Gravity-wave insights to Bianchi type-IX universes

Every Bianchi type-IX universe can be interpreted as a closed Friedmann universe on which is super-imposed circularly polarized gravitational waves with the longest wavelength that will fit into a closed universe. In this paper, I give a new derivation of this result based on the concept of homogeneous tensor fields on the three-sphere. Every homogeneous symmetric traceless tensor field is shown to be a longest-wavelength three-sphere harmonic. Contrary to previous authors, I show that the wavelength of these gravitational waves is one-half the circumference of the universe. In order to maintain homogeneity, the gravitational waves must all have the same polarization. There are five longest-wavelength modes for each polarization. This interpretation is an exact description that is valid for every Bianchi type-IX universe—it is in no way limited to first-order perturbations of a Friedmann universe.