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≡Rf,RR/f,R and r≡-Rf,R/f with f,R≡df/dR. 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 acceleration requires that the variable m satisfies the conditions m(r)≅+0 and dm/dr>-1 at r≅-1. For the existence of a viable late-time acceleration we require instead either (i) m=-r-1, (√(3)-1)/20 and n<-1 and are thus cosmologically unacceptable. Similar conclusions can be reached for many other examples discussed in the text. In most cases the standard matter era is replaced by a cosmic expansion with scale factor a∝t1/2. We also find that f(R) models can have a strongly phantom attractor but in this case there is no acceptable matter era