Weyl R2 inflation with an emergent Planck scale

Abstract

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, ns) is not affected by unknown higher dimensional curvature operators (suppressed by some large mass scale) since these are forbidden by the Weyl gauge symmetry.