Gauging scale symmetry and inflation: Weyl versus Palatini gravity

We present a comparative study of inflation in two theories of quadratic gravity with gauged scale 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 nonmetricity induced by the gauge field (w(mu)) 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(mu)), Planck scale and metricity emerge in the broken phase after w(mu) acquires mass (Stueckelberg mechanism), then decouples. In the presence of matter (phi(1)), non-minimally coupled, the scalar potential is similar in both theories up to couplings and field resealing. For small field values the potential is Higgs-like while for large fields inflation is possible. Due to their R-2 term, both theories have a small tensor-to-scalar ratio (r similar to 10(-3)), larger in Palatini case. For a fixed spectral index n(s), reducing the non-minimal coupling (xi(1)) increases r which in Weyl theory is bounded from above by that of Starobinsky inflation. For a small enough xi(1) <= 10(-3), unlike the Palatini version, Weyl theory gives a dependence r (n(s)) similar to that in Starobinsky inflation, while also protecting r against higher dimensional operators corrections.

Automated summary

This study compares inflation in two theories of quadratic gravity with gauged scale symmetry, presenting differences in the speed of volume expansion in both theories, with one corresponding to the Palatini method. Both theories exhibit novel spontaneous breaking of gauged scale symmetry, with a geometrically-originated scalar field as part of the quadratic action without the need for additional scalar fields.