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As is known, in modified cosmological theories of gravity many of thecosmologies which could not be generated by standard Einstein gravity, can beconsistently described by $F(R)$ theories. Using known reconstructiontechniques, we investigate which $F(R)$ theories can lead to a Hubble parameterdescribing two types of cosmological bounces, the superbounce model, related tosupergravity and non-supersymmetric models of contracting ekpyrosis and alsothe Loop Quantum Cosmology modified ekpyrotic model. Since our method is anapproximate method, we investigate the problem at large and small curvatures.As we evince, both models yield power law reconstructed $F(R)$ gravities, withthe most interesting new feature being that both lead to acceleratingcosmologies, in the large curvature approximation. The mathematical propertiesof the some Friedmann-Robertson-Walker spacetimes $M$, that describesuperbounce-like cosmologies are also pointed out, with regards to the group ofcurvature collineations $CC(M)$.

Superbounce and Loop Quantum Cosmology Ekpyrosis from Modified Gravity