  Using the extended ADM-phase space formulation in the canonical framework we
analyze the relationship between various gauge choices made in cosmological
perturbation theory and the choice of geometrical clocks in the relational
formalism. We show that various gauge invariant variables obtained in the
conventional analysis of cosmological perturbation theory correspond to Dirac
observables tied to a specific choice of geometrical clocks. As examples, we
show that the Bardeen potentials and the Mukhanov-Sasaki variable emerge
naturally in our analysis as observables when gauge fixing conditions are
determined via clocks in the Hamiltonian framework. Similarly other gauge
invariant variables for various gauges can be systematically obtained. We
demonstrate this by analyzing five common gauge choices: longitudinal,
spatially flat, uniform field, synchronous and comoving gauge. For all these,
we apply the observable map in the context of the relational formalism and
obtain the corresponding Dirac observables associated with these choices of
clocks. At the linear order, our analysis generalizes the existing results in
canonical cosmological perturbation theory twofold. On the one hand we can
include also gauges that can only be analyzed in the context of the extended
ADM-phase space and furthermore, we obtain a set of natural gauge invariant
variables, namely the Dirac observables, for each considered choice of gauge
conditions. Our analysis provides insights on which clocks should be used to
extract the relevant natural physical observables both at the classical and
quantum level. We also discuss how to generalize our analysis in a
straightforward way to higher orders in the perturbation theory to understand
gauge conditions and the construction of gauge invariant quantities beyond
linear order.
