Conditions for the cosmological viability of f ( R ) dark energy models

We derive the conditions under which dark energy models whose Lagrangian densities $f$ are written in terms of the Ricci scalar $R$ are cosmologically viable. We show that the cosmological behavior of $f(R)$ models can be understood by a geometrical approach consisting of studying the $m(r)$ curve on the $(r,m)$ plane, where $m\ensuremath{\equiv}R{f}_{,RR}/{f}_{,R}$ and $r\ensuremath{\equiv}\ensuremath{-}R{f}_{,R}/f$ with ${f}_{,R}\ensuremath{\equiv}\mathrm{d}f/\mathrm{d}R$. This allows us to classify the $f(R)$ models into four general classes, depending on the existence of a standard matter epoch and on the final accelerated stage. The existence of a viable matter-dominated epoch prior to a late-time…