation is not justified. However, for those functions adequately describing the present phase of accelerated expansion and satisfying local gravity tests, it provides a correct description for the evolution of perturbations.

Figure 1

${\delta }_k$ with $k=0.2\mathrm{h}{\mathrm{Mpc}}^{-1}$ for $f_{\mathrm{test}}(R)$ model and $\Lambda \mathrm{CDM}$. Both standard quasistatic evolution and Eq. (33) have been plotted in the redshift range from 100 to 0.

Figure 2

${\delta }_k$ with $k=1.67\mathrm{h}{\mathrm{Mpc}}^{-1}$ for $f(R)$ model A evolving according to (36), $\Lambda \mathrm{CDM}$ and quasistatic approximation given by Eq. (34) in the redshift range from 1000 to 0. The quasistatic evolution is indistinguishable from that coming from (36), but diverges from $\Lambda \mathrm{CDM}$ behavior as $z$ decreases.

Figure 3

${\delta }_k$ with $k=1.67\mathrm{h}{\mathrm{Mpc}}^{-1}$ for $f(R)$ model B evolving according to (36), $\Lambda \mathrm{CDM}$ and quasistatic evolution given by Eq. (34) in the redshift range from 1000 to 0. The quasistatic evolution is indistinguishable from that coming from (36), but diverges from $\Lambda \mathrm{CDM}$ behavior as $z$ decreases.

Figure 4

Scale dependence of ${\delta }_k$ evaluated today ($z=0$) for $k/H_0$ in the range from 1000 to 40 000.