Nonlinear quantum cosmology

Abstract

Quantum cosmology has been used to address the singularity problem as the universe approaches a zero size at the initial Big Bang. In our study we will use a simple FRW-Λ model which describes the de Sitter universe. Although there is no singularity at zero size of the universe described by this model, we hope that the avoidance of the `Big Bang?, which we define here to be a zero size universe, will mirror that in other more realistic models that do have a singularity at zero size. The usual approach is based on the quantization of Einstein?s equations which leads to the well known Wheeler DeWitt equation. A small but finite potential barrier is detected near zero size of the universe, which suggests a creation of a quantum universe via quantum tunneling. Hence the universe only comes into existence at a finite size. A contracting classical universe will also experience a bounce due to the barrier before it reaches a zero size. However, this method of quantisation is only successful in overcoming the Big Bang in closed universes (k=1). In the case of other geometries ie flat (k=0) and open (k= -1), no potential barrier is detected. Our approach then is to use non-linear effects, which may be significant at very short distance scales during the beginning of the universe, to arrive at a modified non-linear Wheeler DeWitt equation. We show how this non-linearisation can indeed solve the zero-size problem for flat universes. Previous studies have used perturbation methods to solve the non-linear modified equation as non-linear effects are expected to be small. In our investigation, we explore the use of a more exact method, a non-perturbative method, and compare the results obtained. We also show how this more exact method reveals some new interesting features that were not predicted using the perturbative method, and discuss their implications in the effective classical dynamics.