Effective Actions for Loop Quantum Cosmology in Fourth-Order Gravity
  Loop Quantum Cosmology (LQC) is a theory which renders the Big Bang initial
singularity into a quantum bounce, by means of short range repulsive quantum
effects at the Planck scale. In this work, we are interested in reproducing the
effective Friedmann equation of LQC, by considering a generic $f(R,P,Q)$ theory
of gravity, where $R=g^{\mu\nu}R_{\mu\nu}$ is the Ricci scalar,
$P=R_{\mu\nu}R^{\mu\nu}$, and $Q=R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}$ is
the Kretschmann scalar. An order reduction technique allows us to work in
$f(R,P,Q)$ theories which are perturbatively close to General Relativity, and
to deduce a modified Friedmann equation in the reduced theory. Requiring that
the modified Friedmann equation mimics the effective Friedmann equation of LQC,
we are able to derive several functional forms of $f(R,P,Q)$. We discuss the
necessary conditions to obtain viable bouncing cosmologies for the proposed
effective actions of $f(R,P,Q)$ theory of gravity.
