$\Lambda \mathrm{CDM}$ universe in $f(R)$ gravity

Several different explicit reconstructions of $f(R)$ gravity are obtained from the background Friedmann-Laîmatre-Robertson-Walker expansion history. It is shown that the only theory whose Lagrangian is a simple function of the Ricci scalar $R$, that admits an exact $\Lambda \mathrm{CDM}$ expansion history, is standard general relativity with a positive cosmological constant and the only way to obtain this behavior of the scale factor for more general functions of $R$ is to add additional degrees of freedom to the matter sector.