Cosmological evolution in Weyl conformal geometry

We discuss the cosmological evolution of the Weyl conformal geometry and its associated Weyl quadratic gravity. The Einstein gravity (with a positive cosmological constant) is recovered in the spontaneously broken phase of Weyl gravity; this happens after the Weyl gauge field ($\omega_\mu$) of scale symmetry, that is part of the Weyl geometry, becomes massive by Stueckelberg mechanism and then decouples. This breaking is a natural result of the cosmological evolution of Weyl geometry, in the absence of matter. Of particular interest in the analysis is the special limiting case of Weyl integrable geometry. Both this case and the general one provide an accelerated expansion of the Universe, controlled by the scalar mode of the $\tilde R^2$ term in the action and by $\omega_0$. Their comparison to the $\Lambda$CDM model shows a very good agreement to this model for the (dimensionless) Hubble function $h(z)$ and the deceleration $q(z)$ for redshift $z\leq 3$. Therefore, the Weyl conformal geometry and its associated Weyl quadratic gravity provide an interesting alternative to the $\Lambda$CDM model and to the Einstein gravity.