Palatini formulation of the $R^{-1}$modified gravity with an additionally squared scalar curvature term

In this paper by deriving the Modified Friedmann equation in the Palatini formulation of $R^2$ gravity, first we discuss the problem of whether in Palatini formulation an additional $R^2$ term in Einstein's General Relativity action can drive an inflation. We show that the Palatini formulation of $R^2$ gravity cannot lead to the gravity-driven inflation as in the metric formalism. If considering no zero radiation and matter energy densities, we obtain that only under rather restrictive assumption about the radiation and matter energy densities there will be a mild power-law inflation $a(t)\sim t^2$, which is obviously different from the original vacuum energy-like driven inflation. Then we demonstrate that in the Palatini formulation of a more generally modified gravity, i.e., the $1/R+R^2$ model that intends to explain both the current cosmic acceleration and early time inflation, accelerating cosmic expansion achieved at late Universe evolution times under the model parameters satisfying $\alpha\ll\beta$.