Weyl-Dirac theory predictions on galactic scales
  We consider the Weyl-Dirac theory within the framework of the weak field approximation and show that the resulting gravitational potential differs from that of Newtonian by a repulsive correction term increasing with distance. The scale of the correction term appears to be determined by the time variation rate of the gravitational coupling. It is shown that if the time variation rate of gravitational coupling is adopted from observational bounds, the theory can explain the rotation curves of typical spiral galaxies without resorting to dark matter. To check the consistency of our theoretical model with observation we use Likelihood analysis to find the best-fit values for the free parameters. The mean value for the most important free parameter, $\beta\times 10^{14} (1/yr)$, using the Top-Hat and Gaussian priors are $6.38^{+2.44}_{-3.46}_{-6.71}^{+6.18}$ and $5.72_{-1.18}^{+1.22}_{-2.69}^{+2.90}$, respectively. Although the interval for which $\beta$ is defined is wide, our results show that the goodness of the fit is, by and large, not sensitive to this quantity. The intergalactic effects and gravitational lensing of clusters of galaxies are estimated and seem to be consistent with observational data. 
S. Mirabotalebi, S. Jalalzadeh, S. Movahed and H. R. Sepangi