Gauging scale symmetry and inflation: Weyl versus Palatini gravity

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 ($w_{\mu }$) 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 (${\varphi }_1$), 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 $R^2$ term, both theories have a small tensor-to-scalar ratio ($r\sim 10^{-3}$), 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 ${\xi }_1\le 10^{-3}$, 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.

Comments: 25 pages, 7 figures, LaTeX; v3: Sections 1 and 3 expanded; improved presentation