Palatini formulation of the R-¹-modified gravity with an additional squared scalar curvature term

In this paper by deriving the modified Friedmann equation in the Palatini formulation of R² gravity, first we discuss the problem of whether in Palatini formulation an additional R² term in Einstein's general relativity action can drive an inflation. We show that the Palatini formulation of R² gravity cannot lead to gravity-driven inflation as in the metric formalism. If considering no zero radiation and matter energy densities, we obtain that only under rather restrictive assumptions about the radiation and matter energy densities will there be a mild power-law inflation a(t) ∼ t², 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 + R² model that intends to explain both the current cosmic acceleration and early time inflation, accelerating cosmic expansion is achieved at late universe evolution times under model parameters satisfying α ≪ β.