Conditions for the cosmological viability of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>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 acceleration requires that the variable $m$ satisfies the conditions $m(r)\ensuremath{\approx}+0$ and $\mathrm{d}m/\mathrm{d}r&gt;\ensuremath{-}1$ at $r\ensuremath{\approx}\ensuremath{-}1$. For the existence of a viable late-time acceleration we require instead either (i) $m=\ensuremath{-}r\ensuremath{-}1$, $(\sqrt{3}\ensuremath{-}1)/2&lt;m\ensuremath{\le}1$ and $\mathrm{d}m/\mathrm{d}r&lt;\ensuremath{-}1$ or (ii) $0&lt;m\ensuremath{\le}1$ at $r=\ensuremath{-}2$. These conditions identify two regions in the $(r,m)$ space, one for the matter era and the other for the acceleration. Only models with an $m(r)$ curve that connects these regions and satisfies the requirements above lead to an acceptable cosmology. The models of type $f(R)=\ensuremath{\alpha}{R}^{\ensuremath{-}n}$ and $f=R+\ensuremath{\alpha}{R}^{\ensuremath{-}n}$ do not satisfy these conditions for any $n&gt;0$ and $n&lt;\ensuremath{-}1$ 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 $a\ensuremath{\propto}{t}^{1/2}$. We also find that $f(R)$ models can have a strongly phantom attractor but in this case there is no acceptable matter era.

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