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 .