Cosmological implications and stability of f(Q,T) gravity with pilgrim dark energy model

This manuscript endeavors to construct a pilgrim dark energy framework within the $f\mathrm{(Q,T)}$ gravity theory, employing a correspondence approach aligned with a non-interacting model that incorporates pressureless matter alongside a power-law scale factor. Here $Q$ and $T$ represent the non-metricity and trace of the energy–momentum tensor, respectively. This extended modified gravity framework accurately replicates various epochs in the cosmological history. The $f\mathrm{(Q,T)}$ gravity models are utilized to derive the equation of state parameter, phase planes and squared speed of sound. The analysis reveals that the reconstructed model exhibits an increasing or decreasing trend with the pilgrim dark energy parameter. The equation of state parameter characterizes the phantom regime, while the squared speed of sound parameter provides a stable framework for examining the ongoing cosmic evolution. The ${\omega }_{DE}-{\omega }_{DE}^{\prime }$ plane trajectories reveal the freezing region, while the $r-s$ phase plane shows the Chaplygin gas model. It is important to highlight that our findings align with the most recent observational data.