The Born-Oppenheimer approach to quantum cosmology

The scope of this paper is to compare two different approaches for solving the Wheeler-DeWitt (WDW) equation in the presence of homogeneous matter (inflaton) and perturbations around it. The standard Born-Oppenheimer (BO) decomposition, which consists of factorising out the gravitational wave function and then defining the flow of the time through it, and a more general BO decomposition where the whole minisuperspace wave function (i.e. scale factor and homogeneous scalar field) is factorized out. The two approaches are compared, for simplicity, in the case of a minimally coupled inflaton with a flat potential. Besides being exactly solvable, the model we study is also relevant as a toy model for ultra slow roll and primordial black holes formation. The consistency of the latter decomposition is checked against the former by comparing the resulting perturbation (Mukhanov-Sasaki) equations. Finally a few solutions to the homogeneous WDW not suitable for the traditional BO treatment are presented and the corresponding Mukhanov-Sasaki equations are evaluated.