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

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 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 ($$\phi $$) 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 $$\partial _\mu \ln \phi $$ 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 $$M\sim \langle \phi \rangle $$, 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