Advances in Mathematical Physics

Volume 2017 (2017), Article ID 1056514, 5 pages

https://doi.org/10.1155/2017/1056514

## Quantum Cosmology of Quadratic Theories with a FRW Metric

^{1}Facultad de Ingeniería, Benemérita Universidad Autónoma de Puebla, 72000 Puebla, PUE, Mexico^{2}Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, P.O. Box 165, 72000 Puebla, PUE, Mexico

Correspondence should be addressed to V. Vázquez-Báez

Received 1 April 2017; Accepted 17 May 2017; Published 18 June 2017

Academic Editor: Anzhong Wang

Copyright © 2017 V. Vázquez-Báez and C. Ramírez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the quantum cosmology of a quadratic theory with a FRW metric, via one of its equivalent Horndeski type actions, where the dynamic of the scalar field is induced. The classical equations of motion and the Wheeler-DeWitt equation, in their exact versions, are solved numerically. There is a free parameter in the action from which two cases follow: inflation + exit and inflation alone. The numerical solution of the Wheeler-DeWitt equation depends strongly on the boundary conditions, which can be chosen so that the resulting wave function of the universe is normalizable and consistent with Hermitian operators.

#### 1. Introduction

Since its formulation, general relativity has been a successful theory, verified in many ways and at any scale. However, there are instances where it does not reproduce in a precise way the results of observations, in particular the origin of the universe and the early and present inflationary phases. One way to describe these issues has been given by means of modified gravity theories. Starobisnky [1] has proposed theory as an effective action of gravity obtained by coupling quantum matter fields to gravity, which explains inflation and reheating, and recently it has been used to explain the effects of dark matter and dark energy [2, 3] (see also [4, 5]). Even if these theories appear as effective theories, one appealing feature of them is that they are pure gravity theories, although they are higher order. However, under certain conditions, it is possible to give actions of Horndeski type equivalent to actions [6, 7], which have the advantage of being second-order and consistent with the usual inflationary or dark energy scenarios where there are scalar fields coupled to Einstein relativity.

Another scenario where theory has been considered is the origin of the universe by a tunneling mechanism from “nothing” to the de Sitter phase of Starobinski’s model [8], where a description of the universe in the framework of quantum cosmology is given, from which the tunneling probability, the subsequent curvature fluctuations, and the duration of the inflationary phase were computed, in the WKB approximation. Quantum cosmology of theories has been studied also in [9–12]. One frequent problem related to the wave function of the universe is its interpretation and, related to it, its normalizability, which might depend on the initial conditions [9].

In this work, we consider the quantum cosmology of theory with the FRW metric in the formulation due to O’Hanlon [6], with an auxiliary scalar field , and give a numerical solution for the Wheeler-DeWitt equation in its exact form. Considering that the scale factor is positive, we require that the wave function of the universe vanishes at , in order for the conjugate momentum of to be hermitic. For the numerical computation, we consider a compact domain in and . We get that the solution is consistent to zero on all boundaries, corresponding to normalizability. This feature seems to consistently persist in the noncompact limit. In the second section, we make a short analysis of the classical solutions and in the third section we show the numerical solution of the WdW equation considering values of the parameter for which classically the solutions are qualitatively different. In the last section, we draw some conclusions.

#### 2. Lagrangian Analysis

Let us focus on the model for gravity (see [4]) without matter which has an action given by ; variation with respect to the metric results in which leads to the following equations of motion:where .

Recalling that for a FRW geometry the metric tensor is the following:then the scalar curvature is given by which written in terms of the Hubble factor becomes once we fixed the gauge field to and set . With the scalar curvature as a function of , as stated above, the field equations (2) are where is regarded as a scalar degree of freedom, the so-called “scalaron” [1].

This approach presents some calculational advantages; for instance, it is evident that, treating and as the dynamical fields, the above equations of motion are second-order; choosing as the dynamical variable leads to third-order equations; also, the dynamic of the scale factor is simply given by , provided one can solve for . Nevertheless, the previous statements are based on the one hand on setting and on the other hand on the fixing on the gauge . Let us examine these two conditions.

For , the curvature in terms of reads . Nevertheless, this loss of generality is not a problem since the observations favor an almost flat universe [13]. On the other hand, the choice for the gauge field on the Lagrangian formulation represents no difficulty; however, if we want to implement the Hamiltonian formulation of the theory and thus the quantum formulation, we have to keep the gauge arbitrary. Additionally, for the Hamiltonian formulation of an theory, we should implement Ostrogradsky’s formalism [14–16] to deal with the higher order derivatives in the Lagrangian; for instance, the terms containing derivatives of the lapse function must be integrated out of the action since, being a gauge field, it cannot have dynamics; this in turn produces third-order derivatives for the scale factor and in general it is not possible to eliminate all the derivatives of as can be seen in the following Lagrangian: which corresponds to the quadratic Starobinsky’s model, . The third derivative of can be seen in the first term on the right; also, we can see the presence of a derivative of , which complicates the Hamiltonian formulation. These complications can be avoided in Starobinsky’s model, with the FRW metric plugged in an O’Hanlon type of action [6] and given bywhere is a free parameter. This Horndeski type action resembles the action used in [8], where the definition of the scalar curvature is regarded as a constraint and used to eliminate terms in the action.

A variation with respect to results in which leads toThus, action (7) is completely equivalent to that of the Starobinsky’s model. Once we enter the FRW metric into this action, we are able to partially integrate the second-order derivative terms and all the derivatives of unlike the case mentioned above; thus, we get

After fixing the gauge , the resulting equations of motion of , , and , from this action, areFrom (12), we get which substituted in (10) givesEquation (11) is redundant. Obviously, (14) can be solved only numerically. We fix the parameter in order to produce a consistent profile for the evolution of the scale factor. In general, whatever the value of , we have an inflation stage (see Figure 1), but for sufficiently large values, this inflationary phase becomes an exit, as we depict in Figure 2. Also, in this figure, we can see that as reaches a certain value, its evolution stops and the universe freezes out; this feature of the model should change in the presence of matter.