Assessing the foundation and applicability of some dark energy fluid models in the Dirac-Born-Infeld framework

In this paper, we will deepen the understanding of some fluid models proposed by other authors for the description of dark energy. Specifically, we will show that the so-called (Modified) Berthelot fluid is the hydrodynamic realization of the free Dirac-Born-Infeld theory and that the Dieterici fluid admits a non-relativistic $k$-essence formulation; for the former model the evolution of the scalar field will be written in terms of some cosmographic parameters. The latter model will also be tested using Machine Learning algorithms with respect to cosmic chronometers data, and results about the dynamics at a background level will be compared with those arising when other fluids (Generalized Chaplygin Gas and Anton-Schmidt) are considered. Due to some cosmic opacity effects, the background cosmology of universes filled by these inequivalent fluids, as they arise in physically different theories, may not be enough for discriminating among them. Thus, a perturbation analysis in the long-wavelength limit is carried out revealing a rich variety of possible behaviors. It will also be shown that the free Dirac-Born-Infeld theory cannot account for flat galactic rotation curves, and therefore we derive an appropriate relationship between the scalar field potential and the brane tension for achieving this goal; this provides an estimate for the dark matter adiabatic speed of sound inside the halo consistent with other literature. A certain relationship between the Newtonian gravitational potential within the galaxy and the Lagrangian potential in the non-relativistic regime for the (Modified) Berthelot fluid will also be enlightened.