We study isotropic and anisotropic (Bianchi I) cosmologies in Palatini f (R) and f (R, R µν R µν ) theories of gravity and consider the existence of non-singular bouncing solutions in the early universe. We find that all f (R) models with isotropic bouncing solutions develop shear singularities in the anisotropic case. On the contrary, the simple quadratic model R + aR 2 /R P + R µν R µν /R P exhibits regular bouncing solutions in both isotropic and anisotropic cases for a wide range of equations of state, including dust (for a < 0) and radiation (for arbitrary a). It thus represents a purely gravitational solution to the big bang singularity and anisotropy problems of general relativity without the need for exotic (w > 1) sources of matter/energy.

We consider the early time cosmology of f (R) theories in Palatini formalism and study the conditions that guarantee the existence of homogeneous and isotropic models that avoid the Big Bang singularity. We show that for such models the Big Bang singularity can be replaced by a cosmic bounce without violating any energy condition. In fact, the bounce is possible even for pressureless dust. We give a characterization of such models and discuss their dynamics in the region near the bounce. We also find that power-law lagrangians with a finite number of terms may lead to non-singular universes, which contrasts with the infinite-series Palatini f (R) lagrangian that one needs to fully capture the effective dynamics of Loop Quantum Cosmology. We argue that these models could also avoid the formation of singularities during stellar gravitational collapse.

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