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Exact cosmological solutions of models with an interacting dark sector

A. B. Pavan, Elisa G. M. Ferreira, Sandro M. R. Micheletti, J. C. C. de Souza, E. Abdalla

TL;DR

This work addresses the challenge of obtaining exact cosmological solutions in models where dark energy (a canonical scalar field) interacts with dark matter (a fermionic field). It extends the First Order Formalism by treating the Hubble parameter as a function of the scalar field, $H(t)=W(φ)$, and requiring $φ(t)$ to be invertible, to derive first-order equations and exact solutions. The authors present four explicit examples showing how different choices of $n$, $W(φ)$, and $J(φ)$ yield diverse histories, including de Sitter-like acceleration, early deceleration with late-time acceleration, and Chaplygin-gas–like dynamics, with the interaction playing a variable role across cosmic time. The results illustrate that the FOF provides a robust route to tractable, exact interacting dark-sector cosmologies and establish a foundation for future observational constraints and model refinement. The work highlights how the strength and form of the dark-sector coupling, together with the invertibility constraint in the scalar field, shape the viability of cosmological histories.

Abstract

In this work we extend the first order formalism for cosmological models that present an interaction between a fermionic and a scalar field. Cosmological exact solutions describing universes filled with interacting dark energy and dark matter have been obtained. Viable cosmological solutions with an early period of decelerated expansion followed by late acceleration have been found, notably one which presents a dark matter component dominating in the past and a dark energy component dominating in the future. In another one, the dark energy alone is the responsible for both periods, similar to a Chaplygin gas case. Exclusively accelerating solutions have also been obtained.

Exact cosmological solutions of models with an interacting dark sector

TL;DR

This work addresses the challenge of obtaining exact cosmological solutions in models where dark energy (a canonical scalar field) interacts with dark matter (a fermionic field). It extends the First Order Formalism by treating the Hubble parameter as a function of the scalar field, , and requiring to be invertible, to derive first-order equations and exact solutions. The authors present four explicit examples showing how different choices of , , and yield diverse histories, including de Sitter-like acceleration, early deceleration with late-time acceleration, and Chaplygin-gas–like dynamics, with the interaction playing a variable role across cosmic time. The results illustrate that the FOF provides a robust route to tractable, exact interacting dark-sector cosmologies and establish a foundation for future observational constraints and model refinement. The work highlights how the strength and form of the dark-sector coupling, together with the invertibility constraint in the scalar field, shape the viability of cosmological histories.

Abstract

In this work we extend the first order formalism for cosmological models that present an interaction between a fermionic and a scalar field. Cosmological exact solutions describing universes filled with interacting dark energy and dark matter have been obtained. Viable cosmological solutions with an early period of decelerated expansion followed by late acceleration have been found, notably one which presents a dark matter component dominating in the past and a dark energy component dominating in the future. In another one, the dark energy alone is the responsible for both periods, similar to a Chaplygin gas case. Exclusively accelerating solutions have also been obtained.

Paper Structure

This paper contains 9 sections, 38 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Interacting Dark Energy and Dark Matter Model
  3. First Order Formalism - FOF
  4. Exact Solutions for the Interacting Dark Energy and Dark Matter Models with FOF
  5. $1^{st}$ Example:
  6. $2^{nd}$ Example:
  7. $3^{rd}$ Example:
  8. $4^{th}$ Example:
  9. Conclusions

Figures (4)

  • Figure 1: Density parameter $\Omega$ (left panel) and equation of state parameter $w$ and acceleration parameter (right panel) for the model represented by eqs. \ref{['rho_1']}-\ref{['rho_1_int']}.
  • Figure 2: Density parameter $\Omega$ (left panel) and equation of state parameter $w$ and acceleration parameter (right panel) for the model represented by Eqs.(\ref{['rho_2']}-\ref{['rho_2_int']}) .
  • Figure 3: Density parameter $\Omega$ (left panel) and equation of state parameter $\omega$ and acceleration parameter (right panel) for the model represented by Eqs.(\ref{['phi_3']}-\ref{['rho_3_int']}). The equation of state of dark matter is normalized in order to make the graph easier to see in a way still holds its main features.
  • Figure 4: Density parameter $\Omega$ (left panel) and equation of state parameter $w$ and acceleration parameter (right panel) for the model represented by eqs. \ref{['phi_4']}-\ref{['rho_4_int']}.