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Additionally the effective Hamiltonian may contain modifications due to the inverse scale factor operator both in the gravitational and the matter part which arise by expressing inverse powers of the triad in terms of Poisson brackets between holonomies and positive powers of the triad. These modifications become important at a scale determined by a half-integer parameter j in LQC. This scale is independent of the critical energy density 12 which arises in Eq. 11. For semiclassical states, if the value of j is small, which theoretical considerations indicate, effects due to inverse scale factor modifications become weak in comparison to ρ2 modifications (Eq. 11) and can be neglected. Because of this reason the modified Friedmann equation in this work is different from the early works in LQC (see for e.g. Ref.).
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Interestingly, the authors of Ref. assumed the presence of the outer boundary of the phase space in their analysis and called it the "quantum boundary". But there was no bounce in their solutions since they where interested in GR and considered the boundary as the limit of validity of their treatment. Instead in our case this boundary is the direct consequence of the cosmological equations and have very different and precise meaning, namely, the set of points when the nonsingular bounce occurs.
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This is a well known problem with the potential in cyclic model. In LQC treatment of negative potentials this issue has also been discussed in Ref..
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