Résumé
Phys.Rev.D75:083504,2007 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 in 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) approx+0 and dm/dr>-1 at r approx-1. For the existence of a viable late-time acceleration we require instead either (i) m=-r-1, (sqrt3-1)/2<m<1 and dm/dr<-1 or (ii) 0<m<1 at r=-2. These conditions identify two regions in the (r, m) space, one for the matter era and the other for the acceleration. Only models with a m(r) curve that connects these regions and satisfy the requirements above lead to an acceptable cosmology. The models of the type f(R)=alpha R⁻ⁿ and f=R+alpha R⁻ⁿ do not satisfy these conditions for any n>0 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 propto t^(1/2). We also find that f(R) models can have a strongly phantom attractor but in this case there is no acceptable matter era.