$f(R)$ gravity without a cosmological constant

In this work we consider the possibility of describing the current evolution of the universe, without the introduction of any cosmological constant or dark energy (DE), by modifying the Einstein-Hilbert (EH) action. In the context of the $f(R)$ gravities within the metric formalism, we show that it is possible to find an action without cosmological constant which exactly reproduces the behavior of the EH action with cosmological constant. In addition the $f(R)$ action is analytical at the origin having Minkowski and Schwarzschild solutions as vacuum solutions. The found $f(R)$ action is highly nontrivial and must be written in terms of hypergeometric functions but, in spite of looking somewhat artificial, it shows that the cosmological constant, or more generally the DE, is not a logical necessity.