Power-law cosmic expansion in $f(R)$ gravity models

We show that within the class of $f(R)$ gravity theories, Friedmann-Lemaître-Robertson-Walker power-law perfect fluid solutions only exist for $R^n$ gravity. This significantly restricts the set of exact cosmological solutions which have similar properties to what is found in standard general relativity.

Figure 1

Upper left plot: $V(\varphi )$ for $w=0$ and $3m/2=1.1$ in the case $C_1=C_2=0$. This corresponds to $f(R)\propto R^n$ with $n=1.1$. Upper right plot: Same model as in the upper left plot but with $C_1=C_2={\rho }_0$. Lower left plot: $V(\varphi )$ for $w=1/3$ and $3m/2=1.1\times 3/4$ in the case $C_1=C_2=0$. This corresponds to $f(R)\propto R^n$ with $n=1.1$. Lower right plot: Same model as in the lower left plot but with $C_1=C_2={\rho }_0$.

Figure 2

Left plot: $V(\varphi )$ for $w=0$ and $3m/2=0.9$ in the case $C_1=C_2=0$. This corresponds to $f(R)$ with the first term in (11) proportional to $R^n$ with $n=0.9$. Right panel: Same model as in the left plot but with $C_1=C_2={\rho }_0$.