Study of geodesic focusing in Power-Law Cosmologies within Matter-Geometry Coupled Gravity via the Raychaudhuri Equation

We investigate the viability of power-law cosmologies in matter–geometry coupled modified gravity using the timelike Raychaudhuri equation as a diagnostic of geodesic focusing. Focusing on [Formula: see text] and [Formula: see text] gravity, we examine whether non-minimal matter–geometry coupling can prevent the focusing of timelike geodesic congruences while ordinary matter satisfies the strong energy condition. Two physically relevant power-law regimes are considered: ekpyrotic contraction ([Formula: see text]) and late-time accelerated expansion ([Formula: see text]). In the [Formula: see text] sector, the linear model [Formula: see text] does not support geodesic defocusing, whereas the logarithmic model [Formula: see text] also fails to admit viable non-focusing power-law solutions within the considered parameter space. In the [Formula: see text] framework, both the linear model [Formula: see text] and the nonlinear geometric model [Formula: see text] are unable to simultaneously realize non-focusing ekpyrotic and late-time accelerating cosmologies. In contrast, the quadratic matter-coupled model [Formula: see text] successfully accommodates both cosmological phases while avoiding geodesic focusing without invoking exotic matter. These results demonstrate that nonlinear matter coupling plays a decisive role in modifying the Raychaudhuri convergence scalar and suggest that appropriately chosen matter–geometry interactions provide a viable geometrical mechanism for avoiding Raychaudhuri focusing singularities in cosmological evolution.