Effective actions for loop quantum cosmology in fourth-order gravity

Marcello Miranda,Salvatore Capozziello,Francisco S N Lobo,Daniele Vernieri

doi:10.1140/epjc/s10052-021-09767-5

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.

Highlights

A quantum theory aimed at solving this singularity is Loop Quantum Cosmology (LQC) [1–4]
Rather than discussing all of the fine details of LQC, which is not the aim of this paper, we focus on one of its achievements, namely, the effective Friedmann equation given by [7]: The cosmological model which received a great consensus from the scientific community is the so-called Big Bang theory
We found specific cosmological models, coming from f (R, P, Q) modified theory of gravity, reproducing the effective Friedmann equation of LQC

Summary

Fourth-order gravity and order reduction technique

Let us consider the following class of gravity theories in a 4-dimensional spacetime,. This is not enough due to the presence of the Riemann tensor, and one needs a further simplification To this effect, we assume that our spacetime is conformally flat, i.e., there exists always a local reference frame where the metric is flat (Minkowski metric) up to a conformal factor. We assume that our spacetime is conformally flat, i.e., there exists always a local reference frame where the metric is flat (Minkowski metric) up to a conformal factor The reason for this assumption is that we are interested in studying this modified theory of gravity from a cosmological point of view and, in particular, we will work with the FLRW metric which always fulfils this property. + κ2T μν Tμν , − 1 κ2T 2 + 2κ2Tμν T μν At this point, we have all the ingredients to obtain the order reduced field equations, at first order in , by substituting Eqs. These are the field equations that will be used throughout this work, in order to obtain the effective Friedmann equation (1.1) of LQC in our setting

Modified Friedmann equation and bouncing cosmology

Solution I

Solution II

Solution III

Discussion and conclusions