Gauging scale symmetry and inflation: Weyl versus Palatini gravity

We present a comparative study of inflation in two theories of quadratic gravity withgaugedscale symmetry: (1) the original Weyl quadratic gravity and (2) the theory defined by a similar action but in the Palatini approach obtained by replacing the Weyl connection by its Palatini counterpart. These theories have different vectorial non-metricity induced by the gauge field ( w_μ wμ ) of this symmetry. Both theories have a novel spontaneous breaking of gauged scale symmetry, in the absence of matter, where the necessary scalar field is not added ad-hoc to this purpose but is of geometric origin and part of the quadratic action. The Einstein-Proca action (of w_μ wμ ), Planck scale and metricity emerge in the broken phase after w_μ wμ acquires mass (Stueckelberg mechanism), then decouples. In the presence of matter ( φ ₁ ϕ1 ), non-minimally coupled, the scalar potential is similar in both theories up to couplings and field rescaling. For small field values the potential is Higgs-like while for large fields inflation is possible. Due to their R² R2 term, both theories have a small tensor-to-scalar ratio ( r~ 10⁻³ r∼10-3 ), larger in Palatini case. For a fixed spectral index nₛ ns , reducing the non-minimal coupling ( ξ ₁ ξ1 ) increasesrwhich in Weyl theory is bounded from above by that of Starobinsky inflation. For a small enough ξ ₁≤ 10⁻³ ξ1≤10-3 , unlike the Palatini version, Weyl theory gives a dependence r(nₛ) r(ns) similar to that in Starobinsky inflation, while also protectingragainst higher dimensional operators corrections.