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) ≈ +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)/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) = αR −n and f = R + αR −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 ∝ 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.

We consider the linear growth of matter perturbations in various dark energy (DE) models. We show the existence of a constraint valid at $z=0$ between the background and dark energy parameters and the matter perturbations growth parameters. For $\Lambda$CDM $\gamma'_0\equiv \frac{d\gamma}{dz}_0$ lies in a very narrow interval $-0.0195 \le \gamma'_0 \le -0.0157$ for $0.2 \le \Omega_{m,0}\le 0.35$. Models with a constant equation of state inside General Relativity (GR) are characterized by a quasi-constant $\gamma'_0$, for $\Omega_{m,0}=0.3$ for example we have $\gamma'_0\approx -0.02$ while $\gamma_0$ can have a nonnegligible variation. A smoothly varying equation of state inside GR does not produce either $|\gamma'_0|>0.02$. A measurement of $\gamma(z)$ on small redshifts could help discriminate between various DE models even if their $\gamma_0$ is close, a possibility interesting for DE models outside GR for which a significant $\gamma'_0$ can be obtained.Comment: 8 pages, 8 figures. Results unchanged; clarifying sentence added; one reference adde