Dynamics of Dirac observables in canonical cosmological perturbation theory

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.