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

We study quadratic gravity $R^2+R_{[mu u]}^2$ 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 $v_mu= (tildeGamma_mu-Gamma_mu)/2$, with $tildeGamma_mu$ ($Gamma_mu$) the trace of the Palatini (Levi-Civita) connection, respectively. The underlying geometry is non-metric due to the $R_{[mu u]}^2$ 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 $partial_mulnphi$ 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 $Msim langlephirangle$, 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 $0.007leq r leq 0.01$ 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.