  Bouncing cosmologies are obtained by adding to the Einstein-Hilbert action a
term of the form $\sqrt{-g}f(\chi)$, with $\chi$ a scalar depending on the
Hubble parameter only, not on its derivatives, and which is here shown to arise
from the divergence of the unitary time-like eigenvector of the stress tensor.
At background level, the dynamical equations for a given $f$-theory are
calculated, showing that the simplest bouncing cosmology resulting leads to
exactly the same equations as those for holonomy corrected Loop Quantum
Cosmology (LQC). When dealing with perturbations, the equation for tensor ones
is the same as in General Relativity (GR); for scalar perturbations, when one
uses the $f$-theory which leads to the same background as the standard version
of holonomy corrected LQC, one obtains similar equations (although a bit more
elaborated) as those coming from LQC in the so-called deformed algebra
approach.
