Non-Radial Free Geodesics. I. In Spatially Flat FLRW Spacetime

Abstract

This paper provides an analytical examination of non-radial geodesics within the context of the spatially flat Friedmann Lema\^itre Robertson Walker (FLRW) spacetime. Using the symmetry properties of the system, two constants of motion related to this dynamical system are derived. This treatment enables us to explicitly compute the radial and angular peculiar velocities, as well as the evolution of the comoving radial distance and angle over time. The study introduces three distinct methods to characterize geodesic solutions, and additionally examines the radial geodesic limit, the null geodesic limit, and the comoving geodesic limit. Additionally, the paper investigates both non-radial and radial free geodesics within the Lambda Cold Dark Matter ({\Lambda}CDM) model, presenting detailed results and discussions. Lastly, we explore the total angle swept by a free particle from the initial singularity to infinity as measured by a comoving observer, offering insights into the dynamics of free particles in FLRW spacetime.