  We present further analysis of an anisotropic, non-singular early universe
model that leads to the viable cosmology presented in Dechant et al
(arXiv:0809.4335). Although this model (the DLH model) contains scalar field
matter, it is reminiscent of the Taub-NUT vacuum solution in that it has
biaxial Bianchi IX geometry and its evolution exhibits a dimensionality
reduction at a quasi-regular singularity that one can identify with the
big-bang. We show that the DLH and Taub-NUT metrics are related by a coordinate
transformation, in which the DLH time coordinate plays the role of conformal
time for Taub-NUT. Since both models continue through the big-bang, the
coordinate transformation can become multivalued. In particular, in mapping
from DLH to Taub-NUT, the Taub-NUT time can take only positive values. We
present explicit maps between the DLH and Taub-NUT models, with and without a
scalar field. In the vacuum DLH model, we find a periodic solution expressible
in terms of elliptic integrals. Mapping the vacuum solution over to Taub-NUT
coordinates, recovers the standard (non-periodic) Taub-NUT solution in the Taub
region, where Taub-NUT time takes positive values, but does not exhibit the two
NUT regions known in the standard Taub-NUT solution. Conversely, mapping the
complete Taub-NUT solution to the DLH case reveals that the NUT regions
correspond to imaginary time and space in DLH coordinates. We show that many of
the well-known `pathologies' of the Taub-NUT solution arise because the
traditional coordinates are connected by a multivalued transformation to the
physically more meaningful DLH coordinates. In particular, the
`open-to-closed-to-open' transition and the Taub and NUT regions of the
(Lorentzian) Taub-NUT model are replaced by a closed pancaking universe with
spacelike homogeneous sections at all times.
