On a close relationship between the dressed metric and the hybrid approach to perturbations in effective loop quantum cosmology

The dressed metric and the hybrid approach to perturbations are the two main approaches to capture the effects of quantum geometry in the primordial power spectrum in loop quantum cosmology. Both consider Fock quantized perturbations over a loop quantized background and result in very similar predictions except for the modes which exit the horizon in the effective spacetime in the Planck regime. Understanding precise relationship between both approaches has so far remained obscured due to differences in construction and technical assumptions. We explore this issue at the classical and effective spacetime level for linear perturbations, ignoring backreaction, which is the level at which practical computations of the power spectrum in both of the approaches have so far been performed. We first show that at the classical level both the approaches lead to the same Hamiltonian up to the second order in perturbations and result in the same classical mass functions in the Mukhanov-Sasaki equation on the physical solutions. At the effective spacetime level, the difference in phenomenological predictions between the two approaches in the Planck regime can be traced to whether one uses the Mukhanov-Sasaki variable $Q_{\vec k}$ (the dressed metric approach) or its rescaled version $\nu_{\vec k}=aQ_{\vec k}$ (the hybrid approach) to write the Hamiltonian of the perturbations, and associated polymerization ambiguities. It turns out that if in the dressed metric approach one chooses to work with $\nu_{\vec{k}}$, the effective mass function can be written exactly as in the hybrid approach, thus leading to identical phenomenological predictions in all regimes. Our results explicitly show that the dressed metric and the hybrid approaches for linear perturbations, at a practical computational level, can be seen as two sides of the same coin.