Classical and loop quantum cosmology phase space of interacting dark energy and superfluid dark matter

In this paper we study in detail the phase space of a cosmological system consisting of two coupled fluids, namely, a dark energy fluid coupled with a superfluid dark matter fluid. The dark matter fluid is assumed to have a superfluid equation of state; hence, it is not pressureless and our aim is to find the impact of this nontrivial equation of state on the phase space of the coupled system. We shall use two theoretical contexts, namely, that of classical cosmology and that of loop quantum cosmology. In the classical case, we investigated the existence and stability of fixed points, and as we will show, no de Sitter fixed points occur, however, matter and radiation domination fixed points occur, which arc hyperbolic and unstable. We also show that there exist limited sets of initial conditions for which singular solutions occur in the phase space. With regard to the loop quantum cosmology case, we demonstrate that stable de Sitter fixed points exist, for some values of the free parameters of the theory, and interestingly enough, for the same values, singular solutions corresponding to general sets of initial conditions occur. To our knowledge this feature does not occur so frequently in loop quantum cosmological frameworks. We also demonstrate that nonsingular solutions corresponding to a general set of initial conditions occur; however, these occur when the dark matter superfluid has negative pressure, so it is a rather physically unappealing situation.