Cracking the Taub-NUT

Pierre-Philippe Dechant (1) , Anthony N Lasenby , Michael P Hobson

1 KICC - Kavli Institute for Cosmology [Cambridge]

We present further analysis of an anisotropic, non-singular early universe model that leads to the viable cosmology presented in [ 1 ]. 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 ; 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. PACS numbers: 98.80.Bp (Origin and formation of the universe), 98.80.Cq (Particle-theory and field-theoretic models of the early universe), 98.80.Jk (mathematical and relativistic aspects of cosmology), 04.20.Dw (Singularities and cosmic censorship), 04.20.Jb (Exact solutions), 04.20.dc (Numerical studies of critical behaviour, singularities) 2 Contents I. Introduction 4 II. Bianchi Models 5 III. The DLH model 7 IV. The Taub-NUT model 9 V. Relationship between DLH and Taub-NUT models 11 VI. The reparameterised Taub-NUT model 13 VII. Comparison of DLH and reparamaterised Taub-NUT models 15 A. DLH vacuum solution 15 B. DLH solution with a scalar field 17 C. Taub-NUT vacuum solution 18 1. Mapping the DLH vacuum solution using the diffeomorphism 19 2. Direct solution of Einstein equations for reparameterised Taub-NUT metric 21 3. Comparison of the vacuum solution in different set-ups 22 D. Taub-NUT with a scalar field 26 VIII. Conclusions 30 Acknowledgments 31 A. Mapping of curvature invariants 31 1. Petrov type 32 2. Principal curvatures 33 B. Mapping of geodesics 33 C. Derivation of the elliptic integral solution for the vacuum DLH model 35 References 38 3

Mots clés: scalar fields; Bianchi models; big bang singularity; cosmology; exact solutions; Taub-NUT; pre-Big-Bang scenarios

Pierre-Philippe Dechant, Anthony N Lasenby, Michael P Hobson. Cracking the Taub-NUT. Classical and Quantum Gravity, 2010, 27 (18), pp.185010. ⟨10.1088/0264-9381/27/18/185010⟩. ⟨hal-00625163⟩