Towards a high-energy theory for the Higgs phase of gravity

Spontaneous Lorentz violation due to a time-dependent expectation value for a massless scalar has been suggested as a method for dynamically generating dark energy. A natural candidate for the scalar is a Goldstone boson arising from the spontaneous breaking of a $U(1)$ symmetry. We investigate the low-energy effective action for such a Goldstone boson in a general class of models involving only scalars, proving that if the scalars have standard kinetic terms then at the classical level the effective action does not have the required features for spontaneous Lorentz violation to occur asymptotically $(t\to \infty )$ in an expanding Friedman-Robertson-Walker universe. Then we study the large $N$ limit of a renormalizable field theory with a complex scalar coupled to massive fermions. In this model an effective action for the Goldstone boson with the properties required for spontaneous Lorentz violation can be generated. Although the model has shortcomings, we feel it represents progress towards finding a high energy completion for the Higgs phase of gravity.