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

We give a rigorous and mathematically well defined presentation of the covariant and gauge invariant theory of scalar perturbations of a Friedmann-Lemaître-Robertson-Walker universe for fourth order gravity, where the matter is described by a perfect fluid with a barotropic equation of state. The general perturbations equations are applied to a simple background solution of $R^n$ gravity. We obtain exact solutions of the perturbations equations for scales much bigger than the Hubble radius. These solutions have a number of interesting features. In particular, we find that for all values of $n$ there is always a growing mode for the density contrast, even if the universe undergoes an accelerated expansion. Such behavior does not occur in standard general relativity, where as soon as dark energy dominates, the density contrast experiences an unrelenting decay. This peculiarity is sufficiently novel to warrant further investigation of fourth order gravity models.

Figure 1

Plot of the real part of the exponents of each mode of the solution (107) against $n$ in the case of dust ($w=0$). The continuous and dashed line represent the modes $t^{{\alpha }_{\pm }}$ respectively (note how they coincide when ${\alpha }_{\pm }$ are complex), the dashed-dot line represents the mode $t^{2-(4n/3(w+1))}$ and the dot line the mode $t^{-1}$.

Figure 2

Plot of the real part of exponents of the modes of the Newtonian potential (118) against $n$ in the case of dust ($w=0$). The continuous and dot-dashed line represent the modes $t^{(4n/3)+{\beta }_{\pm }{|}_{w=0}}$ respectively (note how they coincide when ${\beta }_{\pm }$ are complex), the dotted line represents the mode $t^{(4n/3)-3}$.