Gauge invariant canonical cosmological perturbation theory with geometrical clocks in extended phase-space — A review and applications

Abstract

The theory of cosmological perturbations is a well-elaborated field and has been successfully applied, e.g. to model the structure formation in our universe and the prediction of the power spectrum of the cosmic microwave background. 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.