An Inverse $f(R)$ Gravitation for Cosmic Speed up, and Dark Energy Equivalent

To explain the cosmic speed up, brought to light by the recent SNIa and CMB observations, we propose the following: a) In a spacetime endowed with a FRW metric, we choose an empirical scale factor that best explains the observations. b) We assume a modified gravity, generated by an unspecified field lagrangian, $f(R)$. c) We use the adopted empirical scale factor to work back retroactively to obtain $f(R)$, hence the term Inverse $f(R)$'. d) Next we consider the classic GR and a conventional FRW universe that, in addition to its known baryonic content, possesses a hypothetical Dark Energy’ component. We compare the two scenarios, and find the density, the pressure, and the equation of the state of the Dark Energy required to make up for the differences between the conventional and the modified GR models.