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‐fitting values for the free parameters. The mean value for the most important free parameter, β× 1014 (yr−1), using the top‐hat and Gaussian priors are 6.38+2.44−3.46+6.18−6.71 and 5.72+1.22−1.18+2.90−2.69, respectively. Although the interval for which β 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.

We study the spontaneous symmetry breaking in a conformally invariant gravitational theory. We particularly emphasize on the nonminimal coupling of matter fields to gravity. By the nonminimal coupling we consider a local distinction between the conformal frames of metric of matter fields and the metric explicitly entering the vacuum sector. We suppose that these two frames are conformally related by a dilaton field. We show that the imposition of a condition on the variable mass term of a scalar field may lead to the spontaneous symmetry breaking. In this way the scalar field may imitate the Higgs field behavior. Attributing a constant configuration to the ground state of the Higgs field, a Higgs conformal frame is specified. We define the Higgs conformal frame as a cosmological frame which describes the large scale characteristics of the observed universe. In the cosmological frame the gravitational coupling acquires a correct value and one no longer deals with the vacuum energy problem. We then study a more general case by considering a variable configuration for the ground state of Higgs field. In this case we introduce a cosmological solution of the model.

In order to study the mass generation of the vector fields in the framework of a conformal invariant gravitational model, the Weyl-Dirac theory is considered. The mass of the Weyl's meson fields plays a principal role in this theory, it connects basically the conformal and gauge symmetries. We estimate this mass by using the large-scale characteristics of the observed universe. To do this we firstly specify a preferred conformal frame as a cosmological frame, then in this frame, we introduce an exact possible solution of the theory. We also study the dynamical effect of the massive vector meson fields on the trajectories of an elementary particle. We show that a local change of the cosmological frame leads to a Hamilton-Jacobi equation describing a particle with an adjustable mass. The dynamical effect of the massive vector meson field presents itself in the form of a correction term for the mass of the particle.

We have used the results of renormalization of a two-dimensional quantum stress tensor to develop a conformally invariant dynamical model. The model requires the consideration of those conformal frames in which there exists a correspondence between the trace anomaly and a cosmological constant. We apply this model to a two dimensional-Schwarzschild (-de Sitter) spacetime to show that in these conformal frames one may achieve Hawking radiation without recourse to the trace anomaly . *