  In the framework of the theory of scale relativity, we suggest a solution to
the cosmological problem of the formation and evolution of gravitational
structures on many scales. This approach is based on the giving up of the
hypothesis of differentiability of space-time coordinates. As a consequence of
this generalization, space-time is not only curved, but also fractal. In
analogy with Einstein's general relativistic methods, we describe the effects
of space fractality on motion by the construction of a covariant derivative.
The principle of equivalence allows us to write the equation of dynamics as a
geodesics equation that takes the form of the equation of free Galilean motion.
Then, after a change of variables, this equation can be integrated in terms of
a gravitational Schrodinger equation that involves a new fundamental
gravitational coupling constant, alpha_{g} = w_{0}/c. Its solutions give
probability densities that quantitatively describe precise morphologies in the
position space and in the velocity space. Finally the theoretical predictions
are successfully checked by a comparison with observational data: we find that
matter is self-organized in accordance with the solutions of the gravitational
Schrodinger equation on the basis of the universal constant w_{0}=144.7 +- 0.7
km/s (and its multiples and sub-multiples), from the scale of our Earth and the
Solar System to large scale structures of the Universe
