Comparative Analysis of Dark Energy Compact Stars in $f(T,\mathcal{T})$ and $f(T)$ Gravity Theories via Conformally Flat Condition

This manuscript is the first investigation of dark energy celestial phenomena in the modified gravity theory by examining dark energy compact stars within the context of modified $f(T,\mathcal{T})$ gravity. In order to compare the outcomes of the $f(T,\mathcal{T})$ and $f(T)$ gravity theories, the model $f(T,\mathcal{T})=\alpha T(r)+\beta \mathcal{T}(r)+\phi$ is selected. This model is then simplified to $f(T)$ gravity by setting $\beta=0$. The $f(T)$ gravity is torsion-based gravity, while in $f(T,\mathcal{T}$) gravity trace of energy-momentum tensor is coupled with torsion. Moreover, we note that we have more dense object formation in $f(T,\mathcal{T})$ gravity as compared to $f(T$)-gravity. The spherically symmetric space-time inside the internal geometry is analyzed using a conformally flat condition, while the Schwarzschild geometry represents the outer space-time. Numerous characteristics of dark energy stars are examined, including equation of state components, energy conditions, and dark energy pressure components. Empirical evidence for the existence of dark energy in stellar configurations is presented by the results for pressure components in dark energy, which show a significant negative tendency in these stellar parameters. A complete analysis is performed by thoroughly investigating energy conditions, pressure profiles, sound speeds, gradients, adiabatic index, TOV equation, mass function, compactness, and redshift function. The mass-radius relation is also discussed via $M-R$ curves. This confirms that the studied star configuration is realistic and acceptable.