Phase space analysis of an exponential model in $f(Q)$ gravity including linear dark-sector interactions

TL;DR

This work analyzes an exponential $f(Q)$ gravity model within a dynamical-systems framework to map the modified Friedmann equations into an autonomous system and study the background cosmology. Using the Böhmer method, the authors define a two-variable system in terms of $x$ and a compact variable $m$ to capture the exponential modification, and they identify fixed points corresponding to radiation, matter, and a late-time de Sitter attractor, with stability governed by a parameter $b$. The model is extended by a linear DM-DE interaction with kernel $\\mathcal{Q}=\\xi\rho_{DE}$, which shifts the de Sitter attractor to an interacting point $P_{int}$ and modifies the evolution of densities and the effective equation of state. Stability analyses show $R$ is unstable, $M$ is a saddle, and the de Sitter-type points are attracting, with the interacting case preserving an attractor albeit with shifted coordinates and possible phantom-like late-time behavior. Overall, the exponential $f(Q)$ model behaves as a perturbation around $\Lambda$CDM, capable of reproducing standard cosmological epochs while allowing controlled deviations via $b$ and the DM-DE coupling $\xi$; future DESI data could constrain these parameters and test the viability of such dark-sector interactions.

Abstract

We present a cosmological analysis of an exponential $f(Q)$ gravity model, within the dynamical systems formalism. Following the method introduced by Böhmer \textit{et al} [Universe \textbf{9} no.4, 166 (2023)], the modified Friedmann modified equations are successfully reduced to an autonomous system. Given the exponential form of $f(Q)$, the equilibrium conditions result in transcendental equations, which we approximate to identify the critical points. We therefore perform a general stability analysis of these points in terms of the model parameters. Finally, we extend the model by including a linear dark energy-dark matter interaction, where the equilibrium points are found with their stability properties. The model exhibits the three main domination epochs in the Universe, as well as a non-trivial impact on the late-time de Sitter attractor.

Figures (12)

• Figure 1: Values for the de Sitter point of the dynamical system as a function of $b$. The function $m_{DE}$ goes through the point $1/4$ when $b=0$.
• Figure 2: Phase-space portraits of the dynamical system with contributions from the model. For negative values of $b$ [Left]. For positive $b$ values [Right].
• Figure 3: Phase space for $b=1/3$. The de Sitter attractor point disappears, since variable $m$ becomes complex and the point is non-hyperbolic.
• Figure 4: Eigenvalues of the Jacobian $J|_{P_{dS}}$ for negative and positive values of $b$.
• Figure 5: Evolution of density parameters for matter and radiation, additionally the geometrical contribution from the model ($b>0$ [Left] y $b<0$ [Right])