Constraining $f(T,\mathcal{T})$ gravity from dynamical system analysis

The dynamical system analysis of the cosmological models in $f(T,\mathcal{T})$ gravity, where $T$ and $\mathcal{T}$ respectively represents the torsion scalar and trace of the energy-momentum tensor has been investigated. It demonstrates how first-order autonomous systems can be treated as cosmological equations and analyzed using standard dynamical system theory techniques. Two forms of the function $f(T,\mathcal{T})$ are considered (i) one with the product of trace and higher order torsion scalar and the other (ii) linear combination of linear trace and squared torsion. For each case, the critical points are derived and their stability as well the cosmological behaviours are shown. In both the models the stable critical points are obtained in the de-Sitter phase whereas in the matter and radiation dominated phase unstable critical points are obtained. At the stable critical points, the deceleration parameter shows the accelerating behaviour of the Universe whereas the equation of state parameter shows the $\Lambda CDM$ behaviour. Finally the obtained Hubble parameter of the models are checked for the cosmological data sets