Phase space analysis of Bianchi III Universe with $f(R,T)$ gravity theory

The Bianchi type III (BIII) metric is a useful geometry to study cosmic anisotropies. It includes an extra exponential term multiplied by a directional scale factor and recasts the cosmological model as a dynamical system to provide various significant information regarding the evolution, stability of the system, etc. In this study, we have constructed a dynamical system for the BIII metric using $f(R,T)$ gravity theory and performed fixed point analysis in three different $f(R,T)$ models. Here, we have found that the first two models, i.e., $f(R,T) = αR + βf(T)$ and $f(R,T) = R + 2 f(T)$ are agreed with standard $Λ$CDM cosmology but the third one, i.e., $f(R,T) = (ζ+ η\, τ\, T)R$ has the issue of unbounded energy density. Thus, we can remark that some $f(R,T)$ models may not be suitable for studying the evolution of the Universe with an anisotropic background, like using BIII metric, etc. However, all three models agree with the heteroclinic path of radiation-dominated, matter-dominated, and dark energy-dominated phases of the Universe as predicted by standard cosmology.