  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.
