Linear and nonlinear interactions in the dark sector

TL;DR

The paper investigates two interacting dark components in a spatially flat FRW universe with $3H^2=\rho_c+\rho_x$ and $\rho'=-\gamma_c\rho_c-\gamma_x\rho_x$, showing that an effective one-fluid description can capture the combined dynamics and relate interacting models to unified descriptions through a source equation for the total density. It develops a hierarchy of couplings, including linear (Q_l, Q_L, and generalizations) and nonlinear interactions (Q_nL), derives exact solutions for the total density and the scale factor, and identifies stable attractor states $\gamma_s$ (and $\gamma_L^+$) that can alleviate the coincidence problem; these couplings yield a variety of late-time behaviors, from power-law to de Sitter and Chaplygin-gas–like equations of state. The framework provides explicit mappings between unified models (e.g., Chaplygin gas) and interacting two-fluid models, including prescriptions to derive an interaction from a given unified density evolution, and demonstrates how relaxing or constraining the interaction tunes the effective equation of state to reproduce known cosmological scenarios. Overall, the study offers a versatile method to generate and analyze dark-sector cosmologies with stable attractors and ΛCDM–like late-time behavior, while clarifying the relationship between interacting and unified approaches to dark energy and dark matter.

Abstract

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".