Anisotropic Cosmology with Inhomogeneous $SU(2) $ and $SL(2,\mathbb{R})$ Nonlinear $σ$-Models
We investigate the cosmological dynamics of the nonlinear $σ$-model in a locally rotational symmetric Bianchi V background geometry. We employ the $SU(2)$ and the $SL(2,\mathbb{R})$ internal symmetries, for which the kinetic metric of the multi-scalar fields is a space of constant curvature. We assume an inhomogeneous ansatz for the scalar fields, such that the $σ$-model, which allows the existence of nontrivial solutions in the Bianchi V geometry. For an exponential potential, we perform a detailed analysis of the phase space, we determine the stationary points, and we study their stability properties. The scalar field inhomogeneities prevent both the slow-roll and hyperbolic inflationary epochs. The existence of a cosmological constant is required to describe cosmic acceleration.