Scalar field versus hydrodynamic models in homogeneous isotropic cosmology

We study relations between hydrodynamical (H) and scalar field (SF) models of the dark energy in the homogeneous isotropic universe. The focus is on the SF described by the Lagrangian with the canonical kinetic term within spatially flat cosmology. We analyze requirements that guarantee the same cosmological history for the SF and H models at least for special solutions. The differential equation for the SF potential is obtained that ensures such equivalence of the SF and H models. However, if the “equivalent” SF potential is found for a given equation of state (EOS) of the H model, this does not mean that all solutions of this SF model have corresponding H-model analogs. In this view we derived a condition that guarantees an “approximate equivalence,” when there is a small difference between energy-momentum tensors of the models. The equivalent SF potentials and corresponding SF solutions for linear EOS are found in an explicit form; we also present examples with more complicated EOS.