Gauging scale symmetry and inflation: Weyl versus Palatini gravity

TL;DR

This paper compares inflationary models based on gauged scale symmetry in Weyl and Palatini quadratic gravity, highlighting their differences in symmetry breaking, scalar potentials, and predictions for inflationary observables like the tensor-to-scalar ratio.

Contribution

It introduces a novel geometric origin of scalar fields in gauged scale symmetry models and compares inflationary predictions in Weyl versus Palatini formulations with detailed analysis.

Findings

Both theories exhibit spontaneous breaking of scale symmetry with a geometric scalar field.

Inflationary predictions include a small tensor-to-scalar ratio (~10^{-3}), larger in Palatini.

Weyl theory can mimic Starobinsky inflation for small non-minimal coupling, with added protection against higher-dimensional operators.

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 ($\phi_1$), 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 $\xi_1\leq 10^{-3}$, 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.