We show that there exist modified theories of gravity in which the metricsatisfies second-order equations and in which the Big Bang singularity isreplaced by a cosmic bounce without violating any energy condition. In fact,the bounce is possible even for presureless dust. We give a characterization ofsuch theories, which are formulated in the Palatini formalism, and discusstheir dynamics in the region near the bounce. We consider spatially flat andnon-flat homogeneous and isotropic universes.
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