  The expansion of space, and other geometric properties of cosmological
models, can be studied using geometrically defined notions of relative
velocity. In this paper, we consider test particles undergoing radial motion
relative to comoving (geodesic) observers in Robertson-Walker cosmologies,
whose scale factors are increasing functions of cosmological time. Analytical
and numerical comparisons of the Fermi, kinematic, astrometric, and the
spectroscopic relative velocities of test particles are given under general
circumstances. Examples include recessional comoving test particles in the de
Sitter universe, the radiation-dominated universe, and the matter-dominated
universe. Three distinct coordinate charts, each with different notions of
simultaneity, are employed in the calculations. It is shown that the
astrometric relative velocity of a radially receding test particle cannot be
superluminal in any expanding Robertson-Walker spacetime. However, necessary
and sufficient conditions are given for the existence of superluminal Fermi
speeds, and it is shown how the four concepts of relative velocity determine
geometric properties of the spacetime.
