We use a powerful Analytical Model of Igm and GAlaxy formation and evolution (AMIGA) to bracket the reionisation history of the universe. The model includes in a fully consistent way molecular cooling and the formation of Population III stars, their life and feedback effects as well as those of their remnants (black holes and metals) together with the formation and evolution of the subsequent generations of ordinary galaxies formed from those primeval galaxies as well as their feedback effects. We analyse the galaxy formation history predicted by the model that better fits the different observables currently available on the high-z universe (z > 3), namely the cold gas and stellar mass density contents, the star formation rate, the ionisation flux density, the IGM metallicity and temperature, Hydrogen ionized fraction and luminosity functions of Ly-alpha emitters (LAEs) and Lyman-break galaxies (LBGs). We show that all these observables put very severe constraints on the galaxy formation history and on the shape of the luminosity functions of LAEs and LBGs at different z's. Some implications are drawn for the ALBA project.