Scale covariant gravity and equilibrium cosmologies

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.