Vacuum solution of a quadratic red-shift based correction in f(R) gravity

In the present paper, using action derivative with respect to Ricci scalar and its expansion relative to red-shift, to the second-order and consequently without needing dark energy, the history of cosmos expansion is reconstructed in vacuum. Then, employing supernova data, free parameters of the model which are the expansion coefficients, are calculated. It will be seen that if the free parameters of the model are zero, action of general relativity, which is a boundary mode of the gravity, is found. Also the cosmic age for this model in vacuum is calculated. Finally, it is attempted to reconstruct the reference action in terms of its Taylor expansion. Thus, it will be found that the reconstructed action and their actions definitely pass the Solar system tests.