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Cosmological dynamics of interacting dark matter-dark energy in generalized Rastall gravity

Manuel Gonzalez-Espinoza, Ramón Herrera, Giovanni Otalora, Carlos Ríos, Carlos Rodriguez-Benites

TL;DR

This work develops an autonomous dynamical-systems treatment of interacting dark energy and dark matter within generalized Rastall gravity, where non-conservation of the energy–momentum tensor naturally drives the interaction via $Q_1=\alpha\dot{f}$ and $Q_2=-\dot{f}(1+\alpha)$. An explicit closed system in $(\Omega_{de},\Omega_m)$ is derived for three coupling forms $f=\beta\rho_m$, $f=\beta\rho_{de}$, and $f=\beta(\rho_m+\rho_{de})$, revealing standard cosmological phase-space behavior with an unstable radiation point, a transient matter saddle (requiring $\alpha=-1$ for a proper matter era in some cases), and a stable late-time accelerated attractor. The Rastall framework introduces a time-dependent parameter $\lambda_{Ras}$ and a GR-like function $g$, with deviations from GR generally diminishing at high redshift but potentially remaining noticeable at late times, depending on the model and parameters. A joint chi-squared analysis with Cosmic Chronometers, PantheonPlus, and DESI data provides marginalized constraints on $(H_0,\Omega_m,\beta,\omega_{de})$ (and $\alpha$ where applicable), showing mild departures from $\Lambda$CDM in some cases, especially for the $f\propto\rho_{de}$ and $f\propto(\rho_m+\rho_{de})$ models. Overall, the framework offers a coherent, testable path to explore geometry–matter exchanges beyond standard GR, with clear predictions for the background dynamics and implications for the growth of structure.

Abstract

In this work, we investigate late-time interacting cosmologies within the framework of generalized Rastall gravity, where the interaction arises naturally from the non-conservation of the energy-momentum tensor. We formulate the background evolution of the dark sector as an autonomous dynamical system, defining interaction terms $Q_1=α\,\dot{f}$ and $Q_2=-\dot{f}\,(1+α)$, with $α$ a constant parameter and $f$ a time-dependent function. Three interaction cases are studied: $f \propto ρ_m$, $f \propto ρ_{de}$, and $f \propto ρ_m + ρ_{de}$, assuming a constant dark-energy equation of state $w_{de}$. For each scenario, we derive the closed dynamical system in terms of the density parameters $(Ω_{de}, Ω_m)$, identify its fixed points, and analyze their stability across the parameter space. In this context, the phase-space exhibits a standard cosmological dynamics: an unstable radiation point, a transient matter saddle, and a stable late-time attractor with accelerated expansion. In addition, we utilize a joint likelihood analysis with Cosmic Chronometers, PantheonPlus, and DESI data to obtain marginalized parameter estimates at the $68\%$ and $95\%$ confidence levels, constraining the parameter space in each interaction model.

Cosmological dynamics of interacting dark matter-dark energy in generalized Rastall gravity

TL;DR

This work develops an autonomous dynamical-systems treatment of interacting dark energy and dark matter within generalized Rastall gravity, where non-conservation of the energy–momentum tensor naturally drives the interaction via and . An explicit closed system in is derived for three coupling forms , , and , revealing standard cosmological phase-space behavior with an unstable radiation point, a transient matter saddle (requiring for a proper matter era in some cases), and a stable late-time accelerated attractor. The Rastall framework introduces a time-dependent parameter and a GR-like function , with deviations from GR generally diminishing at high redshift but potentially remaining noticeable at late times, depending on the model and parameters. A joint chi-squared analysis with Cosmic Chronometers, PantheonPlus, and DESI data provides marginalized constraints on (and where applicable), showing mild departures from CDM in some cases, especially for the and models. Overall, the framework offers a coherent, testable path to explore geometry–matter exchanges beyond standard GR, with clear predictions for the background dynamics and implications for the growth of structure.

Abstract

In this work, we investigate late-time interacting cosmologies within the framework of generalized Rastall gravity, where the interaction arises naturally from the non-conservation of the energy-momentum tensor. We formulate the background evolution of the dark sector as an autonomous dynamical system, defining interaction terms and , with a constant parameter and a time-dependent function. Three interaction cases are studied: , , and , assuming a constant dark-energy equation of state . For each scenario, we derive the closed dynamical system in terms of the density parameters , identify its fixed points, and analyze their stability across the parameter space. In this context, the phase-space exhibits a standard cosmological dynamics: an unstable radiation point, a transient matter saddle, and a stable late-time attractor with accelerated expansion. In addition, we utilize a joint likelihood analysis with Cosmic Chronometers, PantheonPlus, and DESI data to obtain marginalized parameter estimates at the and confidence levels, constraining the parameter space in each interaction model.

Paper Structure

This paper contains 11 sections, 44 equations, 9 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Generalized Rastall Theory
  3. Dynamical system
  4. Interaction function $f(t) = \beta \rho_m(t)$. Critical points and Stability of critical points.
  5. Interaction function $f(t) = \beta \rho_{de}(t)$. Critical points and Stability of critical points.
  6. Interaction function $f(t) = \beta [\,\rho_m(t)+\rho_{de}(t)\,]$. Critical points and Stability of critical points.
  7. Statistical analysis: $\chi^2$ estimators
  8. Interaction function $f(t) = \beta \rho_m(t)$
  9. Interaction function $f(t) = \beta \rho_{de}(t)$
  10. Interaction function $f(t) = \beta [\,\rho_m(t)+\rho_{de}(t)\,]$
  11. Conclusions

Figures (9)

  • Figure 1: This figure depicts the phase-space evolution $(\Omega_m,\Omega_{de})$ for the first interaction function within generalized Rastall gravity, with parameters given by the mean values of Table \ref{['mean_inter1']} for CC + PantheonPlus + DESI.
  • Figure 2: The upper panel shows the evolution of the function $g$, defined by Eq.(\ref{['GG1']}), as a function of $\log_{10}(1+z)$. The lower panel shows the evolution of the Rastall parameter given by Eq.(\ref{['Ras1']}) as a function of the $\log_{10}(1+z)$. As before, all panels are computed for $\alpha = -1$, considering the mean values (solid line), inferior marginalized values (dashed line) and superior marginalized values (dot-dashed line) of Table \ref{['mean_inter1']}, at the 68% confidence level for CC + PantheonPlus + DESI.
  • Figure 3: This figure depicts the phase-space evolution $(\Omega_m,\Omega_{de})$ for the second interaction function $f\propto\rho_{de}$ within generalized Rastall gravity, with parameters given by the mean values of Table \ref{['mean_inter2']} for CC + PantheonPlus + DESI.
  • Figure 4: As before, the upper panel shows the evolution of the function $g$, defined by Eq.(\ref{['GG2']}), as a function of $\log_{10}(1+z)$. The lower panel displays the evolution of the Rastall parameter given by Eq.(\ref{['Ras2']}) as a function of the $\log_{10}(1+z)$. In all panels we have used the mean values (solid line), inferior marginalized values (dashed line) and superior marginalized values (dot-dashed line) of Table \ref{['mean_inter2']}, at the 68% confidence level for CC + PantheonPlus + DESI.
  • Figure 5: This figure depicts the phase-space evolution $(\Omega_m,\Omega_{de})$ for the first interaction function within generalized Rastall gravity, with parameters given by the mean values of Table \ref{['mean_inter3']} for CC + PantheonPlus + DESI.
  • ...and 4 more figures