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III. Interacting Dark Energy: Summary of Models, Pathologies, and Constraints

Marcel van der Westhuizen, Amare Abebe, Eleonora Di Valentino

TL;DR

This work compiles and extends analytical background solutions for eight IDE kernels, yielding $ρ_{ m dm}$, $ρ_{ m de}$, and $h(z)$ with explicit expressions across linear and non-linear interactions. It systematically catalogs theoretical pathologies (imaginary or negative densities and future big rip scenarios) and prescribes parameter-space conditions to avoid them, while assessing each model's impact on the coincidence problem. The authors introduce practical constraining regimes, including +i$w$CDM and fi$w$CDM, to guide data analyses and priors. They outline a program to extend these results to perturbations and broader interaction forms, enabling robust confrontation with current and upcoming cosmological data.

Abstract

We present an overview of the main results from our two companion papers that are relevant for observational constraints on interacting dark energy (IDE) models. We provide analytical solutions for the dark matter and dark energy densities, $ρ_{\rm dm}$ and $ρ_{\rm de}$, as well as the normalized Hubble function $h(z)$, for eight IDE models. These include five linear IDE models, namely $Q=3H(δ_{\rm dm} ρ_{\rm dm} + δ_{\rm de} ρ_{\rm de})$ 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 $Q=3Hδρ_{\rm de}$, together with three non-linear IDE models: $Q=3Hδ\left( \tfrac{ρ_{\rm dm} ρ_{\rm de}}{ρ_{\rm dm}+ρ_{\rm de}} \right)$, $Q=3Hδ\left( \tfrac{ρ_{\rm dm}^2}{ρ_{\rm dm}+ρ_{\rm de}} \right)$, and $Q=3Hδ\left( \tfrac{ρ_{\rm de}^2}{ρ_{\rm dm}+ρ_{\rm de}} \right)$. For these eight models, we present conditions to avoid imaginary, undefined, and negative energy densities. In seven of the eight cases, negative densities arise if energy flows from DM to DE, implying a strong theoretical preference for energy transfer from DE to DM. We also provide conditions to avoid future big rip singularities and evaluate how each model addresses the coincidence problem in both the past and the future. Finally, we propose a set of approaches and simplifying assumptions that can be used when constraining IDE models, by defining regimes that restrict the parameter space according to the behavior researchers are willing to tolerate.

III. Interacting Dark Energy: Summary of Models, Pathologies, and Constraints

TL;DR

This work compiles and extends analytical background solutions for eight IDE kernels, yielding , , and with explicit expressions across linear and non-linear interactions. It systematically catalogs theoretical pathologies (imaginary or negative densities and future big rip scenarios) and prescribes parameter-space conditions to avoid them, while assessing each model's impact on the coincidence problem. The authors introduce practical constraining regimes, including +iCDM and fiCDM, to guide data analyses and priors. They outline a program to extend these results to perturbations and broader interaction forms, enabling robust confrontation with current and upcoming cosmological data.

Abstract

We present an overview of the main results from our two companion papers that are relevant for observational constraints on interacting dark energy (IDE) models. We provide analytical solutions for the dark matter and dark energy densities, and , as well as the normalized Hubble function , for eight IDE models. These include five linear IDE models, namely and four special cases: , , , and , together with three non-linear IDE models: , , and . For these eight models, we present conditions to avoid imaginary, undefined, and negative energy densities. In seven of the eight cases, negative densities arise if energy flows from DM to DE, implying a strong theoretical preference for energy transfer from DE to DM. We also provide conditions to avoid future big rip singularities and evaluate how each model addresses the coincidence problem in both the past and the future. Finally, we propose a set of approaches and simplifying assumptions that can be used when constraining IDE models, by defining regimes that restrict the parameter space according to the behavior researchers are willing to tolerate.

Paper Structure

This paper contains 14 sections, 28 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Analytical solutions for the energy densities and Hubble function for 8 IDE models.
  3. Linear IDE models
  4. Linear IDE model 1: $Q=3 H (\delta_{\text{dm}} \rho_{\text{dm}} + \delta_{\text{de}} \rho_{\text{de}})$
  5. Linear IDE model 2: $Q=3H\delta( \rho_{\text{dm}}+\rho_{\text{de}})$
  6. Linear IDE model 3: $Q=3H\delta( \rho_{\text{dm}}-\rho_{\text{de}})$
  7. Linear IDE model 4: $Q=3H\delta \rho_{\text{dm}}$
  8. Linear IDE model 5: $Q=3H\delta \rho_{\text{de}}$
  9. Non-linear IDE models
  10. Non-linear IDE model 1: $Q=3H\delta \left( \frac{\rho_{\text{dm}} \rho_{\text{de}} }{\rho_{\text{dm}}+\rho_{\text{de}}} \right)$
  11. Non-linear IDE model 3: $Q=3H\delta \left( \frac{\rho_{\text{de}}^2}{\rho_{\text{dm}}+\rho_{\text{de}}} \right)$
  12. Conditions to avoid pitfalls in the parameter space for each IDE model
  13. Some approaches and simplifying assumptions when constraining IDE models
  14. Conclusions

Figures (1)

  • Figure 1: 2D Portraits of the parameter space for each of the 8 IDE models, using Table \ref{['tab:Com_real']}, \ref{['tab:Com_PEC']} and \ref{['tab:Com_AE_BR']}. Blue areas indicate where the model has positive energy densities throughout all of cosmic evolution. Pink areas indicate that negative energies will occur in either the past or future expansion. The gray overlay indicates the presence of imaginary energy densities, while the purple areas show undefined energy densities. Lastly, the green mesh indicates the presence of future big rip singularities. The purple areas in the bottom two panels may have undefined values at some scale factor, but it is not guaranteed, see equations \ref{['NLID2_dm_BG']}, \ref{['NLID2_de_BG']}, \ref{['NLID3_dm_BG']} and \ref{['NLID3_de_BG']}. For all cases $\Omega_{\rm{(dm,0)}}=0.266$, $\Omega_{\rm{(de,0)}}=0.685$ (implying $r_0=0.388$). For linear IDE model 1, we set $w=-1$.