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More than half a century after the Chapel Hill meeting of 1957, the landscape of general relativity has been transformed. In relativistic cosmology we are entering an era in which we can seriously consider testing some of the most fundamental assumptions that go into our view of the Universe. We propose that precision tests of gravity on cosmological scales may be feasible with up-and-coming surveys of galaxies. To achieve the most accurate general constraints yet we briefly describe a formalism which is applicable on the largest scales: the parametrized post-Friedmann approach. We sketch out the steps that need to be taken to link realistic, cosmological observables with fundamental properties of general relativity. [ABSTRACT FROM AUTHOR]

Euclid is a European Space Agency medium-class mission selected for launch in 2020 within the cosmic vision 2015-2025 program. The main goal of Euclid is to understand the origin of the accelerated expansion of the universe. Euclid will explore the expansion history of the universe and the evolution of cosmic structures by measuring shapes and red-shifts of galaxies as well as the distribution of clusters of galaxies over a large fraction of the sky. Although the main driver for Euclid is the nature of dark energy, Euclid science covers a vast range of topics, from cosmology to galaxy evolution to planetary research. In this review we focus on cosmology and fundamental physics, with a strong emphasis on science beyond the current standard models. We discuss five broad topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions and questions of methodology in the data analysis. This review has been planned and carried out within Euclid’s Theory Working Group and is meant to provide a guide to the scientific themes that will underlie the activity of the group during the preparation of the Euclid mission. [ABSTRACT FROM AUTHOR]