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Loop quantum cosmology has recently been applied in order to extend theanalysis of primordial perturbations to the Planck era and discuss the possibleeffects of quantum geometry on the cosmic microwave background. Two approachesto loop quantum cosmology with admissible ultraviolet behavior leading topredictions that are compatible with observations are the so-called hybrid anddressed metric approaches. In spite of their similarities and relations, weshow in this work that the effective equations that they provide for theevolution of the tensor and scalar perturbations are somewhat different. Whenbackreaction is neglected, the discrepancy appears only in the time- dependentmass term of the corresponding field equations. We explain the origin of thisdifference, arising from the distinct quantization procedures. Besides, giventhe 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 bounceoccurs. We prove that the mass of the tensor perturbations is positive in thehybrid approach when the kinetic contribution to the energy density of theinflaton dominates over its potential, as well as for a considerably largesector of backgrounds around that situation, while this mass is alwaysnonpositive in the dressed metric approach. Similar results are demonstratedfor the scalar perturbations in a sector of background solutions that includesthe kinetically dominated ones; namely, the mass then is positive for thehybrid approach, whereas it typically becomes negative in the dressed metriccase. More precisely, this last statement is strictly valid when the potentialis quadratic for values of the inflaton mass that are phenomenologicallyfavored.

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