Effective actions for loop quantum cosmology in fourth-order gravity

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 $$R=g^{\mu \nu }R_{\mu \nu }$$ R = g μ ν R μ ν is the Ricci scalar, $$P=R_{\mu \nu }R^{\mu \nu }$$ P = R μ ν R μ ν , and $$Q=R_{\alpha \beta \mu \nu }R^{\alpha \beta \mu \nu }$$ Q = R α β μ ν R α β μ ν 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.

Bouncing Cosmology in Fourth-Order Gravity

“…Our analysis generalizes the results obtained in previous works for f (R), R + f (G) and f (R, G) theories of gravity [6][7][8], providing the same results in appropriate limits (see Ref. [10] for more details). In such a context, we have found bouncing solutions assuming specific ansatz and obtaining algebraic constraints on the free parameters of the models.…”

“…However, this type of theories is characterized by higher-order derivatives of the metric tensor which, as a rule, provide spurious degrees of freedom. A way to face this inconvenience is to adopt an order reduction technique, which allows to obtain effective action perturbatively close to GR [6][7][8][9][10]. The reduction technique is based on the redefinition of the Lagrangian as L = √ −g(R + ϕ)/2κ, where g is the determinant of the metric tensor, R is the Ricci scalar, is a perturbative dimensionless parameter which indicates the deviation of the model from GR, and ϕ is a function of the curvature invariants.…”

“…An interesting gener-alization of these cases consists in taking into account ϕ = ϕ(R, P, Q), (see Ref. [10]). In this way, we can deal with a general fourth-order theory of gravity which is perturbatively close to GR.…”

“…In Section 2 we briefly describe the general approach based on the order reduction technique, highlighting its key steps, and in Section 3 we focus on the recent generalization of such an approach to ϕ = ϕ(R, P, Q) [10]. In particular, we will focus on the case w = 1 which corresponds to a massless scalar field in LQC.…”

Bouncing Cosmology in Fourth-Order Gravity

“…However, this type of theories is characterized by higher-order derivatives of the metric tensor which, as a rule, provide spurious degrees of freedom. A way to face this inconvenience is to adopt an order reduction technique, which allows to obtain effective action perturbatively close to GR [6][7][8][9][10]. The reduction technique is based on the redefinition of the Lagrangian as L = √ −g(R+ǫϕ)/2κ, where g is the determinant of the metric tensor, R is the Ricci scalar, ǫ is a perturbative dimensionless parameter which indicates the deviation of the model from GR, and ϕ is a function of the curvature invariants.…”