Dynamical Effects of the Scale Invariance of the Empty Space: The Fall of Dark Matter?
The hypothesis of the scale invariance of the macroscopic empty space, which intervenes through the cosmological constant, has led to new cosmological models. They show an accelerated cosmic expansion and satisfy several major cosmological tests. No unknown particles are needed. Developing the weak field approximation, we find that the here derived equation of motion corresponding to Newton's equation also contains a small outwards acceleration term. The new term is particularly significant for very low density systems. 
A modified virial theorem is derived and applied to clusters of galaxies. For the Coma and Abell 2029 clusters, the dynamical masses are about a factor of 5 to 10 smaller than in the standard case. This tends to let no room for dark matter in these clusters. Then, the two-body problem is studied and an equation corresponding to the Binet equation is obtained. The results are applied to the rotation curve of the outer layers of the Milky Way. Starting backwards from the present rotation curve, we calculate the past evolution of the galactic rotation and find that, in the early stages, it was steep and Keplerian. Thus, the flat rotation curves of galaxies appears as an age effect, a result consistent with recent observations of distant galaxies by Genzel et al. (2017) and Lang et al. (2017). Finally, in an Appendix we also study the long-standing problem of the increase with age of the vertical velocity dispersion in the Galaxy. The observed increase appears to result from the new small acceleration term in the equation of the harmonic oscillator describing stellar motions around the galactic plane. Thus, we tend to conclude that neither the dark energy, nor the dark matter seem to be needed in the proposed theoretical context.