Non-Affine Extensions of the Raychaudhuri Equation in the K-essence Framework

We present a new avenue of the Raychaudhuri Equation (RE) by introducing a non-affine parametrization within the k-essence framework. This modification accounts for non-geodesic flow curves, leading to emergent repulsive effects in cosmic evolution. Using a DBI-type k-essence Lagrangian, we derive a modified RE and demonstrate its ability to address the Hubble tension while predicting a natural emergence of a dynamical dark energy equation of state. Our Bayesian analysis, constrained by cosmological data, supports the theoretical scaling relation of the k-essence field ($\dot{\phi}$) and the cosmic scale factor ($a$). Furthermore, we reinterpret the modified RE as an anti-damped harmonic oscillator, we found a caustic avoidance signature, it may reveal classical or quantum-like effects in cosmic expansion. These results suggest a deep connection between scalar field dynamics and modified gravity, offering new perspectives on the nature of the expansion history of the universe.