Could we orbit the expanding universe in finite time?

According to Einstein's cosmological principle, our universe at any fixed time is homogeneous and isotropic on large scales. It is thus modeled up to scaling by one of the following three three-dimensional maximally symmetric manifolds: the sphere, the Euclidean space, or the hyperbolic pseudosphere, none of which contain any privileged points or directions. We present several arguments which indicate why the last two unbounded manifolds cannot be good models of our physical universe at any fixed time. We shall then concentrate only on the expanding three-dimensional sphere. We prove that if its radius expands linearly, then the total travel time of a shuttle along geodesics around the entire universe is finite. If the shuttle has a constant velocity v > 0, its trajectory on such an expanding sphere is a logarithmic spiral.