  The theory of cosmological perturbations is a well elaborated field. To deal
with the diffeomorphism invariance of general relativity one generally
introduces combinations of the metric and matter perturbations which are gauge
invariant up to the considered order in the perturbations. For linear
cosmological perturbations one works with the so-called Bardeen potentials
widely used in this context. However, there exists no common procedure to
construct gauge invariant quantities also for higher order perturbations.
Usually, one has to find new gauge invariant quantities independently for each
order in perturbation theory. With the relational formalism introduced by
Rovelli and further developed by Dittrich and Thiemann, it is in principle
possible to calculate manifestly gauge invariant quantities, that is quantities
that are gauge invariant up to arbitrary order once one has chosen a set of
so-called reference fields, often also called clock fields. This article
contains a review of the relational formalism and its application to canonical
general relativity following the work of Garcia, Pons, Sundermeyer and
Salisbury. As the starting point for our application of this formalism to
cosmological perturbation theory, we also review the Hamiltonian formulation of
the linearized theory for perturbations around FLRW spacetimes. The main aim of
our work will be to identify clock fields in the context of the relational
formalism that can be used to reconstruct quantities like the Bardeen potential
as well as the Mukhanov-Sasaki variable. This requires a careful analysis of
the canonical formulation in the extended ADM-phase space where lapse and shift
are treated as dynamical variables. The actual construction of such observables
and further investigations thereof will be carried out in our companion paper.
