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

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.

Figure 1

Left panel: Evolution, on the expanding branch of the Universe, of the Mukhanov-Sasaki potentials $\mathcal{U}$ and $\mathcal{V}$, corresponding respectively to the hybrid and dressed metric approaches. Right panel: Relative contribution of the Mukhanov-Sasaki potential to the corresponding time-dependent mass, denoted by $s^{(\mathrm{s})}$ in the hybrid approach and by ${\breve{s}}^{(\mathrm{s})}$ in the dressed metric approach. Here, we consider a quadratic potential, $V(\varphi )=m^2{\varphi }^2/2$. In Planck units, the inflaton field at the bounce and the parameters of the model are taken equal to ${\varphi }_{\mathrm{B}}=0.97$, $m=1.20\times {10}^{-6}$, and $\gamma =0.2375$. We represent the absolute value of the considered quantities, distinguishing between positive and negative values by employing solid and dashed lines, respectively. The black vertical dashed line marks the onset of the slow-roll phase.

Figure 2

Left panel: Evolution on the expanding branch of the Universe of the time-dependent masses for the tensor and the Mukhanov-Sasaki perturbations in the hybrid approach, denoted by $s^{(\mathrm{t})}$ and $s^{(\mathrm{s})}$, respectively, and in the dressed metric approach, denoted by ${\breve{s}}^{(\mathrm{t})}$ and ${\breve{s}}^{(\mathrm{s})}$, respectively. Right panel: Relative difference between the value of the time-dependent mass in the two approaches, both for the tensor and for the Mukhanov-Sasaki perturbations. In this right panel, solid (dashed) lines correspond to positive (negative) values of the quantities for which we represent the absolute value. Here, we consider a quadratic potential, $V(\varphi )=m^2{\varphi }^2/2$. In Planck units, the inflaton field at the bounce and the parameters of the model are taken equal to ${\varphi }_{\mathrm{B}}=0.97$, $m=1.20\times {10}^{-6}$, and $\gamma =0.2375$.

Figure 3

Left panel: The time-dependent Mukhanov-Sasaki mass at the bounce in the dressed metric approach as a function of the value of the potential at the bounce, for the case of the quadratic potential, $V(\varphi )=m^2{\varphi }^2/2$, with $m=1.20\times {10}^{-6}$ in Planck units. Again, we take $\gamma =0.2375$. We consider the two possibilities of positive and negative values of the inflaton at the bounce. Right panel: Zoom of the two regions in which the time-dependent mass changes sign. In these regions, we also plot the polynomials $P_{\pm }$ and their zeros.