On coordinate distances to type Ia supernovae and radio galaxies
  The quantum model of the homogeneous and isotropic universe predicts logarithmic-law dependence of coordinate distance to source on redshift z which is in good agreement with type Ia supernovae and radio galaxies observations for the redshift range z = 0.01 - 1.8. A comparison with phenomenological models with dark energy in the form of cosmological constant and without dark energy component is made. Fluctuations of the cosmological scale factor about its average value which can arise in the early universe produce accelerating or decelerating expansions of space subdomains containing separate sources with high redshift whereas the universe as a whole expands at a steady rate. 
V. E. Kuzmichev and V. V. Kuzmichev (Bogolyubov Institute for   Theoretical Physics)