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The aim of this article is to understand the role of dust reference fields,often also called clocks, on cosmological perturbations around a flat FLRWuniverse. We derive the Mukhanov-Sasaki (MS) equation for the Brown-Kuchar (BK)and Gaussian dust models, which both consider four dust fields as referencefields. The reduced phase space of Dirac observables, that is thegauge-invariant part of the theory, is constructed by means of an observablemap applied to all elementary phase space variables of the coupled system,consisting of gravity, a massive scalar field and the dust degrees of freedom.The evolution of these observables is governed by a so called physicalHamiltonian which can be derived once the set of reference fields are chosenand differs for each model. First, the reduced phase space as well as thecorresponding equations of motion are derived for full general relativity. Thenfrom this, the gauge-invariant version of the equations of motion for thebackground are derived which contain a fingerprint of the dust referencefields. Afterwards we study linear cosmological perturbations around a FLRWmetric using the scalar-vector-tensor decomposition and derive the equation ofmotion for the MS variable in this formalism for a chosen set of variables onthe reduced phase space and expressed in terms of Dirac observables. The MSequation involves additional contributions that can be understood as backreactions from the dust reference fields. These additional dust contributionsto the MS equation were absent if the dust energy and momentum density as wellas their perturbations are vanishing. The nature of the correction termssuggest that BK and Gaussian dust reference fields contribute differently.Using numerical simulations we study the behavior of the dust contributions tothe MS equation during inflation.

Mukhanov-Sasaki equation in manifestly gauge-invariant linearized cosmological perturbation theory with dust reference fields