Gauge invariant variables for cosmological perturbation theory using geometrical clocks

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.