Exploring Cartan gravity with dynamical symmetry breaking

Abstract

It has been known for some time that General Relativity can be regarded as a Yang–Mills-type gauge theory in a symmetry broken phase. In this picture the gravity sector is described by an SO(1, 4) or SO(2, 3) gauge field and Higgs field Va which acts to break the symmetry down to that of the Lorentz group SO(1, 3). This symmetry breaking mirrors that of electroweak theory. However, a notable difference is that while the Higgs field Φ of electroweak theory is taken as a genuine dynamical field satisfying a Klein–Gordon equation, the gauge independent norm V2 ≡ ηabVaVb of the Higgs-type field Va is typically regarded as non-dynamical. Instead, in many treatments Va does not appear explicitly in the formalism or is required to satisfy V2 = const. ≠ 0 by means of a Lagrangian constraint. As an alternative to this we propose a class of polynomial actions that treat both the gauge connection and Higgs field Va as genuine dynamical fields with no ad hoc constraints imposed. The resultant equations of motion consist of a set of first-order partial differential equations. We show that for certain actions these equations may be cast in a second-order form, corresponding to a scalar–tensor model of gravity. One simple choice leads to the extensively studied Peebles–Ratra rolling quintessence model. Another choice yields a scalar–tensor symmetry broken phase of the theory with positive cosmological constant and an effective mass M of the gravitational Higgs field ensuring the constancy of V2 at low energies and agreement with empirical data if M is sufficiently large. More general cases are discussed corresponding to variants of Chern–Simons modified gravity and scalar-Euler form gravity, each of which yield propagating torsion.