Qualitative stability analysis of cosmological models in \(f(T,\phi )\) gravity

Using the dynamical system approach, we investigated the stability condition of two considered models in \(f(T,\phi )\) gravity where T is the torsion scalar of teleparallel gravity and \(\phi \) is a canonical scalar field. In this context, we are concerned with the phenomenology of the class of models with non-linear coupling to gravity and exponential potential. We assume the forms of G(T) as (i) G(T) = \(\alpha T+\frac{\beta }{T}\) and (ii) G(T) = \(\zeta T\) ln\((\psi T)\), where \(\alpha \), \(\beta \), \(\zeta \) and \(\psi \) be the free parameters and G(T) is the function of T. We evaluated the equilibrium points for these models and examine the stability behaviors. For Model I, we found four stable critical points while for Model II, we found three stable critical points. The stable critical points represent the attractors with accelerated expansion. The phase plots for these systems are examined and discussed the physical interpretation. We illustrate all the cosmological parameters such as \(\Omega _{m}\), \(\Omega _{\phi }\), q and \(\omega _{Tot}\) at each fixed points and compare the parameters with observational values. In both Model I and Model II, we found \(\Omega _{de}=1\) which represents the dark energy dominant Universe. Further, we assume hybrid scale factor to develop our model and this model produces a transition phase from deceleration to the acceleration. We transform all the parameters in redshift and examine the behavior of these parameters. From the Figures, it is observed that \(q=- 1\) represents the accelerating stage of the Universe and EoS parameter \(\omega =-1\) represents the \(\Lambda \)CDM model. For Model I, we get \(\omega _{0}=- 0.992\) and for Model II, we get \(\omega _{0}=- 0.883\) which is comparable to the observational data. The energy conditions are examined in terms of redshift while strong energy condition is violated for both models which shows the accelerated expansion evolution of present universe. We also find the statefinder parameters \(\{r,s\}\) in terms of z and discuss the nature of \(r-s\) and \(r-q\) plane. For both models, \(r=1, s=0\) and \(r=1, q=- 1\) represent the \(\Lambda \)CDM model. We observed that our \(f(T,\phi )\) models are stable and it is in accordance with the observational data.