Power-law Bianchi type I inflation with multiple vector fields

TL;DR

We analyze power-law inflation in a Bianchi type I background with a scalar field $\phi$ non-minimally coupled to three vector fields along the coordinate axes, using $V(\phi)=V_0 e^{\lambda\phi}$ and $f_i(\phi)=f_{i0} e^{\rho_i\phi}$. The authors derive an autonomous 6D dynamical system from the background equations and classify all exact power-law solutions according to the number of nonzero vector fields: type 0 (FLRW), type I (one vector), type II (two vectors), and type III (three vectors), with closed-form expressions for the expansion rate $\zeta$, anisotropy parameters $\eta_i$, scalar slope $\xi$, and vector-kinetic components $\omega_i$. A stability analysis via Routh–Hurwitz criteria and complementary numerical tests maps out the stability regions in parameter space $(\lambda,\rho_a,\rho_b,\rho_c)$, revealing that all solution classes can be attractors for suitable couplings. The results show that the vector fields either decay or persist depending on the relative magnitudes of the couplings, and that the spacetime metric approaches a general BI form when multiple vectors survive, a rsBI form when only one survives, or FLRW when none survive, consistent with cosmic no-hair intuition in appropriate limits. These findings extend the KSW anisotropic inflation framework to three vector fields and provide a systematic classification of stable anisotropic power-law solutions with potential implications for early-universe anisotropies and vector-field dynamics.

Abstract

We seek power-law Bianchi type I solutions for an inflationary universe in a model with one scalar field non-minimally coupled to three vector fields aligned along the three axes. As a result, we find four types of power-law solutions that are classified according to the number of non-vanishing vector fields. Moreover, we find that all these solutions are stable for certain conditions on the parameters (stability regions), which are described both quantitatively and qualitatively. The attractor property of these solutions is also confirmed by numerical calculations.

Figures (13)

• Figure 1: Stability region of the fixed point type 0 for $\lambda=1.5$ displayed as the dark blue cube. The red line corresponds to $\rho_a=\rho_b=\rho_c$. The black point is a special point, where $\rho_a=\rho_b=\rho_c=(4-\lambda^2)/(2\lambda)$. The thick black line corresponds to $\rho_b=\rho_c=(4-\lambda^2)/(2\lambda)$.
• Figure 2: Existence and stability region of the fixed point type III for $\lambda=1.5$ displayed as the dark blue region. The red line corresponds to $\rho_a=\rho_b=\rho_c$, while the black point is shown for a special case, in which $\rho_a=\rho_b=\rho_c=(4-\lambda^2)/(2\lambda)$.
• Figure 3: (left) Existence region colored as orange and (right) stability region colored as dark blue of the fixed point type $\text{II}_{bc}$ for $\lambda=1.5$. The red line corresponds to $\rho_a=\rho_b=\rho_c$. The black point is a special point, where $\rho_a=\rho_b=\rho_c=(4-\lambda^2)/(2\lambda)$. The thick black line corresponds to $\rho_b=\rho_c=(4-\lambda^2)/(2\lambda)$.
• Figure 4: Stability regions of the fixed points type $\text{II}_{ac}$ (left) and $\text{II}_{ab}$ (right) colored as dark blue for $\lambda=1.5$.
• Figure 5: Combination of the stability regions (colored as dark blue) of the fixed points type 0, III, $\text{II}_{bc}$, $\text{II}_{ac}$, and $\text{II}_{ab}$ for $\lambda=1.5$. The two figures are identical but viewed from different angles.