# Gauging scale symmetry and inflation: Weyl versus Palatini gravity

Research paper by D. M. Ghilencea

Indexed on: 30 Jul '20Published on: 29 Jul '20Published in: arXiv - High Energy Physics - Theory

#### Abstract

We present a comparative study of inflation in two theories of quadratic gravity with {\it 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 non-metricity induced by the gauge field ($\omega_\mu$) of this symmetry. In both theories, the Einstein-Proca action (of $\omega_\mu$), Planck scale and metricity emerge in the broken phase of this symmetry when $\omega_\mu$ acquires mass by Stueckelberg mechanism and decouples. The scalar potential in the presence of non-minimally coupled matter ($\phi_1$) 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 an $R^2$ term in their action, both theories have a small but different tensor-to-scalar ratio ($r\sim 10^{-3}$), larger in the Palatini case. In both Weyl and Palatini theory with a fixed $n_s$, 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 $\xi_1\leq 10^{-3}$, unlike the Palatini case, Weyl quadratic gravity gives a dependence $r(n_s)$ on the spectral index ($n_s$) similar to that in Starobinsky inflation, while also protecting $r$ against higher dimensional operators corrections.