  Causal structure, inertial path structure and compatibility with quantum
mechanics demand no full Lorentz metric, but only an integrable Weyl geometry
for space time (Ehlers/Pirani/Schild 1972, Audretsch e.a. 1984). A proposal of
(Tann 1998,,Drechsler/Tann 1999) for a minimal coupling of the Hilbert-Einstein
action to a scale covariant scalar vacuum field $\phi$ (weight -1) plus (among
others) a
  Klein-Gordon action term opens the access to a scale covariant formulation of
gravity. The ensuing scale covariant K-G equation specifies a natural scale
gauge vacuum gauge). Adding other natural assumptions for gauge conditions (in
particular Newton gauge, with unchanging Newton constant) the chosen Ansatz
leads to a class of Weyl geometric
  Robertson-Walker solutions of the Einstein equation, satisfying $a''a+a'^2=
const$, analogous to the Friedmann-Lemaitre equation but with completely
different dynamical properties ($a$ the warp function in Riemann gauge). The
class has an asymptotically attracting 1-parameter subfamily of extremely
simple space-time geometries with an isotropic Robertson-Walker fluid as source
of the Einstein equation, discussed as Weyl universes elsewhere (Scholz 2005).
  Under the assumption of a heuristic gravitational self energy binding Ansatz
for the fluid, equilibrium solutions arise, in stark contrast to classical
(semi-Riemannian) cosmology. Weyl universes agree very well with a variety of
empirical data from observational cosmology, in particular supernovae
luminosities and quasar data.
