Stephani–Schutz quantum cosmology

Abstract

We study the Stephani quantum cosmological model in the presence of a cosmological constant in radiation dominated Universe. In the present work the Schutz's variational formalism which recovers the notion of time is applied. This gives rise to Wheeler-DeWitt equations which can be cast in the form of Schrödinger equations for the scale factor. We find their eigenvalues and eigenfunctions by using the Spectral Method. Then we use the eigenfunctions in order to construct wave packets and evaluate the time-dependent expectation value of the scale factor, which is found to oscillate between non-zero finite maximum and minimum values. Since the expectation value of the scale factor never tends to the singular point, we have an initial indication that this model may not have singularities at the quantum level.

Accurate energy spectrum for double-well potential: periodic basis

“…In fact, this simple and efficient method is rarely used by physicists to tackle eigenvalue problems and even they sometimes use some complicated and inaccurate techniques [29]. Although, we have studied only double-well oscillators, but the presented method is applicable quite generally for eigenvalue problems having polynomial potential which may correspond to physically relevant situations [30]. Moreover, this method can be extended and used for two-or three-dimensional cases.…”

On the initial condition in quantum cosmology

“…where prime denotes the derivative with respect to v. To completely determine the initial conditions, we need to specify A n s and B n s. The prescription is that these coefficients have the same functional form [3,4,6,8] i.e.…”

“…In this model k(t) plays the role of the spatial curvature and R(t) is the Stephani version of scale factor. By substituting the metric (15) in the action and choosing the curvature function k(t) as k(t) = βR γ (t) [8,12] and after dropping the surface terms, the final reduced action near r ≈ 0 takes the following form…”

A class of cosmological solutions in induced matter theory with conformally flat bulk space

“…This metric, on each hypersurface with constant ψ, can be rewritten in a more familiar form giving rise to a spherically symmetric Stephani universe [34][35][36][37][38][39][40] …”