Toroidal perturbations of Friedmann-Robertson-Walker universes

Explicit expressions are found for the axisymmetric metric perturbations of the closed, flat and open Friedmann-Robertson-Walker universes caused by toroidal motions of the cosmic fluid. The perturbations are decomposed in vector spherical harmonics on 2-spheres, but the radial dependence is left general. Solutions for general odd-parity l-pole perturbations are given for either angular velocities or angular momenta prescribed. In particular, in the case of closed universes the solutions require a special treatment of the Legendre equation.