Weyl R² inflation with an emergent Planck scale

We study inflation in Weyl gravity. The original Weyl quadratic gravity, based on Weyl conformal geometry, is a theory invariant under the Weyl symmetry of gauged scale transformations. In this theory the Planck scale (M) emerges as the scale where this symmetry is broken spontaneously by a geometric Stueckelberg mechanism, to Einstein- Proca action for the Weyl "photon" (of mass near M ). With this action as a "low energy" broken phase of Weyl gravity, century-old criticisms of the latter (due to non-metricity) are avoided. In this context, inflation with field values above M is natural, since this is just a phase transition scale from Weyl gravity (geometry) to Einstein gravity (Riemannian geometry), where the massive Weyl photon decouples. We show that inflation in Weyl gravity coupled to a scalar field has results close to those in Starobinsky model (recovered for vanishing non-minimal coupling), with a mildly smaller tensor-to-scalar ratio (r). Weyl gravity predicts a specific, narrow range 0.00257 ≤ r ≤ 0.00303, for a spectral index ns within experimental bounds at 68%CL and e-folds number N = 60. This range of values will soon be reached by CMB experiments and provides a test of Weyl gravity. Unlike in the Starobinsky model, the prediction for (r, n_s) is not affected by unknown higher dimensional curvature operators (suppressed by some large mass scale) since these are forbidden by the Weyl gauge symmetry.