Today, thanks in particular to the results of the ESA Planck mission, the concordance cosmological model appears to be the most robust to describe the evolution and content of the Universe from its early to late times. It summarizes the evolution of matter, made mainly of dark matter, from the primordial fluctuations generated by inflation around $10^{-30}$ second after the Big-Bang to galaxies and clusters of galaxies, 13.8 billion years later, and the evolution of the expansion of space, with a relative slowdown in the matter-dominated era and, since a few billion years, an acceleration powered by dark energy. But we are far from knowing the pillars of this model which are inflation, dark matter and dark energy. Comprehending these fundamental questions requires a detailed mapping of our observable Universe over the whole of cosmic time. The relic radiation provides the starting point and galaxies draw the cosmic web. JAXAs LiteBIRD mission will map the beginning of our Universe with a crucial test for inflation (its primordial gravity waves), and the ESA Euclid mission will map the most recent half part, crucial for dark energy. The mission concept, described in this White Paper, GAUSS, aims at being a mission to fully map the cosmic web up to the reionization era, linking early and late evolution, to tackle and disentangle the crucial degeneracies persisting after the Euclid era between dark matter and inflation properties, dark energy, structure growth and gravitation at large scale.
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