Abstract
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 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 (rsim 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 _1le 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.
Highlights
Due to their symmetry, these theories have no mass scales or dimensionful couplings – these must be generated by the vacuum expectations values of the fields and this is the view we adopt here
In this work we present a comparative study of inflation in two theories of quadratic gravity that have a gauged scale symmetry known as Weyl gauge symmetry
The potential depends on φ1 only, see (11). This is the Einstein-Proca action for wμ: this field has become massive of mass m2w = 6α2 γ M2 by absorbing the derivative of the Stueckelberg field ∂μ ln ρ; the radial direction in field space (ρ) is not present anymore in the action. This is a spontaneous breaking of Weyl gauge symmetry; the number n of degrees of freedom other than the graviton (n = 3) is conserved during this breaking: the initial massless scalar ρ and massless vector wμ are replaced by a massive gauge field wμ
Summary
These theories have no mass scales or dimensionful couplings – these must be generated by the vacuum expectations values (vev) of the fields and this is the view we adopt here. The above result is important since it shows a new mechanism of spontaneous breaking of scale symmetry (in the absence of matter) in which the necessary scalar field is not added ad-hoc to this purpose (as usually done); instead, the Stueckelberg field is here of geometric origin, being “extracted” from the R( ̃ , g) term This situation is very different from previous studies that used instead e.g. modified versions of Weyl action that were linear-only in R and/or used additional matter field(s) to generate the Planck scale [19,20,21,22,23,24,25,26,27,28,29]. The Appendix has technical details and an application to inflation
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