Minisuperspace Quantum Cosmology in Metric and Affine Theories of Gravity

Salvatore Capozziello,Francesco Bajardi

doi:10.3390/universe8030177

Abstract

Minisuperspace Quantum Cosmology is an approach by which it is possible to infer initial conditions for dynamical systems which can suitably represent observable and non-observable universes. Here we discuss theories of gravity which, from various points of view, extend Einstein’s General Relativity. Specifically, the Hamiltonian formalism for f(R), f(T), and f(G) gravity, with R, T, and G being the curvature, torsion and Gauss–Bonnet scalars, respectively, is developed starting from the Arnowitt–Deser–Misner approach. The Minisuperspace Quantum Cosmology is derived for all these models and cosmological solutions are obtained thanks to the existence of Noether symmetries. The Hartle criterion allows the interpretation of solutions in view of observable universes.

Highlights

The Arnowitt–Deser–Misner (ADM) formalism was developed in 1962 with the purpose of solving issues occurring in the attempt to merge the formalism of General Relativity (GR) with Quantum Mechanics [1]
The ADM formalism is not considered as the ultimate candidate to solve the quantization problem of GR, both because it does not account for a full theory of Quantum Gravity and because it implies an infinite-dimensional superspace which cannot be handled
We show that the ADM formalism can be suitably applied to cosmology, where the superspace can be reduced to a minisuperspace of configurations where the Wheeler–De Witt (WDW) equation, under some constraints, can be exactly solved

Summary

Introduction

The Arnowitt–Deser–Misner (ADM) formalism was developed in 1962 with the purpose of solving issues occurring in the attempt to merge the formalism of General Relativity (GR) with Quantum Mechanics [1]. The Gauss–Bonnet term can reduce dynamics and provide analytic solutions for the field equations; it naturally emerges in gauge theories of gravity such as Lovelock, Born–Infeld or Chern–Simons gravity [60–63] Another class of alternatives to GR is represented by those models relaxing the assumption of torsionless and metric-compatible connections. It is possible to show that the most general affine connection is made of three different contributions, respectively related to curvature, torsion, and non-metricity. We consider an extension of the TEGR action, containing a function of the torsion This can allow to address the problems suffered by GR at large scales. GR and TEGR are dynamically equivalent, f (R) gravity differs from f (T) gravity As the former leads to fourth-order field equations, the latter provides secondorder equations with respect to the metric.

Quantum Cosmology and Noether Symmetries

Discussion and Conclusions

Methods