Weyl $R^2$ 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 Weyl symmetry of
(gauged) local scale transformations. In this theory 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 $0.00257 \leq r\leq 0.00303$, for a spectral index $n_s$ 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.
