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 Schrödinger 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 Schrödinger equation on the basis of the universal constant w _ { 0 } = 144.7 \pm 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 .