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Models with dust reference fields in relational formalism have proved usefulin understanding the construction of gauge-invariant perturbation theory toarbitrary orders in the canonical framework. These reference fields modify thedynamical equations for perturbation equations. However, important questionsremain open on the relation with conventional perturbation theories of inflatoncoupled to gravity and of multi-fluid systems, and on understandingmodifications in terms of physical degrees of freedom. These gaps are filled inthis manuscript for Brown-Kucha\v{r} and Gaussian dust models, both of whichinvolve three scalar physical degrees of freedom. We establish a relationshipof these models with conventional inflationary and multi-fluid system ofinflation and ordinary dust by introducing a set of gauge invariant variableson the reduced phase space of the dust reference models. We find themodifications due to dust clocks to Bardeen equation in the longitudinal gaugeand Mukhanov-Sasaki equation in the spatially-flat gauge, in terms of physicaldegrees of freedom. This results in a closed system of equations for all thedegrees 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 isa natural choice of the set of gauge invariant variables for each chosen gaugewhich not only offers a direct physical interpretation but also results insimplifications to the dynamical equations.

Relating dust reference models to conventional systems in manifestly gauge invariant perturbation theory