Consider a four-dimensional spacetime satisfying Einstein’s field equations with cosmological constant and pressureless matter, whose stress-energy tensor is Tab=ρuaub, where ρ>0 and u is the unit timelike four-velocity of the matter. Assume that the spacetime is spatially homogeneous, that the space of matter worldlines is compact, that ρ is not constant in time (so the universe expands or contracts), and that the matter congruence has everywhere nonzero vorticity vector ωa. At each event, let the quadric of expansion be the quadric in the three-dimensional rest space orthogonal to u determined by the symmetric spatial expansion tensor of the matter congruence; in this class of rotating universes it is never rotationally symmetric about ωa, so expansion is necessarily accompanied by anisotropic deformation. Construct a quantitative model of the gravitational formation and subsequent evolution of a matter condensation in such a spacetime, and determine whether this anisotropic expansion generically produces an oblong body rotating about a minor axis whose differential rotation bends it into a spiral with morphology and kinematics consistent with observed spiral galaxies.