Quantum Dynamics of the Isotropic Universe in the Metric f(R) Gravity

Abstract

We analyse the canonical quantum dynamics of the isotropic universe, as emerging from the Hamiltonian formulation of a metric f(R) gravity, viewed in the Jordan frame. The canonical method of quantization is performed by solving the Hamiltonian constraint before quantizing and adopting like a relational time the non-minimally coupled scalar field emerging in the Jordan frame. The resulting Schröedinger evolution is then investigated both in the vacuum and in the presence of a massless scalar field, though as the kinetic component of an inflaton. We show that, in vacuum, the morphology of localized wave packets is that of a non-spreading profile up to the cosmological singularity. When the external scalar field is included into the dynamics, we see that the wave packets acquire the surprising feature of increasing localization of the universe volume, as it expands. This result suggests that in the metric f(R) formulation of gravity, a spontaneous mechanism arises for the universe classicization. Actually, when the phase space of the scalar field is fully explored, such an increasing localization in the Universe volume is valid up to a given value of the time, i.e. of the non-minimally coupled mode after which the wave packets spread again. We conclude our analysis by inferring that before this critical transition age is reached, the inflationary phase could take place, here modelled via a cosmological constant. This point of view provides an interesting scenario for the transition from a Planckian Universe to a classical de-Sitter phase, which in the f(R) gravity appears more natural than in the Einsteinian picture.