  The relational formalism based on geometrical clocks and Dirac observables in
linearized canonical cosmological perturbation theory is used to introduce an
efficient method to find evolution equations for gauge invariant variables. Our
method generalizes an existing technique by Pons, Salisbury and Sundermeyer [1,
2] to relate the evolution of gauge invariant observables with the one of gauge
variant quantities, and is applied as a demonstration for the longitudinal and
spatially flat gauges. Gauge invariant evolution equations for the Bardeen
potential and the Mukhanov-Sasaki variable are derived in the extended ADM
phase space. Our method establishes a full agreement at the dynamical level
between the canonical and conventional cosmological perturbation theory at the
linear order using Dirac observables.
