Time-dependent mass of cosmological perturbations in the hybrid and dressed metric approaches to loop quantum cosmology

TL;DR

This paper compares the hybrid and dressed metric approaches in loop quantum cosmology, focusing on their differences in the effective equations for cosmological perturbations, especially regarding the time-dependent mass during the big bounce.

Contribution

It identifies the origin of differences in the effective equations between the hybrid and dressed metric approaches and analyzes the positivity of the perturbation mass at the bounce.

Findings

The discrepancy in the equations arises from distinct quantization procedures.

Tensor perturbation mass is positive in the hybrid approach under certain conditions.

In the dressed metric approach, the tensor and scalar masses are often nonpositive, especially with quadratic potentials.

Abstract

Loop quantum cosmology has recently been applied in order to extend the analysis of primordial perturbations to the Planck era and discuss the possible effects of quantum geometry on the cosmic microwave background. Two approaches to loop quantum cosmology with admissible ultraviolet behavior leading to predictions that are compatible with observations are the so-called hybrid and dressed metric approaches. In spite of their similarities and relations, we show in this work that the effective equations that they provide for the evolution of the tensor and scalar perturbations are somewhat different. When backreaction is neglected, the discrepancy appears only in the time- dependent mass term of the corresponding field equations. We explain the origin of this difference, arising from the distinct quantization procedures. Besides, given the privileged role that the big bounce plays in loop quantum cosmology, e.g. as a natural instant of time to set initial conditions for the perturbations, we also analyze the positivity of the time-dependent mass when this bounce occurs. We prove that the mass of the tensor perturbations is positive in the hybrid approach when the kinetic contribution to the energy density of the inflaton dominates over its potential, as well as for a considerably large sector of backgrounds around that situation, while this mass is always nonpositive in the dressed metric approach. Similar results are demonstrated for the scalar perturbations in a sector of background solutions that includes the kinetically dominated ones; namely, the mass then is positive for the hybrid approach, whereas it typically becomes negative in the dressed metric case. More precisely, this last statement is strictly valid when the potential is quadratic for values of the inflaton mass that are phenomenologically favored.