Asymptotic Rigidity for Timelike Geodesic Expansion in Λ>0 Spacetimes

This paper establishes a geodesic-level rigidity result for timelike congruences in classical General Relativity with a positive cosmological constant (Λ > 0). Using the Raychaudhuri equation and a Riccati comparison argument, we isolate a minimal “eventual Λ-domination” condition on the combined matter–shear term along a future-complete timelike geodesic. Under this hypothesis, the expansion scalar θ is shown to converge to the de Sitter fixed point √(3Λ) and to exhibit only finitely many sign changes. The result complements standard cosmic no-hair theorems by formulating an explicit finite-zero property for the expansion along individual worldlines, without assuming global symmetry (e.g. FLRW). In particular, within the Λ > 0 vacuum-dominated regime considered, infinitely many expansion reversals along a single future-complete timelike geodesic are excluded unless one abandons eventual Λ-domination, future completeness, or the classical Einstein framework. The argument relies only on the Raychaudhuri equation and standard ODE comparison techniques and makes no assumption about specific matter microphysics beyond nonnegativity and eventual smallness of the effective defect term.