Dynamics of the Bianchi~V cosmological model inspired by quintessential $α$-attractors

TL;DR

This work examines scalar-field cosmologies in the anisotropic Bianchi V spacetime with α-attractor-inspired potentials, using dynamical-systems methods and a two-stage averaging (amplitude–phase and coarse-grained) near the potential minimum. The authors derive an exact amplitude–phase system and a reduced averaged system, proving that the averaged dynamics faithfully capture the slow evolution and yield an effective fluid with barotropic index $\overline{\gamma}_\phi=\tfrac{2n}{n+1}$ and dilution $\overline{\rho}_\phi\propto a^{-6n/(n+1)}$, while the oscillations are encoded by a frequency $\omega(A)$ and amplitude decay $\dot A= -\tfrac{3H}{n+1}A$. The reduced system exhibits five generic equilibria, including Kasner vacua $\mathcal{K}_0^\pm$, Matter FLRW $\mathcal{F}$, Scalar FLRW $\mathcal{S}$, and curvature point $\mathcal{K}$, with stability depending on $(n,\gamma)$; notably, $\mathcal{K}_0^\pm$ are sources, $\mathcal{F}$ is typically a saddle, $\mathcal{S}$ can be a sink for certain ranges, and $\mathcal{K}$ becomes a sink for $\gamma>\tfrac{2}{3}$ and $n>\tfrac{1}{2}$. The results show that isotropic FLRW $\alpha$-attractor dynamics extend naturally to anisotropic Bianchi V cosmologies, with inflationary attractors persisting and Milne-type curvature emerging as the late-time state. This provides a robust framework connecting rapid scalar-field oscillations to the macroscopic expansion history in more general cosmologies.

Abstract

We investigate scalar-field cosmologies in the Bianchi V spacetime using a dynamical-systems framework. Motivated by representative $α$-attractor potentials - the E-model and T-model - we apply averaging theorems and amplitude--phase reductions to monomial potentials $\sim φ^{2n}$ of the scalar field, which approximate the attractor models near their minima, in the presence of matter with barotropic index $γ$. The reduced averaged system admits five generic isolated equilibria: Kasner vacua $\mathcal{K}_0^\pm$, the matter FLRW point $\mathcal{F}$, the scalar FLRW point $\mathcal{S}$, and the curvature Milne-type point $\mathcal{K}$, together with special families for tuned $(n,γ)$. We find that $\mathcal{K}_0^\pm$ are always sources, $\mathcal{F}$ is generically a saddle but can act as a sink for $γ<\min\{\tfrac{2n}{n+1},\tfrac{2}{3}\}$, $\mathcal{S}$ is a sink if $0<n<\tfrac{1}{2}$ and $\tfrac{2n}{n+1}<γ\leq 2$, while $\mathcal{K}$ becomes a sink whenever $γ>\tfrac{2}{3}$ and $n>\tfrac{1}{2}$. These results demonstrate that isotropic FLRW $α$-attractor models extend naturally to anisotropic Bianchi~V cosmologies: inflationary attractors remain robust, while the Milne-type curvature solution emerges as the late-time state.

Figures (8)

• Figure 1: Time evolution of the dynamical variables for $n=1$. Solid lines correspond to the averaged system, while dashed lines correspond to the full system. Initial conditions are taken from Table \ref{['tb_1']}.
• Figure 2: Time evolution of the dynamical variables for $n=2$. Solid lines correspond to the averaged system, while dashed lines correspond to the full system. Initial conditions are taken from Table \ref{['tb_1']}.
• Figure 3: 2D phase--space portraits for $n=1$. Projections include $A$ vs. $B$, $\theta$ vs. $\Theta$, $B$ vs. $\theta$, $B$ vs. $\Theta$, $\phi$ vs. $\dot{\phi}$, and $\phi$ vs. $H$. The averaged system reproduces the slow manifold structure of the full dynamics. Initial conditions are taken from Table \ref{['tb_1']}.
• Figure 4: 2D phase--space portraits for $n=2$. Projections include $A$ vs. $B$, $\theta$ vs. $\Theta$, $B$ vs. $\theta$, $B$ vs. $\Theta$, $\phi$ vs. $\dot{\phi}$, and $\phi$ vs. $H$. The averaged system reproduces the slow manifold structure of the full dynamics. Initial conditions are taken from Table \ref{['tb_1']}.
• Figure 5: 3D phase--space portraits for $n=1$. Trajectories in $(\phi,\dot{\phi},H)$ and $(\phi,\dot{\phi},A)$ confirm that the averaged system provides a reliable autonomous approximation to the long-term dynamics. Initial conditions are taken from Table \ref{['tb_1']}.

Theorems & Definitions (10)

• Lemma A.1: Periodic family
• proof
• Lemma A.2: Amplitude–phase modulation
• proof
• Lemma A.3: Virial relation with explicit error
• proof
• Proposition A.4: Near‑identity transform and remainder
• proof
• Theorem A.5: Averaging for monomial potentials
• proof