Effective Actions for Loop Quantum Cosmology in Fourth-Order Gravity

TL;DR

This paper derives effective actions in fourth-order gravity theories that replicate Loop Quantum Cosmology's quantum bounce, providing a bridge between classical modified gravity and quantum cosmological effects.

Contribution

It introduces a method to find $f(R,P,Q)$ gravity models that reproduce LQC's effective Friedmann equations, advancing the understanding of quantum bounce scenarios in modified gravity.

Findings

Derived functional forms of $f(R,P,Q)$ matching LQC dynamics

Identified conditions for viable bouncing cosmologies

Demonstrated perturbative proximity to General Relativity

Abstract

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.