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Spontaneous Lorentz violation due to a time-dependent expectation value for amassless scalar has been suggested as a method for dynamically generating darkenergy. A natural candidate for the scalar is a Goldstone boson arising fromthe spontaneous breaking of a U(1) symmetry. We investigate the low-energyeffective action for such a Goldstone boson in a general class of modelsinvolving only scalars, proving that if the scalars have standard kinetic termsthen at the {\em classical} level the effective action does not have therequired features for spontaneous Lorentz violation to occur asymptotically $(t\to \infty)$ in an expanding FRW universe. Then we study the large $N$ limit ofa renormalizable field theory with a complex scalar coupled to massivefermions. In this model an effective action for the Goldstone boson with theproperties required for spontaneous Lorentz violation can be generated.Although the model has shortcomings, we feel it represents progress towardsfinding a high energy completion for the Higgs phase of gravity.Comment: 20 pages, 5 figures;fixed typos and added reference

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