Effective $f(R)$ actions for modified Loop Quantum Cosmologies via order
  reduction
  General Relativity is an extremely successful theory, at least for weak
gravitational fields, however, it breaks down at very high energies, such as in
correspondence of the initial singularity. Quantum Gravity is expected to
provide more physical insights concerning this open question. Indeed, one
alternative scenario to the Big Bang, that manages to completely avoid the
singularity, is offered by Loop Quantum Cosmology (LQC), which predicts that
the Universe undergoes a collapse to an expansion through a bounce. In this
work, we use metric $f(R)$ gravity to reproduce the modified Friedmann
equations which have been obtained in the context of modified loop quantum
cosmologies. To achieve this, we apply an order reduction method to the $f(R)$
field equations, and obtain covariant effective actions that lead to a bounce,
for specific models of modified LQC, considering matter as a scalar field.
