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

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.

Quantum gravity and gravitational-wave astronomy

“…We excluded that model as an interesting candidate only when combining luminosity-distance observations with constraints on the propagation speed s GW of the tensor perturbation. It may be that other LQC scenarios, such as the hybrid-quantization approach [156,157], could evade speed constraints (because s GW = 1 there) and, at the same time, admit a quantum state with sizable IR corrections.…”

Nonadiabatic evolution of primordial perturbations and non-Gaussinity in hybrid approach of loop quantum cosmology

“…For details about the hybrid quantization of cosmological perturbations, we refer the reader to Refs. [66,68,69,72,77] and references therein.…”

“…In the hybrid quantization approach to cosmological perturbations, the effective equation of motion for the cosmological scalar and tensor perturbations are described, respectively, by [72,77]…”

“…As shown in [77], one of the main characteristics of cosmological perturbations in the hybrid quantization approach is that the time-dependent effective mass of perturbation modes is positive near by the quantum bounce. To see this clearly, let us define the effective mass function for both scalar and tensor perturbations as…”

“…For the dressed metric and hybrid quantization approaches, the non-adiabatic evolutions are generated by an effective time-dependent mass associated with the perturbations. The main discrepancy of the hybrid approach, as shown in [77], is that the effective mass is positive at the quantum bounce for the evolution of the background that is dominated by the kinetic energy of the inflaton field at the bounce, while this mass is always nonpositive in the dressed metric approach. It has been shown in the dressed metric approach that, the nonpositive mass can lead to prominent effects on primordial perturbation spectra at large scales well within current observational constraints [49,50].…”