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Models with dust reference fields in relational formalism have proved useful in understanding the construction of gauge invariant perturbation theory to arbitrary orders in the canonical framework. These reference fields modify the dynamical equations for perturbation equations. However, important questions remain open on the relation with conventional perturbation theories of inflaton coupled to gravity and of multifluid systems, and on understanding modifications in terms of physical degrees of freedom. These gaps are filled in this manuscript for Brown-Kuchar and Gaussian dust models, both of which involve three scalar physical degrees of freedom. We establish a relationship of these models with conventional inflationary and multifluid systems of inflation and ordinary dust by introducing a set of gauge invariant variables on the reduced phase space of the dust reference models. We find the modifications due to dust clocks to the Bardeen equation in the longitudinal gauge and the Mukhanov-Sasaki equation in the spatially flat gauge, in terms of physical degrees of freedom. This results in a closed system of equations for all the degrees of freedom needed to explore the evolution of the scalar perturbations. Our analysis shows for the first time that even for two-fluid systems, there is a natural choice of the set of gauge invariant variables for each chosen gauge which not only offers a direct physical interpretation but also results in simplifications to the dynamical equations.

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Physical Review D