Classical and quantum massive cosmology for the open FRW universe

In an open Friedmann-Robertson-Walker (FRW) space background, we study the classical and quantum cosmological models in the framework of the recently proposed nonlinear massive gravity theory. Although the constraints which are present in this theory prevent it from admitting the flat and closed FRW models as its cosmological solutions, for the open FRW universe it is not the case. We have shown that, either in the absence of matter or in the presence of a perfect fluid, the classical field equations of such a theory adopt physical solutions for the open FRW model, in which the mass term shows itself as a cosmological constant. These classical solutions consist of two distinguishable branches: One is a contacting universe which tends to a future singularity with zero size, while another is an expanding universe having a past singularity from which it begins its evolution. A classically forbidden region separates these two branches from each other. We then employ the familiar canonical quantization procedure in the given cosmological setting to find the cosmological wave functions. We use the resulting wave function to investigate the possibility of the avoidance of classical singularities due to quantum effects. It is shown that the quantum expectation values of the scale factor, although they have either contracting or expanding phases like their classical counterparts, are not disconnected from each other. Indeed, the classically forbidden region may be replaced by a bouncing period in which the scale factor bounces from the contraction to its expansion eras. Using the Bohmian approach of quantum mechanics, we also compute the Bohmian trajectory and the quantum potential related to the system, which their analysis shows are the direct effects of the mass term on the dynamics of the universe.

Figure 1

The figures show the evolutionary behavior of the universes based on Eq. (27). We have used the numerical values $\Lambda =1$ and ${\tau }_{*}=0$.

Figure 2

The square of the wave function for the quantum universe. We take the numerical value $\Lambda =1.5$.

Figure 3

Left: The figure shows qualitative behavior of the classical scale factor in Eq. (34) [solid lines: the left branch for $a^{(\mathrm{I})}(t)$ and the right branch for $a^{(\mathrm{II})}(t)$] and the expectation value of the scale factor of Eq. (62) (dashed line). Right: The same figure in terms of cosmic time. The left and right branches of the solid lines represent $a^{(\mathrm{I})}(\tau )$ and $a^{(\mathrm{II})}(\tau )$, respectively, in Eq. (35) while the dashed line represents the expectation value in Eq. (63).

Figure 4

Left: The Bohmian trajectory of the scale factor. Right: The horizontal line represents a typical energy level. The solid curve is the quantum potential for $\Lambda \ne 0$, while the dashed curve denotes the quantum potential when $\Lambda =0$.