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. 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.

arXiv commentAdded some references, corrected typos

References in corpus (22)

• Planck 2015 results. XIII. Cosmological parameters
• Partial observables
• Partial and Complete Observables for Hamiltonian Constrained Systems
• Partial and Complete Observables for Canonical General Relativity
• Time and a physical Hamiltonian for quantum gravity
• Gauge transformations in the Lagrangian and Hamiltonian formalisms of generally covariant theories
• Algebraic Quantum Gravity (AQG) IV. Reduced Phase Space Quantisation of Loop Quantum Gravity
• Gravity quantized
• Scalar Material Reference Systems and Loop Quantum Gravity
• Reduced Phase Space Quantization and Dirac Observables
• Scalar and Vector Perturbations in Quantum Cosmological Backgrounds
• Revisiting observables in generally covariant theories in the light of gauge fixing methods
• Manifestly Gauge-Invariant General Relativistic Perturbation Theory: I. Foundations
• Manifestly Gauge-Invariant General Relativistic Perturbation Theory: II. FRW Background and First Order
• Gauge invariant perturbations around symmetry reduced sectors of general relativity: applications to cosmology
• Cosmological perturbation theory
• Observables in classical canonical gravity: folklore demystified
• Scalar Perturbations in Scalar Field Quantum Cosmology
• Rigid and gauge Noether symmetries for constrained systems
• The Hole Argument for Covariant Theories
• LTB spacetimes in terms of Dirac observables
• Manifestly Gauge Invariant Perturbations of Scalar-Tensor Theories of Gravity

Cited by in corpus (11)

• Towards a reduced phase space quantization in loop quantum cosmology with an inflationary potential
• Gauge invariant variables for cosmological perturbation theory using geometrical clocks
• Mukhanov-Sasaki equation in manifestly gauge-invariant linearized cosmological perturbation theory with dust reference fields
• Single Graviton Loop Contribution to the Self-Mass of a Massless, Conformally Coupled Scalar on de Sitter Background
• Dynamics of Dirac observables in canonical cosmological perturbation theory
• Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the context of adiabatic vacua and the Lewis-Riesenfeld invariant
• Relating dust reference models to conventional systems in manifestly gauge invariant perturbation theory
• Hamiltonian formalism and gauge-fixing conditions for cosmological perturbation theory
• On consistent gauge fixing conditions in polymerized gravitational systems
• One-loop quantum gravitational backreaction on the local Hubble rate
• Geometrical Quantum Time in the $U(1{)}^3$ Model of Euclidean Quantum Gravity