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 theKretschmann 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 Fried-mann equation in the reduced theory. Requiring that themodified 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.