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) = ((Gamma) over tilde (mu)-Gamma(mu))/2, with (Gamma) over tilde (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 derivative(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 similar to , 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 <= r <= 0.01 for current spectral index ns (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.