Emergent Gravity from a Spontaneously Broken Gauge Symmetry: a Pre-geometric Prospective

We explore the paradigm of pre-geometric gravity, where spacetime geometry and the gravitational field are not fundamental but emerge from the spontaneous symmetry breaking (SSB) of a larger gauge symmetry. Specifically, we consider a gauge theory based on the de Sitter $SO(1,4)$ or anti-de Sitter $SO(3,2)$ group, formulated on a manifold without a prior metric structure. General covariance is maintained by constructing Lagrangian densities using the Levi-Civita symbol. The SSB is triggered by an internal vector field $\phi^A$, which reduces the symmetry to the Lorentz group $SO(1,3)$ and dynamically generates a spacetime metric. We analyze two specific models: the MacDowell-Mansouri formulation, which yields the Einstein-Hilbert action plus a cosmological constant and a Gauss-Bonnet term, and the Wilczek model, which produces a pure Einstein-Hilbert action with a cosmological constant. In both cases, the observed Planck mass and the small cosmological constant emerge from a see-saw mechanism dependent on the symmetry-breaking scale. We then present the Hamiltonian formulation of this pre-geometric theory, demonstrating that it possesses three number of physical degrees of freedom, corresponding to a massless graviton and a massive scalar. Integrating out the massive scalar, the Arnowitt-Deser-Misner Hamiltonian of General Relativity is obtained after SSB. This establishes a foundational bridge between pre-geometric theories and canonical quantum gravity approaches like Loop Quantum Gravity, and allows for the formulation of a pre-geometric Wheeler-DeWitt equation.