We develop a field-theoretic description of large-scale structure formation by taking the non-relativistic limit of a canonically transformed, real scalar field which is minimally coupled to scalar gravitational perturbations in longitudinal gauge. We integrate out the gravitational constraint fields and arrive at a non-local action which is only specified in terms of the dynamical degrees of freedom. In order to make this framework closer to the classical particle description, we construct the corresponding 2PI effective action truncated at two loop order for a non-squeezed state without field expectation values. We contrast the dynamical description of the coincident time phase-space density to the standard Vlasov description of cold dark matter particles and identify momentum and time scales at which linear perturbation theory will deviate from the standard evolution.
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