Late-time acceleration and structure formation in interacting $α$-attractor dark energy models

TL;DR

This work analyzes interacting dark energy within the $\alpha$-attractor framework by performing a background dynamical-systems analysis and a perturbation study. The authors construct an autonomous system in terms of $X$, $Y$, $\xi$, $y$, $\lambda$, and $\Gamma$, identify a set of fixed points including radiation-, matter-, and dark-energy-dominated attractors, and examine their stability—employing Centre Manifold Theory for the non-hyperbolic cases. They show that the model supports a viable cosmic history with transitions from radiation to matter to dark-energy domination and features a scaling matter regime that can modify the growth of structure via a coupling parameter $\beta$. Perturbation analysis under a quasi-static approximation reveals an effective gravitational coupling $G_{mm}=(1+2\beta^2)G$ and yields predictions for the growth rate $f_{\delta}$ and $f\sigma_8$ that can be slightly lower than $\Lambda$CDM, potentially easing the $\sigma_8$ tension; a best-fit to $f\sigma_8(z)$ data yields $\sigma_8=0.750$ and $\beta=0.065$. Overall, the interacting $\alpha$-attractor model provides a coherent framework that ties late-time acceleration to structure formation and motivates further observational constraints. All key equations use the $\alpha$-attractor formalism, with background and perturbation results expressed in terms of $X$, $Y$, $\xi$, $y$, $\lambda$, $\Gamma$, and $\beta$.

Abstract

We investigate the cosmological dynamics of interacting dark energy within the framework of $α$-attractor models. Specifically, we analyze the associated autonomous system, focusing on its fixed points that represent dark energy and scaling solutions, along with their stability conditions. We employ center manifold theory to address cases where some fixed points display eigenvalues with zero and negative real parts. The model reveals attractors describing dark energy, enabling a smooth transition from the radiation-dominated era to the matter-dominated era, and ultimately into the dark-energy-dominated phase. Additionally, we identify a scaling matter solution capable of modifying the growth rate of matter perturbations during the matter-dominated epoch. Consequently, we study the evolution of matter perturbations by obtaining both analytical and numerical solutions to the density contrast evolution equation. Based on these results, we compute numerical solutions for the weighted growth rate $fσ_{8}$, indicating that interacting $α$-attractor dark energy models may provide a better fit to structure formation data than the standard $Λ$CDM scenario.

Figures (15)

• Figure 1: The phase space of $X$ vs. $Y$ for $\beta = 0.01$. We assume $\xi = 0$ and $\lambda = 0$, which are consistent with critical point $g$, to reduce the phase space to two dimensions. The critical points are depicted by red points along with their corresponding labels. Each streamline represents the Universe's evolution under different initial conditions. In particular, the black line presents a Universe with initial conditions $X_0=10^{-11}$, $Y_0=8.6 \times 10^{-13}$, $\xi_0 = 0.999656$, $\lambda_0 = 1.0 \times 10^{-10}$ and $y_0 = 1.0 \times 10^{-11}$. The green region indicates accelerated expansion.
• Figure 2: The phase space of $X$ vs. $Y$ vs. $\xi$ for $\beta = 0.01$. We assume $\lambda = 0$, which is consistent with critical point $g$. The critical points are depicted by red points along with their corresponding labels. Each streamline represents the Universe's evolution under different initial conditions. In particular, the black line presents a Universe with initial conditions $X_0=10^{-11}$, $Y_0=8.6 \times 10^{-13}$, $\xi_0 = 0.999656$, $\lambda_0 = 1.0 \times 10^{-10}$ and $y_0 = 1.0 \times 10^{-11}$.
• Figure 3: Here, we show phase space for the fixed values $\alpha = 1$, $n=2$ and $p=1$ but varying the values of $\beta$, with $\beta=5.0 \times 10^{-3}$ (dot-dashed line), $\beta=1.0 \times 10^{-2}$ (dash line), and $\beta = 0$ (solid line). And initial conditions $X_0=10^{-11}$, $Y_0=8.6 \times 10^{-13}$, $\xi_0 = 0.999656$, $\lambda_0 = 1.0 \times 10^{-10}$ and $y_0 = 1.0 \times 10^{-11}$.
• Figure 4: Here, we show phase space for the values $\beta = 1.0 \times 10^{-2}$, $n=2$ and $p=1$ but varying the values of $\alpha$, with $\alpha=1.0 \times 10^{-1}$ (dot-dashed line), $\alpha=1.0 \times 10^{-2}$ (dash line), and $\beta = 5.0 \times 10^{-3}$ (solid line). And initial conditions $X_0=10^{-11}$, $Y_0=8.6 \times 10^{-13}$, $\xi_0 = 0.999656$, $\lambda_0 = 1.0 \times 10^{-10}$ and $y_0 = 1.0 \times 10^{-11}$.
• Figure 5: The plot illustrates the evolution of the EoS parameters under the same conditions as those presented in Figure \ref{['numerical_phase']}.