Palatini quadratic gravity: spontaneous breaking of gauged scale symmetry and inflation

Abstract

We study quadratic gravity $R^2+R_{[\mu \nu ]}^2$ in the Palatini formalism where the connection and the metric are independent. This action has a {\it gauged} scale symmetry (also known as Weyl gauge symmetry) of Weyl gauge field $v_{\mu }=({\tilde{\Gamma }}_{\mu }-{\Gamma }_{\mu })/2$, with ${\tilde{\Gamma }}_{\mu }$ (${\Gamma }_{\mu }$) the trace of the Palatini (Levi-Civita) connection, respectively. The underlying geometry is non-metric due to the $R_{[\mu \nu ]}^2$ term acting as a gauge kinetic term for $v_{\mu }$. We show that this theory has an elegant spontaneous breaking of gauged scale symmetry and mass generation in the absence of matter, where the necessary scalar field ($\varphi$) is not added ad-hoc to this purpose but is "extracted" from the $R^2$ term. The gauge field becomes massive by absorbing the derivative term ${\partial }_{\mu }\ln \varphi$ of the Stueckelberg field ("dilaton"). In the broken phase one finds the Einstein-Proca action of $v_{\mu }$ of mass proportional to the Planck scale $M\sim ⟨\varphi ⟩$, and a positive cosmological constant. Below this scale $v_{\mu }$ decouples, the connection becomes Levi-Civita and metricity and Einstein gravity are recovered. These results remain valid in the presence of non-minimally coupled scalar field (Higgs-like) with Palatini connection and the potential is computed. In this case the theory gives successful inflation and a specific prediction for the tensor-to-scalar ratio $0.007\le r\le 0.01$ for current spectral index $n_s$ (at $95\%$CL) and N=60 efolds. This value of $r$ is mildly larger than in inflation in Weyl quadratic gravity of similar symmetry, due to different non-metricity. This establishes a connection between non-metricity and inflation predictions and enables us to test such theories by future CMB experiments.

22 pages, 2 figures, LaTeX

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