The dynamical equations for rotational (vector) perturbations of a Friedmann-Robertson-Walker universe containing a perfect fluid of massive matter and radiation together with relativistic collisionless matter are established. These equations have solutions which remain regular as the initial singularity is approached, in contrast to the purely perfect-fluid case, where small rotational perturbations cannot coexist with a Friedmann-type singularity due to the Helmholtz-Kelvin circulation theorem. With collisionless matter present (e.g., gravitons after the Planck era), this obstruction is circumvented, and solutions which exhibit a growing mode of vorticity on superhorizon scales are obtained. The anisotropies in the cosmic microwave background caused by these small vector perturbations are analyzed, and limits on admissible primordial vorticity are derived. In the radiation era, large-scale vorticity gives rise to large-scale primordial magnetic fields, which are shown potentially to have the right magnitude to act as seed fields for galactic dynamo action and thereby to explain the presently observed galactic magnetic fields.
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Anton Rebhan (1992) studied this question.