Cosmological signature change in Cartan Gravity with dynamical symmetry breaking

Abstract

We investigate the possibility for classical metric signature change in a straightforward generalization of the first order formulation of gravity, dubbed "Cartan gravity". The mathematical structure of this theory mimics the electroweak theory in that the basic ingredients are an $SO(1,4)$ Yang-Mills gauge field $A^{ab}_{\phantom{ab}μ}$ and a symmetry breaking Higgs field $V^{a}$, with no metric or affine structure of spacetime presupposed. However, these structures can be recovered, with the predictions of General Relativity exactly reproduced, whenever the Higgs field breaking the symmetry to $SO(1,3)$ is forced to have a constant (positive) norm $V^aV_a$. This restriction is usually imposed "by hand", but in analogy with the electroweak theory we promote the gravitational Higgs field $V^a$ to a genuine dynamical field, subject to non-trivial equations of motion. Even though we limit ourselves to actions polynomial in these variables, we discover a rich phenomenology. Most notably we derive classical cosmological solutions exhibiting a smooth transition between Euclidean and Lorentzian signature in the four-metric. These solutions are non-singular and arise whenever the $SO(1,4)$ norm of the Higgs field changes sign i.e. the signature of the metric of spacetime is determined dynamically by the gravitational Higgs field. It is possible to find a plethora of such solutions and in some of them this dramatic behaviour is confined to the early universe, with the theory asymptotically tending to Einstein gravity at late times. Curiously the theory can also naturally embody a well-known dark energy model: Peebles-Ratra quintessence.