I. Linear Interacting Dark Energy: Analytical Solutions and Theoretical Pathologies

TL;DR

This work analyzes five linear IDE kernels in a flat FLRW setting, focusing on the physical viability of DM and DE densities, the emergence of negative-energy regimes, and the potential for future big rip singularities. Through a detailed dynamical-systems treatment and exact analytical solutions for ρ_{dm} and ρ_{de}, the authors derive conditions that preserve positive densities and avoid abrupt future catastrophes, highlighting that energy transfer from DE to DM generally mitigates pathologies while DM-to-DE transfer worsens them. They reconstruct a dynamical dark-energy equation of state tilde{w}(z) for each kernel and employ statefinders to differentiate models from ΛCDM, finding that observational constraints should favor small DE-to-DM transfer. The study emphasizes that, although IDE models can address certain cosmological tensions, their viability is tightly constrained by positivity, stability, and future-singularity considerations, and it advocates caution in interpreting phantom-crossing signals without a consistent background theory. These results provide a theoretical map of the IDE parameter space, facilitating robust observational tests and guiding future explorations of linear and non-linear kernels in the dark sector.

Abstract

Interacting dark energy (IDE) models, in which dark matter (DM) and dark energy (DE) exchange energy through a non-gravitational interaction, have long been proposed as candidates to address key challenges in modern cosmology. These include the coincidence problem, the $H_0$ and $S_8$ tensions, and, more recently, the hints of dynamical dark energy reported by the DESI collaboration. Given the renewed interest in IDE models, it is crucial to fully understand their parameter space when constraining them observationally, especially with regard to the often-neglected issues of negative energy densities and future big rip singularities. In this work, we present a comparative study of the general linear interaction $Q=3H(δ_{\rm dm}ρ_{\rm dm} + δ_{\rm de}ρ_{\rm de})$ and four special cases: $Q=3Hδ(ρ_{\rm dm}+ρ_{\rm de})$, $Q=3Hδ(ρ_{\rm dm}-ρ_{\rm de})$, $Q=3Hδρ_{\rm dm}$, and $Q=3Hδρ_{\rm de}$. For these five models, we perform a dynamical system analysis and derive new conditions that ensure positive, real, and well-defined energy densities throughout cosmic evolution, as well as criteria to avoid future big rip singularities. We obtain exact analytical solutions for $ρ_{\rm{dm}}$, $ρ_{\rm{de}}$, the effective equations of state ($w_{\mathrm{eff}}^{\rm{dm}}$, $w_{\mathrm{eff}}^{\rm{de}}$, $w_{\mathrm{eff}}^{\rm{tot}}$), and a reconstructed dynamical DE equation of state $\tilde{w}$. Using these results, we examine phantom crossings, address the coincidence problem, and apply the statefinder diagnostic to distinguish between models. We show that energy transfer from DM to DE inevitably produces negative energy densities and make future singularities more likely, while transfer from DE to DM avoids these pathologies and is thus theoretically favored.

Figures (22)

• Figure 1: Interaction strength $Q$ relative to $H \rho_{\rm{tot}}$ as a function of redshift for the interaction $Q = 3H (\delta_{\text{dm}} \rho_{\rm{dm}} + \delta_{\text{de}} \rho_{\rm{de}})$, shown for different special cases. The plot illustrates when the effect of the interaction becomes dominant during cosmic evolution. If $Q \propto \rho_{\rm{dm}}$, the interaction primarily affects the early-time dynamics, whereas if $Q \propto \rho_{\rm{de}}$, it mainly influences the late-time and future expansion. Sign-switching behavior occurs only when $\delta_{\text{dm}}$ and $\delta_{\text{de}}$ have opposite signs.
• Figure 2: Parameter space of the eigenvalues for the critical point \ref{['DSA.Q.ddm+dde.13']} - $Q= 3 H (\delta_{\text{dm}} \rho_{\text{dm}} + \delta_{\text{de}} \rho_{\text{de}})$
• Figure 3: Parameter space of the eigenvalues for the critical point \ref{['DSA.Q.ddm+dde.16']} - $Q= 3 H (\delta_{\text{dm}} \rho_{\text{dm}} + \delta_{\text{de}} \rho_{\text{de}})$
• Figure 4: 3D phase portrait for $Q= 3 H (\delta_{\text{dm}} \rho_{\text{dm}} + \delta_{\text{de}} \rho_{\text{de}})$, showing positive-energy trajectories in the iDEDM regime ($\delta_{\text{dm}}=\delta_{\text{de}}=+0.1$, left panel) and negative-energy trajectories in the iDMDE regime ($\delta_{\text{dm}}=\delta_{\text{de}}=-0.1$, right panel). The trajectories also show radiation, matter (DM+BM) and DE-dominated eras.
• Figure 5: 2D projection of the phase portrait for $Q= 3 H (\delta_{\text{dm}} \rho_{\text{dm}} + \delta_{\text{de}} \rho_{\text{de}})$, showing positive-energy trajectories in the iDEDM regime ($\delta_{\text{dm}}=\delta_{\text{de}}=+0.1$, left panel) and negative-energy trajectories in the iDMDE regime ($\delta_{\text{dm}}=\delta_{\text{de}}=-0.1$, right panel).