The introduction of anisotropy into large scale geometry of the universe has led to the development of Thurston spacetimes possessing an inherent anisotropic curvature in them. Studies related to these anisotropic spacetimes have revealed how they could explain the evolutionary dynamics and light propagation in the universe. Here, we consider these spacetimes under the inflationary epoch to examine whether we could address the possibility of anisotropic inflation in the universe. Researches have proposed inflationary models where a vector field coupled to the inflaton was found violating the cosmic no-hair theorem for the anisotropic Bianchi type I spacetime through the existence of a stable anisotropically inflationary fixed point. Lately, this study was extended to axisymmetric spacetimes of Bianchi type II, III and the Kantowski-Sachs metric and it was inferred that the entire family of spacetimes was attracted to the anisotropic Bianchi I fixed point. By fabricating Thurston spacetimes as inflationary models with its shear sourced by a vector field, we perform dynamical stability and phase space analyses to confer the feasibility of anisotropic inflation. The results obtained for the considered set of Thurston geometries showed the presence and convergence to a unique, stable inflationary fixed point which was found to be similar to those obtained in Bianchi spacetimes, thereby indicating the cosmological viability of inflation with anisotropic hair.
Devika et al. (Sat,) studied this question.