Linear and nonlinear interactions in the dark sector

We investigate models of interacting dark matter and dark energy for the universe in a spatially flat Friedmann-Robertson-Walker (FRW) space-time. We find the "source equation" for the total energy density and determine the energy density of each dark component. We introduce an effective one-fluid description to evidence that interacting and unified models are related with each other, analyze the effective model and obtain the attractor solutions. We study linear and nonlinear interactions, the former comprises a linear combination of the dark matter and dark energy densities, their first derivatives, the total energy density, its first and second derivatives and a function of the scale factor. The latter is a possible generalization of the linear interaction consisting of an aggregate of the above linear combination and a significant nonlinear term built with a rational function of the dark matter and dark energy densities homogeneous of degree one. We solve the evolution equations of the dark components for both interactions and examine exhaustively several examples. There exist cases where the effective one-fluid description produces different alternatives to the $\La$CDM model and cases where the problem of coincidence is alleviated. In addition, we find that some nonlinear interactions yield an effective one-fluid model with a Chaplygin gas equation of state, whereas others generate cosmological models with de Sitter and power-law expansions. We show that a generic nonlinear interaction induces an effective equation of state which depends on the scale factor in the same way that the variable modified Chaplygin gas model, giving rise to the "relaxed Chaplygin gas model".