Cracking the Taub-NUT

Abstract

We present further analysis of an anisotropic, non-singular early universe model that leads to the viable cosmology presented in Dechant et al (2009 Phys. Rev. D 79, 043524). Although this model (the Dechant–Lasenby–Hobson (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; this periodicity is broken in a natural manner as a scalar field is gradually introduced to recover the original DLH model. 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.