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

Research paper by D. M. Ghilencea

Indexed on: 20 Mar '20Published on: 18 Mar '20Published in: arXiv - High Energy Physics - Theory


We study $R^2$ gravity with a dynamical connection ($\tilde\Gamma^\alpha_{\mu\nu}$) in the Palatini formalism where the metric and connection are independent. The action has a gauged scale symmetry (Weyl gauging) of gauge field $v_\mu\propto \tilde\Gamma_\mu-\Gamma_\mu$, with $\tilde\Gamma_\mu$ ($\Gamma_\mu$) the trace of the Palatini (Levi-Civita) connection, respectively. In this case the associated geometry is non-metric. We show that the gauge field becomes massive by a gravitational Stueckelberg mechanism by absorbing the derivative of the dilaton $\partial_\mu\ln\phi$ (where the scalar $\phi$ "linearises" the $R^2$ term). Palatini quadratic gravity with dynamical $v_\mu$ is then a gauged scale invariant theory broken spontaneously. In the broken phase one finds the Einstein-Proca action of $v_\mu$ of mass near the Planck scale ($M$) with a positive cosmological constant. Below this scale $v_\mu$ decouples, the connection becomes Levi-Civita and metricity and Einstein gravity are recovered. The results remain valid in the presence of non-minimally coupled matter. This is similar to recent results by the author for Weyl quadratic gravity, up to different non-metricity effects. When coupled to a scalar field, Palatini quadratic gravity gives successful inflation and a specific prediction for the tensor-to-scalar ratio $0.007\leq r \leq 0.01$ 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 gravity, due to different non-metricity. This establishes a connection between non-metricity and inflation predictions and enables us to test these theories by future CMB experiments.