Non-Radial Free Geodesics. II. In Spatially Curved FLRW Spacetime

This paper presents an in-depth exploration of timelike free geodesics in spatially curved Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetime. A unified approach for these geodesics encompassing both radial and non-radial trajectories across Euclidean, spherical, and hyperbolic geometries is employed. Using the symmetry properties of the system, two constants of motion related to this dynamical system are derived. This treatment facilitates the explicit computation of radial and angular peculiar velocities, along with the evolution of the comoving radial distance and angle over time. The study introduces three distinct methods for characterizing geodesic solutions, further delving into the flatness limit, the null geodesic limit, the radial geodesic limit, the comoving geodesic limit, and the circular orbits. Additionally, we analyze Killing vectors, reflecting the symmetries inherent in the dynamical system. This study provides a more profound understanding of the behavior of free particles as observed from a comoving reference frame.