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Bouncing cosmologies are obtained by adding to the Einstein-Hilbert action aterm of the form $\sqrt{-g}f(\chi)$, with $\chi$ a scalar depending on theHubble parameter only, not on its derivatives, and which is here shown to arisefrom the divergence of the unitary time-like eigenvector of the stress tensor.At background level, the dynamical equations for a given $f$-theory arecalculated, showing that the simplest bouncing cosmology resulting leads toexactly the same equations as those for holonomy corrected Loop QuantumCosmology (LQC). When dealing with perturbations, the equation for tensor onesis the same as in General Relativity (GR); for scalar perturbations, when oneuses the $f$-theory which leads to the same background as the standard versionof holonomy corrected LQC, one obtains similar equations (although a bit moreelaborated) as those coming from LQC in the so-called deformed algebraapproach.

Cosmological perturbations in a class of fully covariant modified theories: Application to models with the same background as standard LQC