An outline of Weyl geometric models in cosmology
  Already the simplest examples of Weyl geometry, the static space-time models of general relativity modified by an additional time-homogeneous Weylian length connection lead to beautiful cosmological models (Weyl universes) . The magnitude-redshift relation of recent supernovae Ia measurements is in perfect agreement with the prediction of decrease of energy flux in the Weyl models. These data allow to estimate the (ex-ante) spacelike curvature of Weyl universes. Quasar frequency data from the SDSS provide strong evidence of a positive ex-ante curvature. Thus an Einstein-Weyl universe, i.e., an Einstein universe endowed with a Weylian length connection, is in good agreement with supernovae and quasar data.   The relative mass-energy density with respect to the critical density of the standard approach, and the relative contribution of the ``vacuum term'' are time-independent in Weyl gauge. Thus the time-evolution anomaly of vacuum energy does not arise. The intervals given in the literature for the dynamically determined matter density parameter are consistent with other data in the Weyl geometric approach. An explanation of the cosmic microwave background by a maximal entropy state of the quantized Maxwell equation, according to a proposal by I.E. Segal for the conformal Einstein universe, translates directly to the Weyl approach. It leads to quantitative estimations for the temperature of the Planck radiation between 2 and 5.3 deg K. Moreover, the anisotropy angles, expected to arise in an Einstein-Weyl universe agrees with the results of multipole measurements of CMB anisotropies. 
Erhard Scholz