Scalar Field Dynamics in Non-Minimally Coupled Theories via the Noether Symmetry and the Eisenhart-Duval Lift

Abstract: This study investigates the dynamics of a non-minimally coupled (NMC) scalar field in modified gravity, employing the Noether gauge symmetry (NGS) approach to systematically derive exact cosmological solutions. By formulating a point-like Lagrangian and analyzing the corresponding Euler-Lagrange equations, conserved quantities were identified, reducing the complexity of the dynamical system. Through the application of Noether symmetry principles, the scalar field potential was found to follow a power-law form, explicitly dependent on the coupling parameter $\xi$, influencing the evolution of the universe. The study further explores inflationary dynamics, showing that for specific values of $\xi$, the potential resembles the Higgs-like structure, contributing to a deeper understanding of early cosmic expansion. To enhance the theoretical framework, the Eisenhart lift method was introduced, providing a geometric interpretation of the system by embedding the dynamical variables within an extended field space. This approach established a connection between the kinetic terms and Killing vectors, offering an alternative perspective on the conserved quantities. The study also derived geodesic equations governing the evolution of the system, reinforcing the link between symmetry-based techniques and fundamental cosmological properties.