Primordial tensor modes of the early Universe

TL;DR

This paper develops a hybrid quantization framework for primordial tensor modes in early Universe cosmology, combining canonical transformations and Born-Oppenheimer ansatz to analyze quantum gravitational effects on tensor perturbations.

Contribution

It introduces a hybrid quantization scheme for tensor modes, integrating Fock and quantum background representations, and compares loop quantum cosmology with other approaches.

Findings

Standard Fock quantization with unitary dynamics achieved for tensor modes.

Derived effective evolution equations for primordial tensor perturbations.

Compared loop quantum cosmology results with alternative quantum gravity approaches.

Abstract

We study cosmological tensor perturbations on a quantized background within the hybrid quantization approach. In particular, we consider a flat, homogeneous and isotropic spacetime and small tensor inhomogeneities on it. We truncate the action to second order in the perturbations. The dynamics is ruled by a homogeneous scalar constraint. We carry out a canonical transformation in the system where the Hamiltonian for the tensor perturbations takes a canonical form. The new tensor modes now admit a standard Fock quantization with a unitary dynamics. We then combine this representation with a generic quantum scheme for the homogeneous sector. We adopt a Born-Oppenheimer ansatz for the solutions to the constraint operator, previously employed to study the dynamics of scalar inhomogeneities. We analyze the approximations that allow us to recover, on the one hand, a Schr\"odinger equation similar to the one emerging in the dressed metric approach and, on the other hand, the ones necessary for the effective evolution equations of these primordial tensor modes within the hybrid approach to be valid. Finally, we consider loop quantum cosmology as an example where these quantization techniques can be applied and compare with other approaches.